Non-Equilibrium Statistical Physics of Queueing-Networks: Theory ...

Non-Equilibrium Statistical Physics of
Queueing-Networks: Theory, Numerics and
Application
vorgelegt von
René Pfitzner
April 2011

Contents
0. How to read this thesis 15
I. Theory 17
1. Non-Equilibrium Statistical Physics 19
1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2. Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3. Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4. The Birth-Death Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2. Queueing Theory 29
2.1. A short glance on history . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2. Theory of Single Queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3. The Single Markovian Queue as Birth-Death System . . . . . . . . . . . . 33
2.4. Networks of M/M/1/∞ queues . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5. Operator technique to solve ME for Queueing Networks . . . . . . . . . . 39
3. Zero-Range Processes and Exclusion Processes 43
3.1. The Zero-Range Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2. The Exclusion Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
II. Application and Numerics 49
4. A Queueing Network based description of General Zero-Range Processes 51
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2. Mapping between queueing networks and zero-range processes . . . . . . . 51
4.3. The general n-dimensional Zero-Range Process . . . . . . . . . . . . . . . 52
4.4. Condensation and renormalization . . . . . . . . . . . . . . . . . . . . . . 55
5. Numerical and analytical results for the n-dimensional ZRP 59
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2. Description and general behavior of a global µ i -model . . . . . . . . . . . 59
6. Conclusion and Further Studies 75
3

Before I start...
Und jedem Anfang wohnt ein Zauber inne,
Der uns beschützt und der uns hilft, zu leben.
- Hermann Hesse
The history of this thesis is somewhat unusual and its existence due to a couple
of people, who I want to thank before I dive into the material of this work. Officially
this diploma thesis is submitted to the Faculty of Physics and Astronomy at
Friedrich-Schiller-University Jena, Germany under the official supervision of Prof. Gerhard
Schäfer. I am inexpressibly thankful to Prof. Schäfer for the freedom he gave me
in my thesis studies and for being a great mentor in the last years.
The research presented in this work was conducted at Los Alamos National Laboratory
in Los Alamos, NM, USA from April 2010 to April 2011 under the guidance of Dr.
Michael ”Misha” Chertkov and Dr. Konstantin ”Kostya” Turitsyn. I am enormously
thankful to Dr. Chertkov for providing me with a very inspiring, diverse and excellent
research environment as well as the necessary funding via NFS grant ”Harnessing Statistical
Physics for Computing and Communication”. I am grateful to both, Dr. Chertkov
and Dr. Turitsyn, for their countless hours of discussion, inspiration and expertise which
always led me the right way. My experience in Los Alamos and the USA in general is
priceless and influenced my thinking and world view in a profound way.
There have been several other people in the past who influenced my thinking as well as
personal and professional development strongly. I especially want to thank Prof. Erik
Aurell, who basically was the seed for my professional development and made my stay
in Los Alamos possible. He was the first one to teach me what interdisciplinarity and
broadness in theoretical research means.
At the end I want to thank my parents, who always encouraged me in my path, provided
me with the necessary resources and always have been and will be there for me.
5

Abstract
Science is not an end in itself, but is there to improve society by improving knowledge.
Hence modern theoretical physics is interdisciplinary - and it has to be. This thesis is an
attempt to make this thinking clear. Not only are a lot of tools and concepts developed
in theoretical physics used in other context, but also concepts from other disciplines can
proof enormously useful when talking about ”native” physics issues. In this thesis I am
buildingexactlysuchaconnection. Theconceptofqueueingnetworksanditsestablished
mathematical theory is transferred into a physics context and is shown to be useful in
extending the non-equilibrium statistical physics framework of zero-range processes. In
detail, the 1-dimensional zero-range process is extended to the n-dimensional case. The
behavior of a special n = 2 model is studied analytically and numerically. In such a
setting new effects, not known in the 1-dimensional case, emerge. This study is far from
exhausted and offers material for further research.
7

Zu aller erst...
Und jedem Anfang wohnt ein Zauber inne,
Der uns beschützt und der uns hilft, zu leben.
- Hermann Hesse
Die Entstehungsgeschichte dieser Arbeit ist etwas ungewöhnlich und ihre Existenz
verschiedenen Menschen zu verdanken, denen ich an dieser Stelle danken möchte. Diese
Diplomarbeit ist offiziell an der Physikalisch-Astronomischen Fakultät der Friedrich-
Schiller Universität Jena, unter der Betreuung von Professor Gerhard Schäfer, eingereicht
worden. Ich bin Herrn Professor Schäfer unsäglich dankbar für die Freiheiten, die
er mir in meinen Studien gelassen hat, sowie dafür in den letzten Jahren ein wertvoller
Mentor gewesen zu sein.
Diese Arbeit wurde am Los Alamos National Laboratory, NM, USA von April 2010
bis April 2011 unter Anleitung von Dr. Michael ”Misha” Chertkov und Dr. Konstantin
”Kostya” Turitsyn angefertigt. Ich bin Dr. Chertkov überaus dankbar dafür, dass ich in
seiner Arbeitsgruppe in Los Alamos eine sehr inspirierende, vielfältige und schlichtweg
exzellente Forschungsumgebung finden durfte, sowie die Bereitstellung finanzieller Mittel
unter US National Science Foundation grant ”Harnessing Statistical Physics for Computation
and Communication”. Ich bin beiden, Dr. Chertkov und Dr. Turitsyn dankbar
für die unzähligen Stunden wissenschaftlicher Diskussionen, Inspirationen und das Weitergeben
ihrer Expertise, die mich stets in die richtige Richtung geführt hat. Meine Erfahrung
in Los Alamos und den USA ganz allgemein ist unbezahlbar und hat mich tief
beeinflusst.
Natürlich haben verschiedene andere Menschen in der Vergangenheit mein Denken sowie
meine persönliche und akademische Entwicklung positiv beeinflusst. An dieser Stelle
möchte ich ganz besonders Professor Erik Aurell danken. Er war der Grundstein für
meine akademische Entwicklung in den letzten zwei Jahren und hat meinen Aufenthalt
in Los Alamos möglich gemacht. Er war der erste von dem ich erfahren durfte, was
Interdisziplinarität und Breite in der theoretischen Forschung bedeuten kann.
Zum Schluss möchte ich auch, und vor allem, meinen Eltern danken. Sie haben mich
stets in meinem Weg unterstützt, waren immer für mich da und werden es immer sein.
9

Zusammenfassung
Wissenschaft hat keinen Selbstzweck. Ihre Aufgabe ist es, unsere Gesellschaft durch die
Erweiterung des menschlichen Wissens voranzutreiben. Daher ist moderne theoretische
Physik interdisziplinär - und sie muss es auch sein. Diese Arbeit ist ein Versuch genau
diesesDenkenvonInterdisziplinaritätexemplarischdarzustellen. Esistnichtnurso, dass
viele in der theoretischen Physik angesiedelte Methoden in vielen anderen Disziplinen
nutzbar und nützlich sind, sondern gibt es auch Methodiken aus anderen Fachbereichen,
die in der theoretischen Physik angewendet werden können. In dieser Arbeit baue ich
genau eine solche Verbindung. Das Konzept der Warteschlangennetzwerke (queueing
networks) und ihre etablierte mathematische Theorie wird in einen physikalischen Kontext
gesetzt. Es wird gezeigt, dass mit dieser Symbiose die Erweiterung der Theorie
der zero-range Prozesse aus der nicht-Gleichgewichts statistischen Physik möglich ist.
Um genau zu sein, die 1-dimensionale Theorie der zero-range Prozesse wird auf eine n-
dimensionale Theorie erweitert. das Verhalten des speziellen n = 2 Falls wird analytisch
und numerisch untersucht. Neue Effekte, bislang unbekannt in der n = 1 Theorie, bilden
sich heraus. Diese Arbeit ist weit davon entfernt vollständig zu sein, sondern bietet
Material für weitere Studien dieser generellen Theorie der n-dimensionalen zero-range
Prozesse.
11

Notations and Conventions
P
usually denotes a probability or probability distribution.
W kk ′ denotes the transition rate from state k ′ to k.
Y
capital roman letters usually denote a random variable.
〈x〉, x denotes the average of x.
X ∼ P if X is a random variable, this denotes that X is distributed according to distribution P.
P ∼ f
denotes that P is (up to a constant) equal to f. This is often used when P is a probability
distribution and f is not yet normalized.
a ≈ b denotes that a is approximately b.
N
bold letters denote vectors or matrices.
∂i denotes the graph neighbor of i.
(a) i = a i denotes the i-th element of vector a.
â
denotes that a is an operator.
|s〉 denotes the ket-vector s.
13

0. How to read this thesis
This thesis consists of two parts and is divided into six consecutive chapters. Part one
provides the necessary theoretical foundations to understand the material presented in
part two.
Part one is mostly a repetition of known concepts and results. We will either show
short proofs/derivations of important results or otherwise cite appropriate references.
• Chapter one begins with a short introduction into the theory of non-equilibrium
statistical physics and proceeds by presenting important results in the field of
stochastic processes. To understand this chapter basic knowledge of probability
theory is necessary.
• Chapter two is devoted to the mathematical theory of queues. Building on the
material presented in chapter one, we will introduce the basic concept thoroughly
and proceed with the more advanced theory of queueing networks. We put special
emphasis on Burke’s theorem, which will provide us with a strong concept to find
the steady-state solution of a queueing network without the necessity to solve the
Master Equation.
• Chapter three is an introduction into the concepts of (1-dimensional) zero-range
processes and exclusion processes, which are famous in non-equilibrium statistical
physics.
Part two presents new ideas and original research conducted for this thesis.
• Chapter four presents the main contribution of this research, which is to establish
a strong connection between the theory of queueing networks and zero-range processes.
Building on that we suggest a possible queueing-network-based treatment
of a general n-dimensional zero-range process.
• Chapter five provides, based on the material of chapter four, numerical studies of
the general 2-dimensional zero-range process.
• Chapter six provides a summary as well as a path forward.
15

Part I.
Theory
17

1. Non-Equilibrium Statistical Physics
1.1. Introduction
The classical theory of thermodynamics is concerned with providing a phenomenological
understanding and mathematical framework to deal with systems of many particles. The
prime example here, and the motivation of developing such a theory, was to understand
thebehaviorofgasesusingtheirmaincharacteristics: temperature, pressureandvolume.
Dating back to the early 19th century and Sadi Carnot’s (1796-1832) theory of the
principles and fundamental limits of the newly emerging steam engine, thermodynamics
was from the beginning a very applied theory, almost an engineering discipline. It
was only with Ludwig Boltzmann (1844-1906), Josiah Willard Gibbs (1839-1903) and
James Clerk Maxwell (1831-1879) that phenomenological thermodynamics got a more
mathematical flavor. Statistical mechanics was used to describe the gas as a manyparticle
system. This newly emerged theory was able to reproduce the phenomenological
results of thermodynamics from a statistics based point of view and is the foundation of
what today is called statistical physics. Statistical physics is a very general theory (or
maybe more precise: a set of tools) to describe many-particle systems of every kind in
a statistical/probabilistic fashion where, due to the vast number of degrees of freedom
and interaction, a classical (mechanical) Hamiltonian approach is not feasible. Most
famously, the Boltzmann-Gibbs distribution links the variables Energy of a state and
Temperature of a system in thermodynamic equilibrium to its microscopic probability
distribution
P(x) = 1
Z(β) e−E(x)β , (1.1.1)
where x denotes the state of the system, P(x) the probability that the system will be
found in that state, Z(β) is the partition function (the normalization factor) and β is
the inverse temperature, scaled by the Boltzmann constant k B
β = 1
k B T . (1.1.2)
This result is remarkable, since it builds the connection between phenomenological thermodynamics
and microscopic statistical physics. Even more remarkable is, that there
exists one distribution which is able to describe the statistics of every system in thermodynamic
equilibrium. It is also this one distribution which is at the very heart of
equilibrium thermodynamics and statistical physics. From it, and specifically the partition
function Z(β), every thermodynamic quantity can be derived. This is, because the
partition function is directly linked to the free energy F = −β −1 ln(Z(β)), which is a
thermodynamic potential and hence contains every thermodynamical information about
19

Chapter 1: Non-Equilibrium Statistical Physics
the system.
In equilibrium settings the Boltzmann distribution provides the correct statistical description
of the system. In systems out of equilibrium (and here we will be mainly
concerned with systems out of ”chemical” equilibrium, i.e. systems with changing particle
number) this elegant approach does not hold - in general there is no energy function
which can be assigned to the system and no universal statistical description exists. Instead,
in such settings one must employ other methods. One such method is to solve for
the Master Equation of the system directly 1 , which often is an exceptionally hard problem
and does not guarantee analytical feasibility[25]. The concept of Master Equations
will play an important role in this thesis and will be outlined in more details later.
Theboundarybetweenequilibriumandnon-equilibriumphenomenaisoftenveryloose
so that we feel to elaborate on this issue in more details. In classical thermodynamics a
system is said to be in equilibrium, if it is in
• thermal equilibrium, i.e. the temperature of the system is constant
• mechanical equilibrium, i.e. the system is mechanically stable
• chemical equilibrium, i.e. the concentration of its compounds is constant
simultaneously. Every completely isolated system will eventually arrive at such an equilibrium
state when time t → ∞. Those are however phenomenological measures. What
does this mean from a statistical point of view? Let’s assume that the process we are
interested in is Markovian, which is true for most ”physical” processes. Then the Master
Equation 2 of the system provides a set of differential-difference equations for the
probability distribution for every state:
dP k (t)
dt
= ∑ k ′ [W kk ′P k ′(t)−W k ′ kP k (t)]. (1.1.3)
Informally, as an easy analogy to classical mechanics, one would expect to find an equilibrium
solution via solving the Master Equation for
dP k (t)
dt
= 0. (1.1.4)
However, this is not a sufficient equilibrium condition but merely the weaker condition
of stationarity. Solving this equation one finds a time-independent, stationary solution,
at which a real-world system eventually arrives in the large t limit. In this case it clearly
holds: ∑
W kk ′P k ′(t) = ∑ W k ′ kP k (t) (1.1.5)
k k
1 The Master Equation exists for systems whose underlying microscopic description as a stochastic
processes shows the Markov property, which is the case for the vast majority of physical systems.
2 For a detailed treatment and derivation of the Master Equation, see the section on Markov Processes.
We merely state it here to be able to introduce the notion of detailed balance in an early stage of this
thesis.
20

Chapter 1: Non-Equilibrium Statistical Physics
To arrive at an equilibrium solution, in the sense defined above, one needs to impose the
stronger condition of detailed balance:
W kk ′P k ′(t) = W k ′ kP k (t) (1.1.6)
on the underlying process. Clearly stationarity is a necessary condition for detailed
balance and an equilibrium state is a special steady state. It can be shown rigorously
(see e.g. [36]) that detailed balance holds in closed and isolated physical systems, which
matches our phenomenological definition of equilibrium. This also means that for all
processes which show detailed balance (and even stronger: only for those) equilibrium
statistical physics can be applied and the Boltzmann-Gibbs distribution is the correct
probability measure. Again, in a non-equilibrium setting, i.e. a setting with detailed
balance being broken, no such universal measure exists, but the Master Equation has to
be solved for its steady state. It is rather rare that in such cases the Master Equation
can be solved exactly. In this work however, we will deal with exactly such a solvable
non-equilibrium system, which hence is often called the Ising-model of non-equilibrium
statistical physics.
Due to its generality, modern Statistical Physics is highly interdisciplinary. Its methods
are used far beyond solid state physics - basically everywhere where a statistical
description of a large number of (interacting) ”particles” or the concept of entropy is
needed. So for example in computer science, the theory of optimization (and all its
applications), complex network theory or engineering disciplines.
For further reading on equilibrium and non-equilibrium statistical physics in general,
see references [27, 28, 25, 16]. For its treatment in modern, interdisciplinary contexts see
especially [34]. For some applications in different disciplines see e.g. [1] (complex networks),
[32] (optimization), [22, 2] (Belief Propagation and distributed message passing)
or [35] (power grid engineering and mitigation of blackouts).
1.2. Stochastic Processes
In this section we will introduce the notion of a stochastic process. This concept will play
an important role in this thesis, especially in the form of so called Markov Processes,
which will be the subject of the next section.
Before talking about stochastic processes, it is necessary to briefly review the idea of
a stochastic variable. We will not go too much into detail, but rather choose only to
present the basic ideas, which are necessary to understand the material of this thesis.
For a very readable (and classic) treatment of probability theory, see e.g. [15].
Stochastic variables are mathematical objects used to capture the outcome of a random
experiment. Associated with a stochastic variable X is a set Ω of possible values x ∈ Ω
the variable can take. Also, for every value (outcome) x there exists a number P(X =
x) ∈ [0,1] (which is called a probability), such that
∑
∫
P(X = x)dx = 1 (1.2.1)
x∈Ω
21

Chapter 1: Non-Equilibrium Statistical Physics
holds 3 . For example, in a standard throw-a-dice experiment, one could define a random
variable X =outcome of the throw. In this case, the value x of the stochastic variable X
can take the values x ∈ {1,2,3,4,5,6}. If the dice is fair P(x) = 1/6 ∀x, since every
outcome of the experiment would be equally probable.
From a mathematical point of view, every function (mapping) of a stochastic variable
is again a stochastic variable: Y = f(X) is a stochastic variable and its probability
distribution is naturally given via:
∑
∫
P(Y = y) = δ(y −f(x))P(X = x)dx, (1.2.2)
which is
x∈Ω
P(Y = y) =
∑
x∈Ω:y=f(x)
P(X = x) (1.2.3)
in the discrete case. Especially, every function of a stochastic variable and a parameter
t, would be a perfectly fine random variable on its own
Y(t) = f(X,t). (1.2.4)
Y(t) is called a random function or a series of random variables (if t is discrete). If t
has the notion of being a parameter measuring time, then Y(t) is said to be a stochastic
process. For every parameter value (t ∗ ), Y(t ∗ ) is a stochastic variable. And of course
there is a probability distribution associated with each of those stochastic variables
P(Y = y,t), (1.2.5)
which now is a function of parameter t as well. It is exactly this last statement, which
makes the use of stochastic processes interesting in a (timely evolving) physical world:
the concept of a stochastic process captures the idea of randomness and time evolution:
P(Y = y,t) is the timely evolving probability distribution of a stochastic variable Y.
How one exactly obtains this quantity for a stochastic process Y(t) will be shown in the
next section for a certain type of stochastic processes, so called Markov processes.
In this work we will basically deal with stochastic processes over discrete states. Specializing
to the discrete case, we will re-formulate P(Y = y;t) as P(k,t) = P k (t), denoting
the probability that a system is in the discrete state k at time t. In some sense, this
is a more general notation, since it allows in an easy way to talk about systems with
more than one stochastic variables at once. A state k is here defined as a particular
realization of the random variable(s) in a system. For example, consider a system which
can be described by two random variables, Y 1 and Y 2 . Let each of those variables take
a value from the finite set Ω = {0,1} 4 . We then define exactly four distinct states of
the system (table 1.1). Of course, if one wants to talk about a set of random variables
in a system one could (instead of talking about states) just replace Y in (1.2.5) by a
3 Here we choose to combine two possibilities: if x is discrete then the summation applies. If x is
continuous we have to integrate.
4 One may think about a system of two spins.
22

Chapter 1: Non-Equilibrium Statistical Physics
k y 1 y 2 P(Y 1 = y 1 ,Y 2 = y 2 ;t)
1 0 0 P 1 (t)
2 1 0 P 2 (t)
3 0 1 P 3 (t)
4 1 1 P 4 (t)
Table 1.1.: A possible mapping of a 2-spin system to a state variable k.
stochastic vector Y = (Y 1 ,Y 2 ), each element Y i being a stochastic variable. However,
here we choose to stick (mainly for notation purposes) with the description in terms of
states.
Stochastic processesareamathematical tool for modelingalotof physical (”real-world”)
phenomena: BrownianMotion(theWienerProcess), changesinthestockmarketorpopulation
dynamics. For a nice textbook on this issue see [36]. For the new emerging field
of Econophysics, in which the theory of stochastic processes plays a central role, see e.g.
[31].
1.3. Markov Processes
We now proceed to describe a special stochastic process and some (for this work) important
properties of it: the Markov Process. A stochastic process is called a Markov
Process, if for the conditional (transition) probabilities the Markov Property holds:
P(X(t n ) = x n |X(t 1 ) = x 1 ,...,X(t n−1 ) = x n−1 ) = P(X(t n ) = x n |X(t n−1 ) = x n−1 )
(1.3.1)
or using the notation of states
P(k n ,t n |k 1 ,t 1 ...k n−1 ,t n−1 ) = P(k n ,t n |k n−1 ,t n−1 ) (1.3.2)
with t n > t n−1 > ... > t 1 . This property is quite remarkable and of big physical
significance. It basically says, that in a process holding this property, the value of the
random variable X (or the state) at time t n will only depend on the previous state of
the process, i.e. the value X obtained at time t n−1 . This is a good model for a lot of
significant real-world processes without memory, like Brownian motion: the place and
momentum of a particle will only depend on the at time t n sampled perturbation in
momentum, given the place and momentum at time t n−1 , and will not depend on its
past.
The Markov property also leads directly to what is formally known as memorylessness.
Memorylessness makes a statement about the distribution of times τ a stochastic process
spends in a state i, specifically it states that a continuous time, memoryless stochastic
process has exponentially distributed transitions times τ:
τ ∼ 1¯τ e−τ¯τ (1.3.3)
23

Chapter 1: Non-Equilibrium Statistical Physics
where ¯τ is the mean transition time. That an exponentially distributed random variable
in a stochastic process is a sign of memorylessness, is due to the fact that the exponential
function locally, i.e. in an ǫ-environment, ”looks the same” everywhere on the whole
domain due to the fact, that f ′ (x) = f(x) whereas f ′ (x) denotes the first derivative of
f with respect to x. Let me demonstrate this statement and make it mathematically
more precise:
The exponential distribution is a sign of memorylessness. Consider a random variable
T, distributed with P(T = τ) = ¯τ −1 e −τ¯τ. If a stochastic process is supposed to be
memoryless, the outcome of sampling the random variable describing it needs to be
independent of its past. In plain language, that means that the probability distribution
P(T) has to be independent of an offset τ 0 . Since for the exponential function the
functional equation
f(x+y) = f(x)·f(y) (1.3.4)
holds, one arrives at
P(T = τ +τ 0 ) = 1¯τ e−τ+τ 0
¯τ (1.3.5)
= 1¯τ e−τ¯τ e
− τ 0¯τ (1.3.6)
= e−τ 0¯τ
¯τ
e −τ¯τ . (1.3.7)
Normalizing this with help of
∫ ∞
τ 0
e −τ¯τ = ¯τe
− τ 0¯τ (1.3.8)
one arrives at the new probability density P ′ (T = τ) function for τ > τ 0
P ′ (X = τ) = 1¯τ eτ 0¯τ e
− τ¯τ (1.3.9)
But this is exactly the initial probability distribution, only with a new normalization
constant. Thisiswhatwemeanby”looksthesame”-thefunctionalrelationshipbetween
the probability and the argument τ is identical.
In this demonstration, it is exactly the functional equation (1.3.4) which leads to
the statement. Since this functional relation is unique to the exponential function [3],
the vice versa statement of the above also holds: it is only the exponential function,
which shows the property of memorylessness. This is an important result in the area of
stochastic processes and the direct link between theory and real-world phenomenon.
A well known result (see e.g. [36]) connects the Markov process and the exponential
distribution of transition times:
Every continuous-time Markov process shows the property of memorylessness. Hence
state-transition times in such processes are exponentially distributed.
24

Chapter 1: Non-Equilibrium Statistical Physics
This is a key result in the theory of stochastic processes and again one reason why
Markov chains and Markov processes are of interest in the physical sciences.
Markov processes have the nice property that there exists a unique differential-difference
equation for every system, which allows to solve for the probability density: the Master
Equation (1.1.3). The Master Equation is a direct consequence of the Chapman-
Kolmogorov equation. The Chapman-Kolmogorov equation itself follows directly from
the Markov property and states:
P(k 3 ,t 3 |k 1 ,t 1 ) = ∑ k 2
P(k 3 ,t 3 |k 2 ,t 2 )P(k 2 ,t 2 |k 1 ,t 1 ). (1.3.10)
This is basically a ”path-integral” kind of logic: the probability to be in state k 3 at time
t 3 , given that the system was in state k 1 at time t 1 , is equal to the sum over probabilities
of all possible paths (states) between t 1 and t 3 . Employing Bayes’ theorem 5 , one finds
that the above equation is equivalent to
P(k 3 ,t 3 |k 1 ,t 1 ) = ∑ k 2
P(k 3 ,t 3 ;k 2 ,t 2 |k 1 ,t 1 ). (1.3.11)
We can use this property of the Markov process to directly derive the Master Equation.
Derivation of the Master Equation. We want to find an expression for
dP(k,t)
dt
= dP(k,t|k 0,t 0 )
dt
P(k,t+∆t|k 0 ,t 0 )−P(k,t|k 0 ,t 0 )
= lim
∆t→0 ∆t
(1.3.12)
where we conditioned on a universal beginning state k 0 at time t 0 . Now, using equation
(1.3.11) and again Bayes’ theorem one finds
P(k,t+∆t|k 0 ,t 0 ) = ∑ k ′ P(k,t+∆t|k ′ ,t)P(k ′ ,t|k 0 ,t 0 ) (1.3.13)
P(k,t|k 0 ,t 0 ) = ∑ k ′ P(k ′ ,t+∆t|k,t)P(k,t|k 0 ,t 0 ). (1.3.14)
Plugging those two identities into equation (1.3.12):
dP(k,t|k 0 ,t 0 )
dt
= ∑ [ P(k,t+∆t|k ′ ]
,t)
lim P(k ′ ,t|k 0 ,t 0 )− P(k′ ,t+∆t|k,t)
P(k,t|k 0 ,t 0 )
k ′ ∆t→0 ∆t
∆t
= ∑ [
Wkk ′P(k ′ ,t|k 0 ,t 0 )−W k ′ kP(k,t|k 0 ,t 0 ) ]
k ′
dP(k,t)
dt
= ∑ k ′ [
Wkk ′P(k ′ ,t)−W k ′ kP(k,t) ] (1.3.15)
5 Bayes’ theorem relates the conditional probability of two random variables X and Y with its joint
probability: P(X;Y) = P(X|Y)P(Y) = P(Y|X)P(X).
25

Chapter 1: Non-Equilibrium Statistical Physics
λ 0 λ 1 λ k−2 λ k−1 λ k λ k+1
0 1 ... k −1 k k +1 ...
µ k−1
µ 1 µ 2 µ k µ k+1 µ k+2
Figure 1.1.: State diagram of a simple birth-death process.
which is exactly the Master Equation. During this derivation we substituted
W kk ′
P(k,t+∆t|k ′ ,t)
= lim
∆t→0 ∆t
W k ′ k = lim
∆t→0
P(k ′ ,t+∆t|k,t)
∆t
(1.3.16)
(1.3.17)
which hence have the meaning of infinitesimal (unit time) transition probabilities (rates).
The Master Equation (1.3.15) can be nicely interpreted as an ”equality of probability
flows”: the temporal change of the probability of being in state k equals the sum of
probability flows into that state minus the sum of probability flows out of that state.
This interpretation provides at the same time a rule for the construction of the Master
Equation of a Markov Process, given the rates (unit time conditional probabilities) for
changing a state. An example will be given in the next section on birth-death processes.
For a nice text book on Markov processes see [36].
1.4. The Birth-Death Process
A special Markov process, and of interest in understanding queueing theory later, is
the Birth-Death process. Figure 1.1 shows the state diagram of such a process. State
diagrams are a very useful tool to visualize a Markov process: it shows every possible
state of the system and its unit transition probabilities (rates) W kk ′. Here the rates are
labeled λ i = W i,i+1 and µ i = W i,i−1 . As opposed to a general Markov process, there are
only allowed transitions between two neighboring states. If k is the occupation of the
state, then an allowed transition can go to k+1, which is called a birth and k−1, which
is called a death. The transition rates from occupation state k to k ± 1 are in general
dependent on the state of departure. λ k is called the birth rate, µ k the death rate. From
a physics point of view, the connection to (second quantization) ladder operator techniques
famous is fairly obvious and will play a role later, when looking at the treatment
of queueing networks by Massey [33] and Chernyak et al. [6].
To answer the question what the density profile/probability distribution of the states k
in such a system is, one looks for the Master Equation of the system. Using the probabil-
26

Chapter 1: Non-Equilibrium Statistical Physics
ity flow interpretation of the Master Equation and the state diagram as a visualization
tool, one finds the following Mater Equation for a general Birth-Death process:
dP k (t)
dt
dP 0 (t)
dt
= −(λ k +µ k )P k (t)+µ k+1 P k+1 (t)+λ k−1 P k−1 (t) (1.4.1)
= −λ 0 P 0 (t)+µ 1 P 1 (t) (1.4.2)
To solve this equation, the technique of z-transforms is very useful [24]. However, since
this is not really a necessary result for this thesis, we will not do the calculation and
rather look at an important special case of the birth-death process.
The uniform pure birth-process, i.e. a birth-death process with µ k = 0, is of special
interest since it is a model for a Markovian, memoryless process. The Master Equation
for this process reads
dP k (t)
dt
dP 0 (t)
dt
= −λP k (t)+λP k−1 (t) (1.4.3)
= −λP 0 (t). (1.4.4)
This system is easily solved iteratively. For k = 0 the solution is (simply via integration)
For k = 1 the differential equation reads
dP 1 (t)
dt
P 0 (t) = e −λt . (1.4.5)
+λP 1 (t) = λe −λt (1.4.6)
which is a non-homogeneous and not-separable ordinary differential equation, which can
be solved by the method of introducing an integrating factor (see e.g. [3]). Thus one
arrives at
P 1 (t) = λte −λt . (1.4.7)
Iterating this procedure, one arrives again at a differential equation of the same type
and, via an integrating factor, yields the iterative equation
P k (t) = e −λt ∫
λe λt P k−1 (t)dt (1.4.8)
with solution
P k (t) = (λt)k e −λt . (1.4.9)
k!
Hence the occupation numbers of every Markovian birth process are Poisson distributed.
An interesting and important feature of the Poisson distribution is, that a Poisson distributed
stochastic process, like here the process of births, has exponentially distributed
inter-birth times. This means, that the time-span between two consecutive births is an
exponentially distributed random variable and can be seen via the following consideration.
27

Chapter 1: Non-Equilibrium Statistical Physics
The inter-birth time is exponentially distributed. Let τ be the inter-birth time, i.e. the
time between to consecutive births and let X(τ 0 ,τ 0 +τ) define the random variable of
the number of births between time span (τ 0 ,τ 0 +τ). Then, according to the dynamics
of a Markovian birth process with rate λ, it holds:
P(X(τ 0 ,τ 0 +τ) = k) = (λτ)k e −λτ (1.4.10)
k!
Hence, letting k be the parameter and τ the free variable, the probability of having
exactly one birth after duration τ is nothing but P(X(τ 0 ,τ 0 +τ) = 1). Hence letting T
denote the random variable of having exactly one birth after time T = τ one finds
for the distribution of inter-birth time T 6 .
P(T = τ) = λe −λτ (1.4.11)
Later the birth process will be used to describe any memoryless, Markovian process
of arrivals of customers to the queuing system.
In this chapter we introduced the notion of non-equilibrium statistical physics on a
formal level. We furthermore provided the concept of stochastic and especially Markov
processes. In the next chapter we will use these concepts to address the mathematical
concept of queues and queueing networks.
6 The notation of T and τ as inter-birth time seems eventually a bit confusing. But T denotes the
random variable, whereas τ denotes the value this variable can take. In (1.4.10) τ is a parameter and
k the free variable, the value the random variable X can take. Whereas in this equation τ is the free
variable denoting the value the random variable T can take.
28

2. Queueing Theory
2.1. A short glance on history
Queueing Theory is traditionally an Operations Research discipline. It was developed
andisemployedtooptimizeandanalyzeprocessesincludingwaitinglines(queues), sofor
example telecommunication processes, production processes or network traffic in applied
computer science. For a nice introduction see e.g. [24], for a more advanced treatment
see e.g. [5].
2.2. Theory of Single Queues
This chapter is devoted to introducing the basic mathematical object queue as well as
presenting general features and theorems. Especially Burke’s theorem will play a central
role here and later. As stated earlier, one of the goals of this thesis is to establish
a connection between the Operations Research native theory of queues and a possible
analogy in physics. However, in this chapter we will mostly use the Operations Research
terminology and will later build the desired connection.
A queue is a mathematical object combining two ideas:
a) the idea of a service facility. Its purpose is to complete jobs/serve customers which
are arriving at this facility.
b) the idea of a storage, in which arriving jobs can be queued if the facility is busy
(Fig. 2.1).
Such a general system is normally notated as a G/G/m/n system. So what does this
mean? To describe a queueing system completely one has to specify four parameters:
1. A parameter to quantify how jobs enter the system.
2. A parameter to quantify how jobs are completed in the facility.
3. A parameter to quantify the ”parallelness” of job completion, i.e. how many jobs
can a facility complete at the same time.
4. A parameter to specify the capacity/storage space of a system, i.e. how many jobs
can be queued before being processed in the facility.
The above notation is used to denote a very general queue: jobs enter the system in a
General fashion, are completed in a General fashion, at most m jobs in parallel and the
29

•
•
•
•
•
•
Chapter 2: Queueing Theory
A(t)
n
m
• • • • • •
B(t)
Figure 2.1.: Illustration of a single A(t)/B(t)/m/n queue. Red squares denote jobs entering
the system, green squares denote completed jobs leaving the system.
storage capacity is n jobs.
However, the notion of ”how jobs enter the system” and ”how jobs are completed in the
facility” is not very precise. Every model and every specification in a model is based
on the desired result. In the case of queueing systems one is mostly interested in the
number of total jobs in the system, the time a job spends in the system and the number
of jobs waiting in the queue. Of course, from a physics point of view a Hamiltonian
standpoint would be totally acceptable and a solution of the problem: specify the exact
times at which jobs arrive to the system as well as the microscopic dynamics of how jobs
are completed. Then construct the Hamiltonian of the system and solve a general kind
of equations of motion. However, such an approach is certainly very limited. In most
systems, especially those arising in Operations Research, one has to deal with a large
number of unpredictable jobs arriving at the system. In such cases a statistical approach
is much more preferable. Hence it is useful to specify the distribution of inter-arrival
times t i of jobs entering the system
P(t i ≤ t) = A(t) (2.2.1)
and the distribution of service times t s of completing a job in the facility
P(t s ≤ t) = B(t). (2.2.2)
NotethatA(t)andB(t)arecumulativedistributions, tostickwiththestandardnotation
in queueing theory [24]. The inter-arrival time t i is defined as the time interval between
the consecutive arrival of two jobs to the system.
We specified the system in terms of constant integer-valued variables m and n as well
as real-valued, positive random variables t i and t s , which we will call the free variables
of our system. As is well known, every function of one or more random variables is
itself again a random [15]. Hence the results of the queueing network analysis will be
randomvariablesaswellandweendupwithacompletelyprobabilisticdescriptionofour
underlying real-world phenomenon. This is important to keep in mind for interpretation
30

Chapter 2: Queueing Theory
of results. It also opens the way to a statistical physics treatment of the problem, which
will be presented later.
After defining the basic quantities of a single queue it is rather straight forward to get
some first interesting results. First let us define a couple of more quantities, following
directly from the basic free variables already defined. Denoting the absolute arrival time
to the system of some job n with τ n , it is clear that the inter-arrival time t n between job
n−1 and n is
t i (n) = τ n −τ n−1 . (2.2.3)
Defining the waiting time t w (n) of a job in the system, i.e. the time it waits in the
queue before the service facility will start processing it, the system time t Σ (n) of job n,
is defined as
t Σ (n) = t s (n)+t w (n). (2.2.4)
The waiting time t w and its distribution will be a property of the system, following from
the free variables in a non-trivial way.
Calculating some basic average properties, like the average inter-arrival time 1
∑
∫
¯t i = t i P(t i )dt i (2.2.5)
t i
and the average service time 1 ∑
∫
¯t s = t s P(t s )dt s (2.2.6)
t s
one defines the average arrival rate
λ = ¯t i
−1
(2.2.7)
and the average service rate
µ = ¯t s −1 . (2.2.8)
This definition is, from a physics point of view, quite natural. Additionally, it will
proof useful later when talking about continuously exponentially distributed inter-arrival
times, since in this case the inverse of the distribution parameter exactly equals the
average in such ∫
¯t = tλe −λt dt = λ −1 . (2.2.9)
Another interesting quantity is the distribution P(α(t ′ )) of customers α(t ′ ) arriving to
the system within the time interval [0,t ′ ]. Here α(t ′ ) is a random variable with parameter
t ′ . Obviously the probability of arrival of exactly one customer is exactly the probability
of having inter-arrival time t ′ P(α(t ′ ) = 1) = P(t i = t ′ ). (2.2.10)
1 Using ∑ or ∫ depending on whether we talk about discrete or continuous time variables respectively.
31

Chapter 2: Queueing Theory
The probability of having exactly two arrivals in [0,t ′ ] is given via the convolution of the
inter-arrival time distribution
∫
P(α(t ′ ) = 2) = dτ 1 P(t i = τ 1 )P(t i = t ′ −τ 1 ). (2.2.11)
Thus the probability of having exactly n arrivals in [0,t ′ ] is given by the following
recursion
∫
P(α(t ′ ) = n) =
dτ 1
∫
∫
dτ 2 ...
n−1
∏
dτ n−1
j=1
n−1
∑
P(t i = τ j )P(t i = t ′ −
k=1
τ k ). (2.2.12)
Note that, in our interpretation, t ′ is a parameter and n is the free variable. If the
inter-arrival times are exponentially distributed with parameter λ, the average number
of arrivals within [0,t ′ ] is given by
¯ α(t ′ ) = λt ′ , (2.2.13)
The most interesting queueing theoretical quantity for this thesis will be the number
of customers in the system at time t, which we will denote by
N(t). (2.2.14)
Again, this quantity is a random variable. We are most interested in a possible steadystate
solution of the system, i.e. the limit
lim N(t) = N. (2.2.15)
t→∞
Such a limit does not necessarily exist and even if it does, its calculation can be far from
trivial, since generally one would have to solve for the Master Equation of the system.
An important queueing theorem correlates the average number of customers in the system
(in the t → ∞ limit), ¯N, to the average system time, t¯
Σ :
¯N = λt Σ (2.2.16)
This result is known as Little’s result in queueing theory and can be translated as
”average customers in the system=(average arrival time to the system)×(average system
time)”. This seems from a physics point of view fairly obvious. However, the proof for
this theorem was only established in 1961 by John D. C. Little’s[30].
In a system containing a simple queue with only one facility, the following measure is
an important quantity:
ρ = λ µ . (2.2.17)
In queueing theory this is called the utilization factor. Its meaning is important for
deciding of whether or not a stable steady-state solution for the system exists. If
ρ > 1 (2.2.18)
32

Chapter 2: Queueing Theory
holds, the system will not be stable, in the sense that the average waiting time of a
customer will go to infinity, since lim t→∞ N(t) = ∞ will hold. This is easily seen, since
in this case (via (2.2.7) and (2.2.8)) the service rate would exceed the arrival rate and
hencecongestioninthewaitingqueuewillresult 2 . Anotherinterpretationofρisinterms
of “fraction of time the facility is busy”. We will show later that (quite remarkably)
equation (2.2.18) holds as a stability condition in a much more general setting than the
one-queue system.
2.3. The Single Markovian Queue as Birth-Death System
In the previous chapter we said that the number of customers in the system is an important
quantity for this thesis. However, its computation seems not so straight forward.
The knowledge of this quantity is also important to be able to make a theoretical statement
about the number of customers leaving the system, which clearly is a function of
how many customers are present in the system.
Developing an universal description of the general G/G/m/n system of one queue seems
to be a bit too optimistic. Also, from a physics perspective, this generality is mostly
not needed. Instead, a very common queueing system which has been studied a lot is
denoted by M/M/n/m. This system is one with Markovian input, Markovian service
completion, n servers and storage of size m. As pointed out earlier, Markovity and
memorylessness are highly coupled and the underlying principles governing a vast number
of real-world stochastic processes. From the things pointed out earlier, it is directly
possible to specify the stochastic processes underlying an M/M/n/m system in more
detail using the analogy to Birth-Death processes. In such a mapping the arrival of a
customer to the system is modeled as birth of a customer to the system and the completion
of service as death. In a M/M/n/m system with customer arrival rate λ, this rate
is directly related to the birth-rate λ of a corresponding pure birth process. Since in this
Markovian queueing system the arrival of customers is supposed to be Markovian (i.e.
independent of the history of arrivals), we find (transferring the results from section 1.4)
that the arrival process to the system is a Poisson distributed stochastic process
with exponentially distributed inter-arrival times
P(N arr = k,t) = (λt)k e −λt (2.3.1)
k!
P(t i = t) = λe −λt . (2.3.2)
Also, if we denote the completion rate of every facility in this system with µ, the Master
Equation for the distribution of number of customers in the system for every given time
2 For illustration, the reader may think of its last visit to whatever (German) public administrative
office.
33

Chapter 2: Queueing Theory
t is in analogy to equation (1.4.2) given by
dP k (t)
dt
dP 0 (t)
dt
= −(Θ(m−k)λ+min(n,k)µ)P k (t)+min(n,k +1)µP k+1 (t)
+Θ(m−k +1)λP k−1 (t) (2.3.3)
= −λP 0 (t)+min(n,1)µP 1 (t)
Here Θ is a Heaviside-like function with Θ(x) = 1, if x ≥ 0 and Θ(x) = 0, otherwise.
WhatdistinguishesthisMasterEquationfromtheoneofthepurebirth-deathprocessare
the multiplicities of the facilities n. This basically leads to a linear increase in customer
service completion and is accounted for by multiplying the death-rates by min(n,k).
The restriction of storage capacity is here modeled via the Θ-function.
We will not solve this Master Equation but rather look at the important case of the
M/M/1/∞ queue, i.e. a system with one facility and infinite storage. In this case the
Master Equation is just (1.4.2) and as mentioned earlier the time-dependent solution
rather complicated. However, considering the steady-state case
0 = −(λ+µ)Pk ss +µPss k+1 +λPss k−1
0 = −λP0 ss +µP1
ss
finding a solution is possible iteratively and the steady-state solution of the distribution
of customers N in the M/M/1/∞ system is given by
( ) λ k
P ss (N = k) = P0
ss , (2.3.4)
µ
as one easily checks via substituting back into the homogeneous Master Equation. This
result is a power-law distribution. P 0 serves as the normalization constant and can be
found, via the convergence of the geometric series, to be
( ∞
) −1
∑
P 0 = ρ k
k=0
= 1−ρ, (2.3.5)
where
ρ = λ µ
(2.3.6)
is the previously introduced utilization factor. Probability distribution (2.3.4) is only
normalizable, if the geometric series in (2.3.5) converges. Convergence is assured only
for 0 < ρ < 1 and hence a steady-state solution for this system only exists if λ < µ,
which matches the criterium pointed out in (2.2.18) and provides for this system a
strict mathematical explanation. Having obtained this solution, the average number of
customers in the steady-state is given by
¯N = ρ
1−ρ . (2.3.7)
34

Chapter 2: Queueing Theory
In the steady-state the distribution of customers N out leaving the facility can also be
computed. On a first thought one might guess that the P(N out ) will be highly coupled
to the number of customers N in the system and will be a complicated result. However,
the following consideration shows that the solution is in fact rather simple - albeit very
surprising:
Burke’s theorem. Consider a M/M/1/∞ system of a single queue with customer arrival
rate λ and job-completion rate µ. As established earlier, in a Markovian system, the
inter-arrival time and facility completion time are exponentially distributed. As seen
earlier, if there is a constant Markovian birth process with rate λ, the number of births
in every time period is Poisson distributed with rate λ. Since the facility completion
process is such a process, the number of customers leaving the queue would be Poisson
distributed with rate µ, iff there would be a constant supply of customers. However,
in this model the supply of customers is regulated by the number of customers in the
system N, which is in the steady state case distributed with (2.3.4). Hence the supply
of customers to the service facility is constrained exactly by this distribution. Let me
derive the statistics of the output:
P(N out = k,t) ∼ e −µt(µt)k
k!
∼ e −µt(µt)k
k!
·P(N ≥ k)
∞∑
(1−ρ)ρ k (2.3.8)
l=k
∼ e −µt(µt)k ρ k .
k!
Using (2.3.6) and normalizing correctly we finally get
P(N out = k,t) = e −λt(λt)k . (2.3.9)
k!
This rather surprising result says, that in the steady-state case the output of the
M/M/1/∞ system is, as the input, Poisson distributed with the arrival-rate λ. This
result is often known as Burke’s theorem[4]. Interestingly, there is (in physics terms)
something like a first order phase transition in the following sense. As we have shown,
a solution for the distribution of the number of customers N in the system only exists if
ρ ≤ 1. In a M/M/1/∞ system with infinite storage space of arriving jobs, this means
that in the case ρ ≥ 1 the number of customers will diverge, i.e. go to N → ∞. Restating
this fact, there is an infinite number of supply for the facility, keeping it constantly busy.
Hence the output of the queueing system will, opposed to the ρ ≤ 1 case, solely depend
on the dynamics of the service facility and not anymore on the input. Hence Burke’s
theorem will not hold in this scenario. Instead the output of the system will be Poisson
distributed with rate µ (since the service time is exponentially distributed with rate µ).
So with ρ → 1 there is an abrupt (first order) change (phase transition) in the output of
the system from a Poisson distribution with rate λ to a Poisson distribution with rate
µ.
35

Chapter 2: Queueing Theory
λ 31
Q1
λ 13
Q3
λ 01
λ 20
λ 30
λ 12 λ 23
Q2
Figure 2.2.: Illustration of a queueing network with 3 nodes.
2.4. Networks of M/M/1/∞ queues
A straight forward extension of the theory of a single queue is to consider networks of
queues. Figure 2.2 illustrates a possible queueing network. These systems are more
elaborate and complex than the former one. The input of one queue in the network
might not only depend on customers arriving from outside of the system, but will also
depend on transitions of customers from the output of one queue to an adjacent queue 3 .
Thisleavesuswithahighlycoupledsystemwhichneedstobeanalyzed. Thistaskisvery
much simplified by Burke’s theorem: even queues, which are being fed by transitioning
customers, i.e. customers just leaving another queue, will have Poissonian input and
hence can be treated as as a single, independent M/M/1/∞ queue, as presented in the
previous chapter:
The sum of two independent Poisson distributed variables is again Poisson distributed.
Consider two consecutive M/M/1/∞ queues Q1 and Q2. Assume that Q1 has, with
Burke’s theorem, Poissonian output with rate λ 12 . Furthermore assume that Q2 has
customers arriving externally with rate λ 02 and customers arriving from Q2 with exactly
its leaving rate λ 12 . Considering the sum S = K+L of two independently Poisson
distributed random variables K and L:
P(S = s) = (P(K)∗P(L))(s)
∞∑
= P(K = k)P(L = s−k) (2.4.1)
=
k=0
∞∑
k=0
e λ 12t (λ 12t) k
e λ 02t (λ 02t) (s−k)
k! (s−k)!
(2.4.2)
3 The term adjacent here is meant in the graph-theoretical way, i.e. neighboring in the sense of being
connected by an edge.
36

Chapter 2: Queueing Theory
The infinite sum on the rhs is convergent and the sum is given by 4
∞∑ (λ 12 t) k
k=0
k!
(λ 02 t) (s−k)
(s−k)!
= (λ 12t+λ 02 t) s
s!
(2.4.3)
which leaves us with
P(S = s) = e (λ 12+λ 02 )t (λ 12t+λ 02 t) s
(2.4.4)
s!
showing that the sum of two independently Poissonian distributed variables is again
Poisson distributed with the sum of the single rates.
Having shown that and keeping Burke’s theorem in mind, the steady-state transition
rates λ ij in a network of queues can be calculated via
λ ij = (λ 0i −λ i0 )−
k∈∂i/jλ ∑
ik + ∑ λ ki (2.4.5)
k∈∂i
where ∂i/j denotes the set of all neighbors of i without j. This is a very general
expression which, thinking of systems of flows, basically equates influx and outflux for
every queue in the system. That such an ”average flux” conversation holds is not at all
obvious, sinceoursystemisequippedwiththepossibilityofstoringjobsinaqueue. That
this result nevertheless holds is due to the double-Markovity in the system (Markovian
input and Markovian service completion) and the resulting Burke’s theorem.
In a case as described here, it is clear that transition rates λ ik for all neighbors k of i
and λ i0 must be interrelated. Concretely, denoting the effective service rate at queue i
with λ i < µ i , the following equation must hold:
λ i = ∑ k∈∂iλ ik +λ i0 . (2.4.6)
Assuming λ ik = λ i p ik with ∑ k p ik = 1 and p ik being the probability that a served
customer is transferred from node i to node k or leaving the network (k = 0), one finds
for the local, effective steady-state service rates
λ i = λ 0i + ∑ k
λ k p ki . (2.4.7)
This is a well-posed system of n variables and n equations which can in principal be
solved using standard methods. However, if the system is large n >> 1, the solution
of such a linear system is far from trivial and special computational algorithms need to
be employed. However, in most cases the underlying network of queues will be weekly
connected, i.e. the linear system very sparse and efficient sparse-matrix algorithms like
the Conjugate Gradient method or Gaussian Belief Propagation can be used to do the
job.
The first one to mathematically describe such kinds of systems was James R. Jackson
4 Using e.g. Mathematica.
37

Chapter 2: Queueing Theory
in his seminal 1963 paper [21]. He even generalized the ideas presented above to a less
restrictive setting, with arrival and completion rates not constant but dependent on the
number of customers in the queue.
After the effective rates of the system have been calculated and (as we have shown
previously) in the steady state every queueing system in the network can be treated as
an independent queueing system with the new effective rates, the solution for the system
P(N) has to be a factorized form of the independent random variables N i :
P(N) = ∏ i
P(N i ) (2.4.8)
Again, for the i−th subsystem there is an effective Poissonian arrival rate λ i and an
inherent service rate µ i associated. We define the i−th utilization factor via
ρ i = λ i
µ i
. (2.4.9)
Let us note that it is also possible to introduce the nominal transition rate from queue i
to queue j as µ ij = µ i p ij , for which then the following relation to the effective transition
rate holds:
λ ij = ρ i µ ij (2.4.10)
This leads us directly to
and with 2.4.8 to the joint probability distribution
P(N i = k i ) = (1−ρ i )ρ k i
i
(2.4.11)
P(N = k) = ∏ i
(1−ρ i )ρ k i
i . (2.4.12)
This result is quite remarkable. There are very few non-equilibrium systems for which
the solution is nicely factorized. However, we want to stress that it is Burke’s theorem
and hence the double-Markovian nature of the system which ensures this nice property.
We also want to stress that we were able to obtain this solution without solving the
Master Equation for this coupled system but just via employing Burke’s theorem.
Using the above introduced utilization factors, one can write (2.4.5) in a convenient
form as:
∑
M ij ρ j = λ 0i (2.4.13)
with
M ij =
j
{
−µ ji i ≠ j
µ i0 + ∑ k µ ik i = j.
(2.4.14)
If there is no misunderstanding possible, we will use the Einstein-convention and e.g. in
(2.4.13) drop the sum sign and assume summation over double indices.
As shortly mentioned earlier, the result obtained by Jackson [21] is more general.
More specifically, he does not restrict himself to constant service rates µ i , but consider
38

Chapter 2: Queueing Theory
service rates which can be dependent on the current number of jobs present in the single
queue. However, the result is remarkably similar and again factorized:
with
P(N = k) ∼ ∏ i
˜ρ k i
i
∏k i
l
µ −1
i
(l). (2.4.15)
λ 0i = ˜M ij˜ρ j
{
(2.4.16)
−p ji i ≠ j
˜M ij =
p i0 + ∑ k p ik i = j.
(2.4.17)
Often it is more convenient to write equation (2.4.16) as a matrix equation:
λ = ˜M˜ρ˜ρ˜ρ
where we defined the vector if arrival rates λ, the vector ˜ρ˜ρ˜ρ and matrix ˜M. We want to
note that in the standard case the diagonal elements of ˜M will be ˜M ii = 1.
2.5. Operator technique to solve ME for Queueing Networks
In this chapter we want to introduce a more theoretical physics based description of the
openqueueingnetwork,suggestedbyChernyaketal. [6],basedonthesocalledDoi-Peliti
[12] technique and Massey’s operator theory treatment of such systems [33]. Since we
already solved the steady-state of the Queueing network in the previous section 5 , we will
not go too much into detail here but rather restrict ourselves to present the basic idea.
We choose to present this material mainly for purposes of interest and because it again
outlines a very nice connection between a theoretical physics based theory/methodology
and the seemingly far distant field of Operations Research.
In general, the so called Doi-Peliti technique is a very useful operator theory to describe
classical many-particle systems and the production/annihilation of particles in
such systems. It is based on the second quantization method in quantum theory, which
is exhaustively used in high-energy particle physics to describe the creation and annihilation
of elementary particles as described by the standard model of particle physics.
The value of such a second quantization technique for classical systems is that one does
not have to solve the Master Equation by using classical approaches to solve differentialdifference
equations. Since this is often a hard and eventually not feasible problem using
the standard methods known in the theory of stochastic processes, the Doi-Peliti technique
offers for birth-death processes a second path to the solution. Since a queue can
be described as a birth-death process, this technique can be applied in such systems.
In a discrete stochastic system, like the queueing network, the joint probability distribution
of having N i customers accumulated at queue Qi i = 1...n is the quantity
of interest. From a quantization point of view, on can hence define a base state
5 Albeit not by solving the Master Equation directly but by employing Burke’s theorem.
39

Chapter 2: Queueing Theory
|N〉 = |N 1 = k 1 ,N 2 = k 2 ,...N L = k L 〉 = |k 1 ,k 2 ,...,k L 〉 of the system to have exactly k i
customers in queue Qi. Since here we are dealing with a stochastic system, the actual
state |s〉 of the system is only given by a stochastic linear combination of the base states
(mixed state)
|s〉 = ∑ P(N)|N〉 (2.5.1)
with P(N) being the probability of finding the system in state |N〉 6 . The idea in this
approach is to transform the Master Equation of a queueing network (here we choose to
present the ME for the M/M/1/∞) network)
dP(k 1 ,...,k L ;t)
dt
= ∑
+
−
(i,j)∈E
µ ij Θ(k j )P(...,k i +1,..,k j −1,...;t)−µ ij Θ(k i )P(...,k i ,...,k j )
L∑
P(...,k i −1...)µ 0i Θ(k i )+P(...,k i +1...)µ i0
i=1
L∑
P(...,k i ,...)µ 0i +P(...,k i ,...)µ i0 Θ(k i )
i=1
(2.5.2)
with Θ(k) being a Heaviside-like function with
{
1 if k > 0
Θ(k) =
0 else
(2.5.3)
and E the set of all edges in the system, into a time-dependent Schrödinger-like equation
d
|s〉 = Ĥ|s〉. (2.5.4)
dt
In order to find the steady state we are interested in the time-independent solution
0 = 0|s ss 〉 = Ĥ|s ss〉 (2.5.5)
which hence constitutes the problem of finding the eigenstate |s ss 〉 of the Hamiltonian
Ĥ with eigenvalue 0 7 . However, we need to transform the rhs of the Master Equation
into a second-quantization Hamiltonian. Following the notation of [6], this can be done
in the following way. Define particle creation (birth) and particle annihilation (death)
operators in such a way, that the standard algebra for ladder operators (see e.g. [8]) is
fulfilled
â † i |...,k i,...〉 = |...,k i +1,...〉 (2.5.6)
â i |...,k i ,...〉 = k i |...,k i −1,...〉, (2.5.7)
6 This idea is clearly borrowed from the theory of many-particle quantum systems where e.g. the effect
of entanglement arises.
7 This would be the state with lowest energy or ”base state”.
40

Chapter 2: Queueing Theory
which is the usual definition. Since in the ME we have to deal with a couple of Heavisidelike
functions, it makes sense to introduce a ”skewed”[6] annihilation operator ˆb, which
is in some sense a ”logical” operator:
ˆbi |...,k i ,...〉 = Θ(k i )|...,k i −1,...〉. (2.5.8)
Using the so defined operators, one sees that the Master Equation translates into equation
(2.5.4) with Hamiltonian
Ĥ = ∑
(i,j)∈E
µ ij (â † j −↠i )ˆb i +
L∑
i=1
(µ 0i (â † i −1)+µ i0(1−â † i )ˆb i
)
, (2.5.9)
which then will be used to solve the eigenvalue problem. We will not solve this problem
here but merely state the solution and refer for details to references [6, 33]. One finds
that the eigenstate elements |s ss 〉 i = ∑ k i
P(N i = k i )|N i = k+i〉 of the Hamiltonian are
eigenstates of the corresponding annihilation operator ˆb i |s ss 〉 i = ρ i |s ss 〉 i to eigenvalue ρ i
and that they have the form
|s ss 〉 i =
1
1−ρ i â † |0〉 =
i
∞∑
(ρ i â † i )n |0〉 (2.5.10)
n=0
which translates for the complete state |s ss 〉 into
|s ss 〉 = ∏ i
= ∑ k
|s ss 〉 i (2.5.11)
∏
i
ρ k i
i
|k〉 (2.5.12)
which yields, after normalization and comparing with (2.5.1):
P(N = k) = 1 Z
= ∏ i
∏
i
ρ k i
i
(2.5.13)
(1−ρ i )ρ k i
i . (2.5.14)
This is is exactly result (2.4.15) from the previous section.
For further literature on this technique in connection to queueing networks, see especially
[6] and the Massey paper [33]. For a general introduction into the Doi-Peliti
technique see [12].
In this chapter we have introduced the concept of queues and queueing networks. We
have shown that it is possible for these kind of systems to obtain a closed analytical
form of the steady-state solution. Quite remarkably this solution turned out to be nicely
factorized due to Burke’s theorem.
41

3. Zero-Range Processes and Exclusion
Processes
3.1. The Zero-Range Process
As mentioned in the introduction, the purpose of non-equilibrium thermodynamics is to
solve the Master Equation of the system under consideration. Often knowledge of the
steady state distribution is sufficient for the theoretician, hence solving the homogeneous
form of the Master Equation is enough. But even obtaining this special solution in closed
analytical form is very often not possible. It is, however, well known that the so called
zero-range process (ZRP) can (at least in its 1 dimensional form) be solved exactly and
exhibits an amazingly simple factorized form of the steady-state distribution. The ZRP
is defined as follow: consider a 1-dimensional lattice such that there are exactly L sites
in the lattice. There are N = ∑ L
i=1 N i particles on the lattice, with site i containing N i
particles. If N is fixed, then the system is said to be closed. If N is variable, due to
particles entering and leaving the network, the system is said to be open. Each site can
potentially hold an infinitely number of particles. Particles can hop from site i to one
of its neighboring sites i+1 or i−1 with probability p i,i+1 and p i,i−1 , respectively. For
the probabilities it is generally assumed that
p i,i+1 +p i,i−1 = 1 ∀i (3.1.1)
holds. In this particular process the hopping dynamics are zero-range, meaning that the
rate u i of one particle at site i to jump to one of its neighboring sites is solely a function
of the occupation of the departure site:
u i = u i (N i ). (3.1.2)
Figure 3.1 illustrates the dynamics. In the zero-range process literature it is mostly
assumed that every site is identical and hence
u i (N i ) = u(N i ) (3.1.3)
holds. A closed system corresponds to p i0 = u 0i = 0 ∀i. Also, a common choice is to
assume that in this 1-dimensional system p i,i+1 = p and p i,i−1 = q for all i holds (see
e.g. [29]).
If we want to describe this system from a statistical point of view, we need to know
the underlying stochastic processes, so for example how the number of arriving particles
43

Chapter 3: Zero-Range Processes and Exclusion Processes
p 10 u 1
p 21 u 2 p 32 u 3 p 43 u 4 u 04
u 01
p 12 u 1 p 23 u 2 p 34 u 3 p 40 u 4
site 1 site 2 site 3 site 4
Figure 3.1.: Illustration of the 1 dimensional zero-range process with associated rates. The
probability p i0 and the rate u 0i represent the probability that a particle will leave
the grid at site i and the rate of particles entering the grid at site i, respectively.
or the number of particles leaving a site is distributed. The rate u of a physical process,
as macroscopic property, is defined as average quantity per time:
u = 〈n〉
t . (3.1.4)
This however implies < n >= ut and is a property of the Poisson process. Also, it is
physical to assume that the observed processes are Markovian and memoryless, which
again is a striking property of the Poisson distribution. Hence one chooses to model the
number of particles n i hopping from site i as a Poisson distributed random variable
n i ∼ Pois(u i ) (3.1.5)
and equivalently the number of particles arriving to the system.
The Master Equation of a system as shown in figure 3.1 can be seen to be:
dP(N 1 = k 1 ,...,N L = k L ;t)
dt
=
−
+
−
+
−
L−1
∑
P(...,N i = k i +1,N i+1 = k i+1 −1,...)p i,i+1 u i Θ(k i+1 )
i=1
L−1
∑
P(...,N i = k i ,N i+1 = k i+1 ,...)p i,i+1 u i
i=1
L−1
∑
P(...,N i = k i −1,N i+1 = k i+1 +1,...)p i+1,i u i+1 Θ(k i )
i=1
L−1
∑
P(...,N i = k i ,N i+1 = k i+1 ,...)p i+1,i u i+1
i=1
L∑
P(...,N i = k i −1...)u 0i Θ(k i )+P(...,N i = k i +1...)p i0 u i
i=1
L∑
P(...,N i = k i ,...)u 0i +P(...,N i = k i ...)p i0 u i .
i=1
(3.1.6)
44

Chapter 3: Zero-Range Processes and Exclusion Processes
and can be straight forwardly generalized to a case with general underlying geometry
and L sites:
dP(N 1 = k 1 ,...,N L = k L ;t)
dt
=
−
+
−
+
−
L−1
∑
∑
P(...,N i = k i +1,N i+1 = k i+1 −1,...)p i,i+1 u i Θ(k i+1 )
i=1 j∈∂i
L−1
∑
∑
P(...,N i = k i ,N i+1 = k i+1 ,...)p i,i+1 u i
i=1 j∈∂i
L−1
∑
∑
P(...,N i = k i −1,N i+1 = k i+1 +1,...)p i+1,i u i+1 Θ(k i )
i=1 j∈∂i
L−1
∑
∑
P(...,N i = k i ,N i+1 = k i+1 ,...)p i+1,i u i+1
i=1 j∈∂i
L∑
P(...,N i = k i −1...)u 0i Θ(k i )+P(...,N i = k i +1...)p i0 u i
i=1
L∑
P(...,N i = k i ,...)u 0i +P(...,N i = k i ...)p i0 u i .
i=1
(3.1.7)
where ∂i denotes the set of all direct neighbors of site i. Here again, Θ(x) is a Heavisidelike
logical function with Θ(x) = 1 iff x > 0 and Θ(x) = 0 else to guarantee that only
states with occupation numbers k ≥ 0 are counted in the sum.
As to our knowledge, a solution to this very general system is in the physics literature
not known. However, it is known that the closed general-topology system is equivalent to
a ”weighted” random-walk process and its steady-state distribution is known and given
by [14, 29]
P(N) =
L∏
i
z N i
i
∏N i
n
u i (n) −1 . (3.1.8)
with the fugacities z i
z i = ∑ j
p ji z j . (3.1.9)
Now, solvingaclosedZRPonlycorrespondstosolvingthelastequationforthefugacities
z i .
We will show later, using the connection to queueing networks, that (3.1.8) is the steadystate
distribution even for a general n-dimensional, opened or closed zero-range process.
The fugacities z i however will in such a general case not be given by (3.1.9) but a more
evolved equation.
45

Chapter 3: Zero-Range Processes and Exclusion Processes
The open 1-dimensional ZRP
Levine et al. [29] considered the one dimensional, driven, open zero-range process. In
this setting there exists a preferred direction of particles 1 modeled via the following
parameters
⎧
p j = i+1
⎪⎨
q i = j −1
p ij =
(3.1.10)
β i = L, j = 0
⎪⎩
γ i = 1, j = 0
u 01 = α (3.1.11)
u L0 = δ. (3.1.12)
The authors were able to derive the steady-state solution using a grand-canonical distribution
as Ansatz for the Master Equation. The solution again factorizes and is, as
expected, given by (3.1.8) with the fugacities satisfying
z i =
) Ns−1
i−1 (
[(α+δ)(p−q)−αβ +γδ](
p
q)
−γδ +αβ
p
q
) L−1
. (3.1.13)
γ(p−q −β)+β(p−q +γ)(
p
q
Once the steady-state solution is known, every thermodynamic quantity can be derived
via the partition function.
Some known results
ZRP’s have been studied quite a lot in the last years. Here we quickly review some of
the previous work and results.
In [29] the authors study a 1-dimensional ZRP model with open boundary conditions.
They divide their study into two seemingly important cases: (i) the totally asymmetric
ZRP (sometimes called a driven system), in which particle transition is only possible
in one direction, e.g. q = γ = δ = 0 and (ii) the partially asymmetric case with
p ≠ q, p+q = 1. Both are studied in a special ”condensation model” with
u n = 1+ b n . (3.1.14)
The authors show that in both cases condensation is possible, in the sense that in the
large-time limit an infinite number of particles gather on at least one site. In this case
no complete steady-state solution exists since at least one fugacity is z ≥ 1. The authors
also find that in the totally asymmetric case different condensation regimes, depending
on parameter b exist. In most of these condensation regimes the average number of
particles at a condensate site either increases linearly or as a power law.
1 One might think of an external electromagnetic field.
46

Chapter 3: Zero-Range Processes and Exclusion Processes
q
p
q
q
p
site 1 site 2 site 3 site 4
Figure 3.2.: Illustration of the 1 dimensional exclusion process with associated rates.
In [19] the authors study current fluctuations in the 1-d ZRP. Their findings coincide
with a more general treatment of large deviations functions for currents in queueing
networks as presented in [6]. These results are inter-transferable between the disciplines
using the analogy which will be established in this thesis.
In [23] the authors study the influence of ”quenched disorder” on the 1-d ZRP, i.e. the
effect of non-homogeneous, temporally constant hopping rates from site to site. Whereas
in the asymmetric cases p ≠ q holds for every site, here the authors assign different p i ’s
and q i ’s to every site i, with p i +q i = 1. This treatment will be included in the general
n-dimensional ZRP model, which we propose in this work.
3.2. The Exclusion Process
When talking about the zero-range process, a seemingly different process often studied
in non-equilibrium statistical physics has to be mentioned: the exclusion process (EP).
This kind of process has been studied a lot by e.g. Bernard Derrida (see his lecture notes
[11] for a nice summary). The EP is defined as follows. Consider ”fermionic” particles
on a 1-dimensional lattice, i.e. each site of the lattice can be occupied by at most one
particle. Assume again that with each time step dt there is a probability pdt associated,
that a particle will jump to the site to its right/left, given that this site is empty (see
figure 3.2). These dynamics imply the following:
• The waiting time for a particle leaving the site it occupies is exponentially distributed
with parameter p.
• In contrast to the ZRP, this process is not ”zero-range” since the possibility for a
particleofleavingitscurrentsiteisdependentontheoccupationoftheneighboring
sites.
Because of the last point, this process seems to be much more evolved than the ZRP
and one would not expect a factorized solution for the Master Equation of this system.
However, as Evans and Hanney point out [14], there exists a unique mapping between
the 1-d EP and the 1-d ZRP (see figure 3.3):
P EP (s 1 ,s 2 ,s 3 ,...,s L ) = P ZRP (N 1 = s 1 −1,N 2 = s 2 −s 1 −1,...,N L = s L −s L−1 −1)
(3.2.1)
47

Chapter 3: Zero-Range Processes and Exclusion Processes
exclusion process
1 2 3
site 1 site 2 site 3 site 4
zero-range process
site 1 particle 1
site 2 particle 2 site 3 particle 3
Figure 3.3.: Illustration of the mapping from a 1-d EP to a 1-d ZRP.
ThebasicideaistotreatparticlesintheEPcaseassitesintheZRP.Thentheoccupation
of site i with k i particles in the ZRP case corresponds to the fact that between particle
i and particle i − 1 in the EP case there are k i vacant sites. Denoting the site which
is occupied by particle i in the EP case with s i , then the occupation number of site N i
in the ZRP case will be given via N i = s i − s i−1 − 1. This leads to the above stated
mapping. Evans and Hanney point out that this mapping only holds for the case where
the order of particles is reserved. However, we want to stress that this mapping also only
seems valid, if in the EP case the amount of particles present on the lattice is preserved,
otherwise the mapping would have to go to a ZRP model with changing grid-size. At
this point we do not see a simple way to treat such a case. The general 1-d exclusion
process, with open boundaries, can be very elegantly solved using Derrida’s ”Matrix
Ansatz”[10].
The exclusion process, as a certain particle hopping process, has been utilized as a
model for a variety of transport phenomena. Most popular is certainly its use as a
agent-based (microscopic) approach for traffic modeling, as nicely reviewed by Helbing
in [20]. Also in the biological sciences exclusion process can be employed, e.g. for the
description of certain phenomena in mRNA translation as suggested by [7].
It also has been pointed out, independently by Krug and Ferrari [26] as well as by
Evans [13], that disordered 1-dimensional exclusion processes, i.e. exclusion processes
with non-uniform transition rates, are highly coupled to the theory of Bose-Einstein
condensation.
After having introduced the concept of zero-range processes and exclusion processes,
we proceed in the next chapter with the main statement of this thesis. We will show
that the theory of 1-dimensional zero-range processes can be extended to a theory of
general n-dimensional zero-range processes. This will be done via building a connection
to the previously studied concept of queueing networks.
48

Part II.
Application and Numerics
49

4. A Queueing Network based description
of General Zero-Range Processes
4.1. Introduction
So far we have presented the theory of queues and networks of queues, in particular the
M/M/1/∞ queue, as applied in Operations Research and the physics-native theory of
zero-rangeprocesses. However, thesetwonon-equilibriumprocessesarehighlylinked. To
beprecise: thezero-rangeprocessisaspecialqueueingsystem. Theunderlyingstochastic
processesarethesame. Themathematicaldescriptionsaspresentedareslightlydifferent,
due to the different origins. The results are the same: a factorized, universal solution
of the steady-states. Here we will describe the connection in detail and will present
limitations of this correspondence as well as a straight-forward extension of the theory
of zero-range processes.
4.2. Mapping between queueing networks and zero-range
processes
Taking into account account the things presented so far, the formal connection is easily
established: the zero-range process in 1-d corresponds to the M/M/1/∞ queuing network.
In the queueing system case one talks about customers arriving at a system and
getting processed by the service facilities. In the zero-range terminology customers are
referred to as particles, who enter the grid and proceed along the sites, due to the underlying
dynamics of a zero-range process. Both cases have in common that there is no
precise (deterministic) microscopic description for the actual processes at the service facilities/
sites provided. Instead the assumption is that the processes acting there can be
sufficiently described via a stochastic approach. In both models the underlying processes
are supposed to be Markovian and memoryless. The rates of customers getting served
corresponds to the rate particles leave a site and does in both cases only depend on the
number of customers/particles present in the queue/ at the site. The number of particles
N i waiting in the queue i to get served corresponds to the number of particles N i site i
is occupied by. The possibility of congestion in the queueing system, i.e. the divergence
of the number of customers (achieved when ρ i ≥ 1), can be interpreted as a certain
kind of condensation in the zero-range process. However, in the queueing network (as a
very application-oriented model) congestion is often equivalent to failure of the system,
since in the large time limit newly arrived customers will never get processed through
the complete network. In the zero-range process, considering particles and with no spe-
51

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
Queueing networks
Zero-Range process
customers
particles
local Markovian and memoryless process local Markovian and memoryless process
service rate µ i
transition rate u i
congestion
condensation
Table 4.1.: Correspondence Queueing - ZRP
cific application in mind, condensation is not an exit-criteria and one will naturally ask
what happens after one of the sites experiences condensation. This question leads us to
suggest a ”renormalization” procedure and will be discussed more formally in one of the
next chapters. Table 4.1 provides a comprehensive list of the formal analogy between
the queueing network model and the zero-range process. This very strong formal correspondence,
as strong as to the point of the underlying stochastic principles, leads of
course also to a similar quantitative description and equivalent results.
• Because of the Markovity and memorylessness underlying in both processes, the
times between the arrival/departure of a customer/particle are exponentially distributed
(1.4.11).
• The number of customers/particles arriving at a site during an interval τ = dt is
Poisson distributed (1.4.10).
• Burke’stheoremholdsinbothmodels. Hence, PoissonianinputleadstoPoissonian
output.
• The steady state solutions to the master equations of both systems are factorized.
Compare equations (3.1.8) and (2.4.12).
• The fugacities z i in the zero-range case are given by the generalized utilization
factors ˜ρ.
Hence, results obtained for each of those two models can be easily transferred to the
other. It is rather surprising to us that this equivalence has not been explicitly pointed
out before. Also, until now the description and research on zero-range processes was
mainly limited to the 1-dimensional case. Here, with the direct correspondence to the
queueing network, it is easily possible to generalize to the n-dimensional zero-range
process.
4.3. The general n-dimensional Zero-Range Process
Herewewillgeneralizetheclassicalnotionofazero-rangeprocess(definedin1-dimension)
to a general n-dimensional zero-range process.
Letusdefinethegeneral n-dimensional zero-range process asfollows: consideragraph/network
G = (V,E) with V and E denoting the set of vertices and edges of the graph, respectively.
Allow particles to enter the network at node i in a Poisson distributed fashion
52

•
•
•
Chapter 4: A Queueing Network based description of General
Zero-Range Processes
•
•
•
1 2 3 4
5 6 7
• • •
•• •
8 9 10 11
12 13 14 15
•
•
•
Figure 4.1.: Illustration of a possible underlying graph structure of the n-dimensional zerorange
process. Here the graph structure is a 4×4 grid with one defect. Transitions
of particles are only allowed along the graphs of the edge. Particles can enter the
grid at sites 5, 7 and 14 and can leave the network at sites 3 and 7.
with rate λ 0i . We will refer to those rates as the arrival rates. Let also be the time
between the departure of two particles from node i be exponentially distributed with
rate µ i (departure rates). Let also be a number 0 ≤ p ij ≤ 1 associated to every directed
edge, representing the probability that a particle departing at node i will transfer to site
j. Here j is a graph-neighbor of site i. Let p i0 be the probability that a particle at node
i will leave the network (figure 4.1). Then the Master Equation of this system is given
by equation (3.1.7):
dP(N 1 = k 1 ,...,N L = k L ;t)
dt
= ∑
(i,j)∈E
− ∑
(i,j)∈E
+ ∑
(i,j)∈E
− ∑
+
−
(i,j)∈E
P(...,N i = k i +1,N i+1 = k i+1 −1,...)p i,i+1 u i Θ(k i+1 )
P(...,N i = k i ,N i+1 = k i+1 ,...)p i,i+1 u i
P(...,N i = k i −1,N i+1 = k i+1 +1,...)p i+1,i u i+1 Θ(k i )
P(...,N i = k i ,N i+1 = k i+1 ,...)p i+1,i u i+1
L∑
P(...,N i = k i −1...)u 0i Θ(k i )+P(...,N i = k i +1...)p i0 u i
i=1
L∑
P(...,N i = k i ,...)u 0i +P(...,N i = k i ...)p i0 u i .
i=1
(4.3.1)
53

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
which we quickly introduced in chapter 3.1. This equation is, however, equivalent to the
Master Equation of a queueing-network with site-dependent hopping rates (equation
(2.5.2)). To be precise, building the connection to the queueing network with constant
hopping rates µ i (N i ) = µ i , equation (3.1.7) transforms to equation (2.5.2) with
u i p ij = µ ij Θ(k i )
= µ i p ij Θ(k i )u i = µ i Θ(k i ).
That the hopping rate (function) in the ZRP will be given by a function of form (4.3.2)
and hence be truncated for the case that the departing site is empty, is a natural assumption.
Hence we can conclude that a general ZRP process as described above can be
directly and without loss of generality mapped to a M/M/1/∞ queueing network. The
steady-state ( dP
dt = 0) of the general case is hence given by1 equation (2.4.15)
P(N = k) ∼ ∏ i
˜ρ k i
i
∏k i
l
µ i (l) −1 .
with equation (2.4.16):
λ 0i = ˜M ij˜ρ j
{
−p ji i ≠ j
˜M ij =
p i0 + ∑ k p ik i = j.
This is, from a physics point of view, again a remarkable result. Like the 1-dimensional
ZRP, the general (open or closed) n-dimensional ZRP (a non-equilibrium system) shows
a factorized steady-state with the universal measure (2.4.15). This result also shows that
the previously known steady-state solution of the closed n-dimensional ZRP (3.1.8) holds
for the open case as well, with fugacities z i = ˜ρ i however given by the above equation.
This formalism also allows for the treatment of locally different departure rates µ i (l),
which has to our knowledge not been studied in the ZRP-literature. In general, finding
a solution to a given setting now ”only” corresponds to solving (2.4.16) for ˜ρ˜ρ˜ρ, i.e.
˜ρ˜ρ˜ρ = ˜M −1 λ. (4.3.2)
As said earlier, this corresponds to inverting matrix ˜M, which today can be most effectively
solved using the Coppersmith-Winograd algorithm in O(L 2.376 ) time (when the
system is of size L) and is in general conjectured to be O(L 2 ) in computational complexity
[9]. Hence, in a big system this procedure can be very resource consuming. However,
often the system will be sparse and sparse-matrix algorithms can be employed.
To connect this result, and to justify its correctness, we prove that the fugacities computed
by Levine et al. [29] and given in (3.1.13) can be obtained using the here presented
formulation.
1 Here and henceforth we will stick with the queueing-network notation, but with table 4.1 the transfer
is straight forward.
54

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
The 1-d open ZRP can be treated with the general n-dimensional theory. In[29]equation
(3.1.13) was derived as the unique solution of the recursion relation
pz k −qz k+1 = α−γz 1 = βz L −δ (4.3.3)
In the following we will show that this recursion relation is equivalent to (4.3.2) for the
1-d asymmetric setting, which means solving
−z k (p+q)+z k−1 p+z k+1 q = 0 k ≠ 1,L, (4.3.4)
−z 1 p+z 2 q +α−z 1 γ = 0 k = 1, (4.3.5)
−z L q +z L−1 p+δ −z L β = 0 k = L. (4.3.6)
To show the equality of (4.3.3) and (4.3.4)-(4.3.6) we use a simple proof-by-induction
technique.
a) Assumption: Let (4.3.3) be valid, i.e. pz k −qz k+1 = α−γz 1 .
b) Induction begin: For k = 1, (4.3.3) obviously is identical to (4.3.5).
c) Induction step: For k → k +1, (4.3.3) yields
pz k+1 −qz k+2 = α−γz 1 (4.3.7)
= pz k −qz k+1 (assumption) (4.3.8)
and renaming the indices k +1 → k leads to (4.3.4).
d) Induction end: For k = L (4.3.3) obviously is identical to (4.3.6).
Hence, we have shown that in the n = 1 case, the here presented general n-dimensional
theory connects to previously known results.
4.4. Condensation and renormalization
If one considers the case of constant µ i (l) = µ i , the steady state solution reads
P(N = k) ∼ ∏ i
(˜ρi
µ i
) ki
. (4.4.1)
In such a case, if for at least one i
(˜ρi
µ i
)
≥ 1 (4.4.2)
holds, the system is non-ergodic since the probability distribution is not normalizable.
However, keeping in mind that the above equation is a factorized probability distribution,
the condition above does not necessarily mean that the whole system diverges.
Instead, if (4.4.2) holds only for a finite number of sites i, there will only be condensates
(i.e. a diverging number of particles gathering) at those particular sites. The occupation
numbers of other sites might still be given by a proper probability distribution. However,
55

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
in the current framework it is not obvious how to obtain a ”renormalized” 2 steady-state
probability distribution, given that one or more sites will form a condensate in the above
sense. We propose the following general scheme to obtain such a renormalized solution.
Renormalizing the model
We propose a renormalization procedure to calculate the steady-state distribution in the
case that condensates emerged at a finite number of sites. This procedure builds on the
idea that a congested site is occupied by virtually an infinite number of particles in the
large-time limit, hence acting as a source. This means that the effective rate of particles
leaving site i to arrive at a neighboring site j is independent of the size of the queue (the
reservoir is always big enough), hence Burke’s theorem does not apply and the rate of
particles leaving that site is equivalent to the intrinsic departure rate µ i :
λ R ij = p ij µ i . (4.4.3)
We use this fact to renormalize the model in the following way:
1. Move the congested site ξ to spatial infinity, i.e. exclude it from the model.
2. Assign to every former neighbor k of the congested site a new external in-link, i.e.
increase the external arrival rate at this node to λ R 0k = λ 0k +µ ξ p ξk .
3. Assign to every former neighbor k of the congested site a new external out-link,
maintaining the former transition rate µ kξ = µ k p kξ .
Thesoconstructedmodelisnormalizablebydefinition(Weremovedallnon-normalizable
parts). Figure 4.2 shows an illustration of the renormalization procedure. We want to
stress that this procedure is only applicable if
• the rates µ i are constants.
• the rates functions µ i (n i ) are globally defined, i.e. µ i (n i ) = µ(n i ).
Verification of this renormalization procedure will be illustrated by numerical simulations
in the next chapter.
Since the renormalized model is a proper M/M/1/∞ queue again, the probability distribution
is factorized. However, the fugacities ˜ρ i will have changed, which we denote
by
˜ρ i → ˜ρ R i . (4.4.4)
Hence the renormalized solution is given via:
P(N = k) R ∼ ∏ {i R }(˜ρ
R
i
µ i
) ki
. (4.4.5)
2 Here we use the term ”renormalization” in its general meaning of finding a new base, such that there
are no infinities in the system anymore.
56

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
p 10
0
0
p 21
p 32
µ 03
1 2 3
µ 1
p 12 µ 2
p 23 µ 3 p 30
µ 01
0
p R 10 = p 10 +p 12
µ R 03 = µ 03 +µ 2 p 23
0 1 3
µ R 01 = µ 01 +µ 2 p 21
µ 1 µ 3 p R 30 = p 30 +p 23
Figure 4.2.: Illustrationofthegeneralrenormalizationprocedureforapossiblesub-graph, given
that there is a condensate at site 2. Top figure shows the pure (not-normalizable
model) with infinite number of particles at site 2. This model is transformed
into the new renormalized model (bottom), without infinity of the congested site
entering the analysis.
Here the product goes over all nodes present in the new model. If it is clear from the context,wechoosetodropthesuperscriptRdenotingtherenormalizedsolution/parameters.
In a model described by a product measure of geometric distributions as the here
presented, the mean number of particles to occupy site i is given by
〈N i 〉 =
=
=
∞∑
P(N i = k i )k i
k i =0
∞∑
k i =0
(
1− ˜ρ i
µ i
)(˜ρi
µ i
) ki
k i
(˜ρi
µ i
)(
1− ˜ρ i
µ i
) −1
. (4.4.6)
This can be easily seen using the sum-formula for the geometric series.
Similarly one finds that the variance Var(N i ) = σ 2 (N i ) holds:
Var(N i ) =
(˜ρi
µ i
)(
1− ˜ρ i
µ i
) −2
. (4.4.7)
There are also statements which can be made about the currents of particles over
every link in the model. As we showed earlier, the distribution of particles leaving a
site is a Poisson distributed variable with effective rate λ ij = µ ij ρ i and in this model
57

Chapter 4: A Queueing Network based description of General
Zero-Range Processes
λ ij = ˜ρ ij . So that we find for the current J ij between sites i and j:
J ij = −J ji = ˜ρ i − ˜ρ j . (4.4.8)
In this chapter we introduced the notion of a general zero-range process. We showed that
it is from a mathematical point of view equivalent to the Jackson network, i.e. a network
of M/M/1/∞ queues. We provided the necessary mathematical formalism and showed
that the steady-state solution is given by a universal product measure. The fugacities
entering this product measure can be calculated by solving a linear system of equations
of size L, where L is the number of sites in the zero-range process. We also introduced
a renormalization technique to solve n-dimensional zero-range processes in the case of
condensation. In the next chapter we will provide numerical evidence for the validity of
the introduced theory and will study some interesting effects of condensation.
58

5. Numerical and analytical results for the
n-dimensional ZRP
5.1. Introduction
In this chapter we will present numerical results for the n-dimensional zero-range process/
M/M/1/∞-queueing network to get a general impression of how such systems
can behave and eventually differ from the well-studied 1-d model. The dynamics and
steady-state solution of a n-dimensional ZRP will be highly dependent on the topology
of the network which we choose to study. Here it is not our aim to conclude any general
statements, rather we choose to study the easiest possible n > 1−dimensional structure,
which is the 2-dimensional grid as in figure 4.1. Our results include:
• a phase transition/condensation phenomenon
• the influence of boundary conditions
• symmetry with respect to in- and outflux in the model
• the influence of in- and outflux.
5.2. Description and general behavior of a global µ i -model
We choose to study the general behavior of a zero-range process with constant and
globally defined µ i = µ = const on a 5 × 5 homogeneous, finite grid (figure 5.1). Also
we define a global rate of particles arriving to the grid λ 0i = λ. Then, looking at the
steady-state probability distribution, we have for this system
P(N = k) ∼ ∏ i
) ki
(˜ρi
(5.2.1)
µ
with
˜M −1 λ = ˜ρ˜ρ˜ρ (5.2.2)
or in a more compact form
P(N = k) ∼ ∏ i
∼ ∏ i
( ˜M −1 e λ ) k i
i
( λ
µ) ki
( ˜M −1 e λ ) k i
i κk i
(5.2.3)
59

•
•
•
•
•
•
Chapter 5: Numerical and analytical results for the n-dimensional ZRP
1 2 3 4
5
6 7 8 9 10
11
12 13 14 15
16
17 18 19 20
21
22 23 24 25
Figure 5.1.: Illustration of a 5×5 grid with particle influx solely at node 1 and outflux solely at
node 3. We define the sets I = {1} and O = {3} as the sets of nodes with particle
influx and outflux, respectively. The transition probabilities in this model do only
depend on the number of neighbors and out-links of a site. See table 5.1 for some
transition probabilities in this structure.
where we defined
κ = λ µ
(5.2.4)
and e λ as the unitary vector of all in-links
e λ = λ 1 λ
(5.2.5)
Hence the local utilization factor, controlling the convergence of the sum, for every site
i is give by
ρ i = ( ˜M −1 e λ ) i κ, (5.2.6)
and the average steady-state occupation number
〈N i 〉 = 1
1−ρ i
. (5.2.7)
The parameter κ is the ratio between arrival and hopping rate and is independent of
the structure of the underlying network. If one fixes the underlying network structure,
i.e. matrix ˜M which defines the structure of the underlying process and the ”routing”
of particles (the probabilities of a particle leaving site i to transfer to site j), hen κ
is the only adjustable parameter influencing the steady-state solution and whether or
not condensates will emerge. Here we choose to study a homogeneous model, with
probability p ij of transferring from site i to j given as
p ij = 1
|∂i| , (5.2.8)
where |∂i| denotes the cardinality of the set of all graph-neighbors of site i, including
edgestoleavethenetwork. Forexampleinfigure5.1theprobabilityforaparticleleaving
60

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
a site in the bulk of the system 1 to one of the 4 neighboring sites would be 1/4 = 0.25.
For illustration purposes, some more transfer probabilities are shown in table 5.1.
departure site i arrival site j p ij
13 {8,12,14,18} 1/4
11 {6,12,16} 1/3
25 {20,24} 1/2
3 {0,2,4,8} 1/4
1 {2,6} 1/2
Table 5.1.: Some transition probabilities according to the ZRP on a 2-dimensional 5×5 closed
grid as shown in figure 5.1. Site index 0 denotes the environment.
Dynamical modeling of n-dimensional ZRP
We now compare simulation results with our theory for the 2-d ZRP with global µ i .
For the simulations it is necessary to sample a valid trajectory of the ongoing stochastic
processes in the ZRP. In general, in a model with total L sites and I sites with particle
in-flux, we need to sample a consistent trajectory of L + I independent and parallel
stochastic processes (L processes of particles leaving any of the L sites and I processes
of particle influx). From a sequential programming point of view, it is not directly clear
how to achieve this task. However, sampling such a trajectory can be done via the
Gillespie-algorithm [17]. The idea of this algorithm is the following:
a) Sample a global time τ, which is the time span between the last stochastic event
(at time t) and the next stochastic event in the complete system. In the ZRP as
considered here, an event can either be the arrival of a new particle to the system
at site i (we denote this event by A i ) or the departure of one particle from site
i (which we denote by D i ). Here the waiting times between any two consecutive
events of the same type is exponentially distributed with rate λ 0i for events A i (i.e.
particles arriving at site i) and with rate µ i for event D i (i.e. particles leaving site
i). Then the global time τ for the next event will be exponentially distributed
τ ∼ µ Σ e −µ Σt
(5.2.9)
with
µ Σ = ∑ i
(µ i +λ 0i ). (5.2.10)
b) Sample the kind of event, which will take place at t+τ according to the rates of
the single event. Specifically, the probability p Ai that event A i will take place is
p Ai = λ 0i
µ Σ
(5.2.11)
1 The bulk is defined as all sites which are not at the boundary. In the case of figure 5.1 this would be
the 3×3 square at the center of the grid.
61

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
i ρ i 〈N i 〉 theo σ theo 〈N i 〉 exp
1 0.6467 1.83 2.28 2.20
5 0.3615 0.57 0.94 0.60
7 0.9125 10.42 10.91 11.10
12 0.8905 8.13 8.61 9.77
17 0.8761 7.07 7.55 7.08
Table 5.2.: Comparisonofmeansiteoccupationsobtaineda)viadynamicalsimulation(〈N i 〉 exp )
and b) as analytical result using equations (5.2.6) and (5.2.7).
10.5
0
9.0
1
7.5
6.0
2
4.5
3
3.0
4
1.5
0 1 2 3 4
0.0
Figure 5.2.: Average occupation numbers of a ZRP on a 5×5 grid as in figure 5.1 with κ = 0.15.
Average particle numbers obtained using the Gillespie-algorithm over 10.000 time
units after allowing to converge to the steady-state solution in 200.000 time units.
and equivalently for events of the D-type:
p Di = µ i
µ Σ
. (5.2.12)
Using this algorithm, one obtains a valid trajectory for the multiple parallel stochastic
processes.
For illustration purposes, we do not show detailed long-term dynamics of the model
obtained using the described algorithm, but restrict ourselves to show the average number
of particles occupying every site for λ = 0.15, µ = 1, κ = 0.15 in Figure 5.2. In
this example, sites 7,12 and 17 show the highest average occupations. For this case it
is possible to calculate the ρ i ’s using equation (5.2.6). Comparing the numerical and
analytical results (table 5.2) obtained using equation (4.4.6) one concludes that the
theoretical prediction holds (within the standard deviation).
62

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
WethusillustratethatthetheoreticalframeworkseemstodescribestheZRPcorrectly.
We will now study the influence of the parameter κ and the topology.
Influence of parameter κ
In this model the global parameter κ is certainly the most important and easy-to-study
parameter which influences the steady-state behavior of the system. To get an impression
of its impact, figures 5.3 (a)-(h) show the average number of particles occupying
the sites, obtained analytically from results of equation (5.2.6) for different κ (after
correct renormalization). The following general statements can be made based on the
observation:
1. With increasing κ the total number of particles in the complete system seems to
increase.
2. With increasing κ the number of condensates seems to increase until the system
reaches a stationary state, where further increase in κ does not lead to more condensates.
The explanation for the first observation is directly given by equation (4.4.6) - with
increasing κ the average number of particles at every site increases. The first part
of the second observation is somewhat intuitive: if we pump more particles into the
system, more sites will get congested. However, the second part of this observation is
somewhat counter intuitive. It seems that the system falls into a stable state, where no
more condensates emerge if κ is increased. A rigorous explanation for this observation
will be given later. If one takes a look at figure 5.3 again, it appears that in general
(independent of κ) more particles gather in average in the bulk of the system than on
the rim. However, this is certainly more a topology-effect and will be discussed later.
Condensation and phase transition
In the description of a global µ and λ ZRP model, introduced previously, the convergence
factors ρ i are given by (5.2.6). From this equation it is possible to calculate the critical
κ c ’s at which condensates (in the aforementioned sense) will emerge, given matrix ˜M.
Let us illustrate this. As said earlier, the first condensate will emerge at some site ξ, if
for the utilization factor it holds ρ ξ ≥ 1. This happens for the first time, if
ρ i = ( ˜M −1 e λ ) i κ = 1 (5.2.13)
for any i. Hence we find the first critical value of κ after which at least one of the sites
will experience condensation to be
κ c = min
i
{
}
1
. (5.2.14)
( ˜M −1 e λ ) i
63

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0
1.50
1.35
0
10
9
1
2
1.20
1.05
0.90
1
2
8
7
6
5
3
0.75
0.60
3
4
3
4 0.45
4
2
1
0 1 2 3 4
0 1 2 3 4
(a) κ = 0.10
(b) κ = 0.15
200
0
1
2
.
120
105
90
75
60
0
1
2
.
.
175
150
125
100
3
45
30
3
75
50
4
15
4
25
0 1 2 3 4
0 1 2 3 4
(c) κ = 0.20
(d) κ = 0.25
0
1
2
3
.
.
.
360
320
280
240
200
160
120
0
1
2
3
. .
.
.
54
48
42
36
30
24
18
80
12
4
40
4
6
0 1 2 3 4
0 1 2 3 4
(e) κ = 0.30
(f) κ = 0.34
0
1
2
3
.
. .
.
.
27
24
21
18
15
12
9
0
1
2
3
.
. .
.
.
27
24
21
18
15
12
9
6
6
64
4
0 1 2 3 4
3
4
0 1 2 3 4
3
(g) κ = 0.40
(h) κ = 0.80
Figure 5.3.: Average occupation of sites for different κ on the setting shown in figure 5.1. The
dark red, dotted sites are condensates. The first condensate emerges at κ = 0.164.
Note: the color maps are always normalized for every instance and not necessarily
inter-comparable.

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
One can indeed say more: if the values in the above set are not degenerated, exactly one
condensate will emerge when approaching κ = κ c and the critical site ξ is obviously
{ }
1
ξ = argmin . (5.2.15)
i ( ˜M −1 e λ ) i
Here we defined the critical value κ c as the one where the first of (eventually) multiple
condensates will emerge. In a sense, one can say that a phase transition takes place at
κ c , transforming the system from the uncongested phase to the 1-site-congested phase.
However, if one looks at the steady-states obtained for different κ (figure 5.3) it seems
that there are more than just one condensates in the large κ limit, hence multiple phase
transitions took place. Interestingly, the system seems to have a stable phase as well.
Once the system reached that phase, increasing κ does not lead to the emergence of
further condensates.
Calculating the critical κ values for every of those phase transitions is an extension of
the method presented above, utilizing the former described re-normalization procedure.
However, (5.2.14) and (5.2.15) will not hold any longer in a renormalized model, since
a new in-link at one of the neighbor nodes k of ξ will not have the universal arrival rate
λ, but arrival rates
λ R 0k = λ 0k +µp ξk . (5.2.16)
However, to account for this effect the scheme presented above can be generalized in the
following way. Let be λ R = e R λ λ+wR µ µ, where e R λ
denotes the unit vector containing
all in-links in the renormalized model with global arrival rate λ
(e R λ ) k = λ 0k
1
λ
and w R µ the weighted unit vector of new in-links due to the re-normalization
(5.2.17)
(w R µ ) k = p ξk . (5.2.18)
Using this more general formulation, the utilization factors read
ρ i = ( ˜M −1 e R λ ) iκ+( ˜M −1 w R µ ) i (5.2.19)
and equation (5.2.14) generalizes to
{ }
1−( ˜M −1 wµ R ) i
κ c = min
i ( ˜M −1 e R λ ) . (5.2.20)
i
Again, the site at which the condensate emerges is obtained by replacing min by argmin
in the equation above.
To show that the method works, we calculate the critical values κ c and the corresponding
sites at which the condensates emerge for the model studied earlier. The results are
shown in table 5.3. Comparing with the sequence earlier (figures 5.3 (a)-(h)), our prediction
seems to hold. After the threshold κ c = 0.355 there are no more condensates,
65

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
# condensates κ c site
1 0.164 7
2 0.218 12
3 0.279 16
4 0.303 6
5 0.355 1
Table 5.3.: Order of emerging condensates, with critical values κ c , for the zero-range process
on a 5×5 structure as in figure 5.1.
since in this case the denominator of equation (5.2.20) becomes zero, meaning that the
next such phase transition will occur for κ → ∞. To explain this behavior, let us look
at the denominator
( ˜M −1 e λ ) i (5.2.21)
and determine when this quantity is zero for all i. This will obviously be the case, when
the unit vector e λ is the null vector. This however is exactly the case, when there are no
more in-links into the system with global rate λ, which can happen when all sites with
such an in-link in the base (not renormalized) scenario of the system form condensates.
This explanation fits exactly the data obtained in table 5.3 and figure 5.3 since here the
last site which experiences condensation is site 1, the only site with an λ-in-link. Hence
we have the following general statement.
Stable configuration In a global µ i , n-dimensional zero-range process, the stable configuration
is exactly then achieved, when all sites with λ-in-links formed condensates.
Mathematically, this results is clear. From a high-level point of view it is rather remarkable
for the following reason. To increase parameter κ we have two possibilities:
a) either choose to increase λ or b) choose to decrease the global parameter µ. That
increasing λ has no further effect if all λ-in-link sites already formed islands (and hence
provide the system in the large time limit with infinite supply of particles) is easily
seen. That decreasing µ has no effect is more subtle: since there are still out-links of
the system, one would suggest that decreasing µ and hence decreasing the effective rate
of particles leaving the network will lead to more congestion in the system. This seems
numerically and theoretically not to be the case. One has to keep in mind that in the
stable configuration µ is not only the rate of particles leaving the network but is also the
rate of particles entering the uncongested part of the system and the rate of particles
departing at a site and transferring to one of its neighboring sites. Hence the local utilization
factors ρ i , which basically are the ratio between local in-rate and local effective
departure-rate will stay the same. This is true, since all effective departure-rates and
in-rates are multiples of the global rate µ. Thus, if one scales (decrease/increase) µ by
a factor of α, the utilization factors are not changed:
ρ i = µ in
µ out
→ α·µ in
α·µ out
= µ in
µ out
= ρ i (5.2.22)
66

•
•
•
•
•
•
Chapter 5: Numerical and analytical results for the n-dimensional ZRP
1 2 3 4
5
6 7 8 9 10
11
12 13 14 15
16
17 18 19 20
21
22 23 24 25
Figure 5.4.: Illustration of a 5 × 5 torus with possible particle influx at site 1 and outflux
at site 3. On a torus periodic boundary conditions are imposed, i.e. every site
has 4 nearest neighbors (plus eventual in- or out-links). As an illustration of this
periodicity, we hint to the fact that in such a setting e.g. site 25 has not only sites
20 and 24 as graph neighbors, but also sites 5 and 21.
Hence we directly conclude, that in the stable configuration not only the number of
condensates will be stable, but also the utilization factors ρ i and hence the steady-state
distribution of the system (including the mean number of particles present at a notcongested
site) will stay the same. This is a very interesting observation since it implies
robustness of the system after a critical value of κ is passed. Once in the stable phase,
local fluctuations in λ and global fluctuations in µ will not perturbate the system out of
the stable solution.
Study of some topological influence
After we have shown the influence of parameter κ, we will now take a look at the second
factor which influences the non-equilibrium steady state of such a global µ i system: the
topology.
The torus
In the former studied realization we chose to consider a finite grid (figure 5.1). Here we
deviate from this setting and introduce the ZRP on a torus, i.e. a periodic grid (figure
5.4). The torus is more homogeneous than the closed grid since there is no rim and bulk
- every site has 4 nearest neighbors and hence particle routing is equivalent. The only
deviation from the transfer probabilities p ij = 0.25 is seen at nodes with out-links, where
p i0 = p ij = 0.2. Figures 5.5 (a)-(c) show the average particle occupations of the ZRP
on a torus. The first striking difference is that, compared to the finite grid, the stable
solution is achieved at a much lower value of κ and with only one condensate emerging
at the site which hosts the in-link. This is a very interesting result for which we unfortunately
lack intuitive understanding at this point. What is also very apparent is the
67

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0
1.9
1.8
0
.
7.0
6.5
1
1.7
1.6
1.5
1
6.0
5.5
2
1.4
2
5.0
1.3
4.5
3
1.2
3
4.0
4
0 1 2 3 4
1.1
1.0
4
0 1 2 3 4
3.5
3.0
(a) κ = 0.10
(b) κ = 0.20
0
.
7.0
6.5
1
6.0
5.5
2
5.0
3
4.5
4.0
4
0 1 2 3 4
3.5
3.0
(c) κ = 0.80
Figure 5.5.: Average occupation of sites for different κ on the 5×5 torus setting as in figure 5.4.
The dark red, dotted site is the condensate. The condensate emerges at κ = 0.152
.
symmetry in the solution, due to the perfect symmetry of the underlying grid. Basically
in this easy setting of one in- and one out-link, the local solution is solely dependent
on the distance and relative position of the local site to the in-link site and the out-link
site. For example sites 7 and 22 are equivalent in this regards and hence show the same
steady-state. There are no boundary effects in such a setting.
Position of in- and out-links:
Since the base models studied so far are highly symmetric (torus) or somewhat symmetric
(closed grid), changing the in- and out-link from figure 5.1 and 5.4 in a ”correlation”
preserving way, should not change the quantitative behavior of the system. To illustrate
thisprinciple, figures5.6(a)and(b)showtheinfluenceofmirroringthesettingofin-and
out-links for the ZRP on a finite 5×5 grid (figure 5.1) along the central vertical axis as
well as rotating by π 2
around site 13. As expected, the steady-states stay quantitatively
the same and are only a mirrored/rotated version of the former solution. However, in
68

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0
1
2
3
.
.
.
.
.
27
24
21
18
15
12
9
0
1
2
3
. .
.
.
.
27
24
21
18
15
12
9
6
6
4
3
4
3
0 1 2 3 4
0 1 2 3 4
(a) mirrored
(b) rotated
0
1
2
3
4
.
.
22.5
20.0
17.5
15.0
12.5
10.0
7.5
5.0
2.5
0 1 2 3 4
(c) translated
Figure 5.6.: Stable configuration with mirrored, rotated and translated in- and out-links of the
ZRP on a finite 5×5 grid with boundaries, as in figure 5.1.
this special case with boundary there is no general rotational or translative symmetry,
as can be seen in figure 5.6 (c), where the in- and out-links have been translated to the
right by a distance of one. That this symmetry does not hold is because in this model
the routing dynamics are different between bulk and rim sites. In general, a change of inand
out-links will only give an equivalent solution if their positions have been changed
using such a transformation, which is a symmetry-preserving transformation for the underlying
routing-network (as is clearly the case with the previous mirror and π 2 -rotation
for the boundary case). The torus is completely symmetric and so is a transformed
solution, given that the relative distances between all in- and out-links stay the same.
Figure 5.7 (a)-(c) show again the steady-states of the torus setting with mirrored (a),
rotated (b) and translated (c) in- and out-links.
Number of in- and out-links
Certainly, the number of in- and out-links will have an influence on the steady-states
and the stable configuration as well. Just to illustrate this influence, figure 5.8 shows
69

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0
.
7.0
6.5
0
.
7.0
6.5
1
6.0
1
6.0
5.5
5.5
2
5.0
2
5.0
4.5
4.5
3
4.0
3
4.0
4
3.5
4
3.5
0 1 2 3 4
3.0
0 1 2 3 4
3.0
(a) mirrored
(b) rotated
0
.
7.0
6.5
1
6.0
5.5
2
5.0
4.5
3
4.0
4
0 1 2 3 4
3.5
3.0
(c) translated
Figure 5.7.: Stable configuration with mirrored, rotated and translated in- and out-links of the
ZRP on a periodic 5×5 grid (torus), as in figure 5.4.
the stable configurations for one random assignment of 3 in- and 4 out-links in the finite
grid and the torus. The steady-states in the stable configurations are very different from
the so far studied model of one in- and one out-link. A common observation for the
torus is however that in this model with 3 sites having in-links, the stable configuration
again consists of exactly three condensates. Nevertheless, due to the vast possible combinations
of assignments of in- and out-links, we are not able to conclude any general
statements from this example. Hence we choose to perform Monte-Carlo sampling to
study several relations:
a) Phase diagram: no condensate → one condensate
An interesting transition in the studied models is certainly the transition of the phase
without any condensates to the phase with one condensate. This transition occurs at
a (for every model) typical value κ c . Hence we choose to show in figure 5.9 the phase
diagram of this transition in a plot spanning the space of parameters in our model: the
number of in-links I, the number of out-links O and the parameter κ. Since the number
70

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0
1
2
3
.
.
.
.
.
.
72
64
56
48
40
32
24
0
1
2
3
.
.
.
7.2
6.6
6.0
5.4
4.8
4.2
3.6
16
3.0
4
8
4
2.4
0 1 2 3 4
0 1 2 3 4
(a) grid
(b) torus
Figure 5.8.: Stable configuration for one randomly chosen combinations of 3 in-links (at sites
1, 4 and 17) and 5 out-linkes (at sites 4, 10, 19 and 25) for the closed 5 grid (a)
and the 5×5 torus.
of combinations of possible in- and out-link assignments is huge 2 we choose to sample for
every pair (O,I) over 100 random samples. Although the sampling has been random,
there appears to be a smooth phase-dividing surface, with the condensate-phase lying
above it. Hence κ c (O,I) seems to be a self-averaging quantity in this kind of system.
Also, somehow surprisingly, the qualitative behavior of the phase diagram seems to be
equivalent for the grid and the much more symmetric torus. However, looking at the
standard deviations for both diagrams, the statistics in the torus case is more smooth
than for the closed grid (table 5.4), as can be seen by the smaller maximum relative
standard deviation. In general the relative standard deviation is biggest for those samcase
I O 〈κ c 〉
σ
〈κ c〉
grid (1,25) (1,25) (0.005,0.75) (0.04,0.24)
torus (1,25) (1,25) (0.005,0.75) (0.005,0.09)
Table 5.4.: Range of means for the first critical value of κ and their relative standard deviations
for the data shown in figure 5.9.
ples involving a lot of disorder, i.e. systems with small I and O. One observed difference
though is that whereas in the torus case the relative standard deviation decreases most
significantly in the direction of increasing I, in the closed grid case this quantity decreases
most in direction of increasing O.
b) Maximum number of condensates/stable configuration
As we have illustrated earlier, the number of maximum condensates (i.e. the number
of condensates in the stable configuration) seems to be much closer to the number of
2 To
(
be precise, in a 5×5 model the number of combinations for I in-links and O out-links is exactly
25
)·( 25
I O)
, which is maximal at O = I = {12,13} with sim10 13 combinations.
71

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
0.7
0.6
0.5
0.4
0.3
0.2
0.1
kappa_c
0.7
0.6
0.5
0.4
0.3
0.2
0.1
kappa_c
5
10
15
in nodes
20
5
10
15
20
out nodes
5
10
15
in nodes
20
5
10
15
20
out nodes
(a) grid
(b) torus
Figure 5.9.: 2-Phase diagram over the space of free parameters O, I and κ for the closed 5×5
grid (a) and the 5×5 torus (b). Each data point was obtained via Monte-Carlo
sampling over 100 instances.
in-links in the case of a totally symmetric torus, than in the case of a finite grid with
boundary. To study this effect further, we performed again Monte-Carlo sampling in
the 5 × 5 grid and torus and show the results in figure 5.10. Figure 5.10 (b) clearly
reinforces the observation that in the completely symmetric case of a torus, the number
of condensates, C, in the stable configuration equals the number of sites with in-links:
C ≈ I. (5.2.23)
The model only deviates from this behavior for the case of relatively large numbers
of in-links and small number of out-links. The closed grid on the contrary shows the
opposite behavior (see also figure 5.10 (c)). In the vast majority of cases the number
of condensates in the stable solution is larger than the number of sites having in-links:
C > I. It is only in the case of large amounts of in-links that the number of condensates
is somewhat equal. However, this might only be due to the fact that in those cases
the maximum number of condensates is anyways bounded from above by the number of
sites in the system, hence no large deviations are possible. Also, in both cases (torus
and finite grid), with increasing number of out-links the maximal number of condensates
seems in average to be closer to the number of sites with in-links
C ≈ I if O ≈ 25, (5.2.24)
hence providing more evidence for above statement positively.
Before we close this chapter, we want to stress that although most effects in this
chapter were studied on a relatively small toy-example with two different boundary conditions,
we expect a lot of the statements to hold in more general settings. For grids in
size much larger than 5×5, their behavior should be very close to the here studied torus
model, since in those cases the boundary is virtually only present at (spatial) infinity.
Hence boundary effects will not have a big influence on the dynamics and statistics in
72

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
20
15
10
max. condensates
20
15
10
5
max. condensates
5
5
10
15
in nodes
20
5
10
15
20
out nodes
5
10
15
in nodes
20
15
10
5
20
out nodes
(a) grid
(b) torus
25
20
max. condensates
15
10
5
0
0 5 10 15 20 25
in nodes
(c) grid - projection
Figure 5.10.: Number of maximal condensates C as a function of O and I for the closed 5×5
grid (a) and the 5×5 torus (b). Each data point was obtained via Monte-Carlo
sampling over 100 instances. (c) shows a projection of (a) on the C-I subspace.
the bulk of the system.
Disordered n-dimensional ZRP
Although here we studied a global µ i = µ and λ 0 i = λ model, the analysis can be easily
extended to a case with quenched disorder in the rates. If one defines the locally different
rates µ i with respect to some base µ as
µ i = α i µ (5.2.25)
and the same with λ 0i
λ 0i = β i λ (5.2.26)
equation (5.2.6) will be transformed into
ρ i =
˜M
−1
( ˜M −1 e λ ) i
κ. (5.2.27)
α i
73

Chapter 5: Numerical and analytical results for the n-dimensional ZRP
with the definition
e λ = λ 1 λ
unchanged, but now due to λ i = β i λ, will be evaluated as e λi = β i . Hence our model
provides a natural way of treating disordered n-dimensional zero-range processes. It
should also be clear that the study of topological irregularities is easily possible within
the presented framework via routing matrix ˜M. In general, the framework offers a wide
variety of possible future studies, may it be theoretical or application-close.
In this chapter we gave some numerical evidence for the correctness of our proposed
n-dimensional ZRP framework. We also studied the influence of a global parameter κ
and established that the system eventually falls into a stable configuration, where any
increase of κ will not have an influence on the steady-state of the system. We also
showed that there are eventually multiple phase transitions with the appearance of multiple
condensates before the stable configuration is achieved. We also showed that the
behavior of the finite and periodic 5×5 grid is rather different, due to boundary effects.
74

6. Conclusion and Further Studies
Inthisworkweestablishedastrongconnectionbetweentheconceptofqueueingnetworks
and the zero-range process. We showed that it is possible to extent the 1-dimensional
theory of zero-range processes to a more general n-dimensional theory and that one is
still able to solve this non-equilibrium problem for the steady-state solution. Based on
this theory we studied the ZRP on a 2-dimensional grid, especially focusing on the influence
of an external parameter κ as well as several topological properties. We furthermore
observed an interesting condensation phenomenon, which is similar but in detail distinct
from the 1-dimensional case.
The here presented treatment and study of the n-dimensional ZRP allows for a number
of natural extensions and further studies:
• Percolation. As we have shown, the n-dimensional ZRP may exhibit condensation
at eventually a large number of sites. We have also shown, that such sites can
basically be considered as ”excluded” from the model. Hence starting from a ZRP
onaconnectedgraph, thiseffectmayleadtoseparationofthegraph(breakdownof
the giant connected component). This can be considered an ”inverse” percolation
problem[18], starting in the supercritical phase and via increasing κ leading to
undergo the percolation threshold. If one denotes the (percolation) critical value
of κ with κ pc , the following expression for the percolation threshold should hold:
p c = N −N c(κ pc )
. (6.0.1)
N
Here N is the total number of sites and N c (κ) the number of congested sites for
given κ. To prove percolation, one would have to study the sub-critical phase
(which here is the phase with κ > κ pc ) and whether or not the cluster size decays
exponentially with increasing κ. Based on preliminary results, we suspect that the
2-dimensional ZRP, as presented here, undergoes a percolation transition. This
however work in progress and provides an interesting research question for further
studies.
• Disorder. As mentioned in the last chapter, the treatment of a disordered n-
dimensional ZRP model is easily possible within the proposed framework and provides
an interesting direction for further research.
• Different topologies. In this thesis we focused in the last chapter on an exemplary
treatment of the 5×5 grid to illustrate the power of the proposed framework. This
can however be naturally extended to the study of ZRP’s on other topologies, for
75

Chapter 6: Conclusion and Further Studies
example also including the study of zero-range processes on (3-dimensional) cubes
or classes of famous network types like random graphs.
• Applications. It would be interesting to see if the here presented theory of n-
dimensional ZRP’s can be applied e.g. as a straight forward extension of an 1-d
ZRP application. In the field of porous materials this could be of interest. Also
one could ask if the recent 1-dimensional EP/ZRP based treatment of traffic flows
can be enhanced[20].
76

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