Using the Pareto Distribution in Queueing Modeling - Classweb ...

Abstract
Submitted to Journal of Probability and Statistical Science, Mar. 2005.
Using the Pareto Distribution in Queueing Modeling
John Shortle Martin Fischer
Donald Gross Denise Masi
George Mason University Mitretek Systems
The Pareto distribution was first formulated in the late 1800s by the Italian economist Vilfredo
Pareto. He argued that the distribution of income could be described by the formula log(N) =
log(A) + m log(x), where N is the proportion of income earners who receive incomes higher than x,
and A and m are constants. Over the years Pareto’s Law has held up in empirical studies. The
Pareto distribution has recently been used as a model for file sizes in the Internet and insurance
losses. It has various forms; here, we consider a 1-parameter form and a 2-parameter form. The
purpose of this paper is to show that using different forms of the Pareto distribution can result in
drastically different congestion measures in a queueing system. This is true even if the means and
variances between two Pareto distributions are the same. In other words, even within the same
family of distributions, a matching of the first two moments is not sufficient to guarantee similar
performance measures. In addition, we show that different forms of the Pareto can yield
qualitatively different behavior in a queueing system, with respect to analogous Markovian
models. We specifically consider P/M/1 and M/P/1 queueing systems.
Keywords: Pareto Distribution; Queueing Modeling
1. Introduction
The Pareto distribution was named after Italian economist Vilfredo Pareto (1848 – 1923).
In“Cours d’economie politique professe a l’universite de Lausanne” (three volumes, 1896, 1897),
Pareto argued that the distribution of income in a country can be described by the formula (called
Pareto’s Law)
log( N ) = log( A)
+ m log( x)
,
where N is the proportion of income earners who receive incomes higher than x, and A and m are
m
constants. Pareto’s Law is equivalent to the probabilistic statement P ( X > x)
= Ax , where m < 0,
−1/
m
A > 0, and x ≥ A .
Pareto distributions play an important role in models of Internet traffic and financial insurance
claims. A key property of the Pareto distribution is that its complementary cumulative distribution
function F ( x)
P(
X x)
decays asymptotically as a power law ( , for some
constants and k). Since decays asymptotically more slowly than any exponential
function, the Pareto distribution is also said to be heavy-tailed. Throughout the literature,
researchers have used a variety of different forms of the Pareto distribution.
c
α c
≡ >
x F ( x)
→ k
F (x)
c
DRAFT

The purpose of this paper is to show that using different forms of the Pareto distribution can result
in drastically different performance measures in a queueing system. This is true even if the means
and variances between two Pareto distributions are the same. Previous research has compared the
impact of using different families of heavy-tailed distributions – specifically the Weibull, Pareto,
and lognormal distributions – in M/G/1, G/M/1, and G/M/c/c queues [5], [6], [14]. In this paper,
we show that qualitatively different performance measures can result from using different
distributions within the same Pareto family of distributions – even when the first two moments are
matched.
Table 1 summarizes various forms of the Pareto distribution that have been used by researchers.
α c
All forms follow an asymptotic power law with shape parameter ( x F ( x)
→ k ). Form (2c)
seems to be used most widely in the literature. Many textbooks use this form (e.g., [13]), as does
the popular distribution fitting package, ExpertFit. This form is used in many papers dealing with
Internet traffic (e.g., [4], [7], [15]) and insurance claims (e.g., [1], [11]). This form also directly fits
α
Pareto’s law given above, with γ and
−
A = m = −α
.
Form (3) in the table is the most general, involving three parameters – shape , scale , and
location (or shift). The other forms can be obtained from (3) by setting appropriate values for
and/or , as shown in the table. In our previous work, (e.g., see [9], [10], [16], [17]), we have used
form (1) due to its simplicity and because it allows the random variable to start at zero, instead of a
minimum threshold.
Number of
Parameters
Table 1. Forms of the Pareto Distribution
Reference
Number Parameter Restrictions F ( t)
1 F(
t)
c
= −
3 (3) α, β > 0, γ ≥ 0
2
(2a) Set γ = 0
(2b) Set β =1
(2c) Set β = γ
1 (1) Set γ = 0, β =1
⎛ β ⎞
⎜ ⎟
⎝t
+ β − γ ⎠
⎛ β ⎞
⎜ ⎟⎠
⎝t
+ β
α
γ ⎟ ⎛ 1 ⎞
⎜
⎝t
+1 − ⎠
⎛γ
⎞
⎜ ⎟⎠
⎝ t
α
⎛ 1 ⎞
⎜ ⎟
⎝t
+1⎠
α
α
α
(t ≥ γ ≥ 0)
(t ≥ 0)
(t ≥ γ ≥ 0)
(t ≥ γ > 0)
(t ≥ 0)
A key issue is that several forms of the Pareto distribution (and in particular, form (2c)) have a
minimum value strictly greater than zero. In other words, the random variable cannot take on
values between zero and some value > 0. In a queueing system, this could imply, for example, a
minimum time between successive arrivals, or a minimum service time. We show that this has
important consequences on performance measures of a queueing system. Thus, care should be
taken when using these forms.
2

In this paper, we investigate how the form of the Pareto influences congestion measures of a
queueing model. We specifically consider the P/M/1 and M/P/1 queues (where P stands for
Pareto).
2. The Pareto Distributions Compared in this Study
In this paper, we compare two forms of the Pareto distribution: the popular (2c) form and the
single parameter (1) form. The other forms reduce either to form (1) or (2c) when matching
moments or shape parameters, as discussed in the next paragraph, so it suffices just to compare
these forms. The key difference is that form (2c) has a minimum value of , while form (1) has a
minimum value of 0. Table 2 gives the means and variances of both distributions, and also gives
necessary and sufficient conditions for the first two moments to exist. (In general, for the k th
moment to exist, the shape parameter must be greater than k.)
Form
(1)
(2c)
F (x)
c
⎛ 1 ⎞
⎜ ⎟
⎝t
+1⎠
⎛γ
⎞
⎜ ⎟⎠
⎝ t
α
2
Table 2. Comparison of moments of Pareto distributions
α
Mean Variance
1
α − 1
α 2γ
α − 1
2
2
1
−
( α − 2)(
α − 1)
( α − 1
2
α 2γ
⎛ α ⎞ 2γ
− ⎜ ⎟
α − 2 ⎝ α − 1⎠
2
2
2
2
)
Conditions for Existence of
First Two Moments
t > 0, α > 2
t > γ, α2 > 2
We specifically look at two different ways of matching distributions:
Case I: Match the mean and variance,
Case II: Match the mean and shape parameter
In Case I, the coefficients of variation (CV = the standard deviation divided by the mean) are
equal, but the asymptotic tail behavior is different. For Case II, the tail behavior is similar (i.e., a
similar asymptotic power law), but the variances are not equal. The relationships between
parameters that achieve the appropriate matches are:
α 2 −1
Case I: α 2 = 1+
2 − 2 / α , γ =
(assuming > 2 and 2 > 2),
α ( α −1)
Case II: α = α
2 , γ
2
1/
α
= (assuming > 1 and 2 > 1).
For Case I, 2 is monotonically increasing as a function of , starting at 2 and bounded above by
1+ 2 . In addition, we can show that 2 when Thus, for Case I, we have 2
and also 1+ 2 > 2 > 2. This implies that the 1-parameter Pareto could have more than just the
first two moments but the 2-parameter Pareto will only have its first two moments. For Case II, is
a monotonically decreasing function of and is always less than 1.
1
Let α 2 = f ( α ) ≡ 1 +
2
2 − 2 / α . Then f ′ ( α ) = ( α
−1
2 − 2 / α ) < 1 when α > 2 . Therefore, since
and f ′ ( α ) < 1,
α > 2 , thenα = f ( α ) < α , α > 2 .
2
f ( 2)
=
2
3

Table 3 shows a numerical example of both cases. For Case I, form (2c), the minimum value of the
random variable is about 0.256. This is more than half of the mean value of 0.476. In Case II, form
(2c), the minimum value of the random variable is about 0.323, which is almost 75% of the mean
value. In contrast, form (1) in both cases has a minimum value of 0. Thus, there is a significant
difference between the two distributions with regard to their support.
3. The P/M/1 Model
Table 3. Numerical example of matching two forms of the Pareto
(or ) Mean Variance CV
Pareto Form Case I: Matching Mean and Variance
(1) 3.1 -- 0.47619 0.63904 1.6787
(2c) 2.1639754 0.2561369 0.47619 0.63904 1.6787
Case II: Matching Mean and
(1) 3.1 -- 0.47619 0.639043 1.6787
(2c) 3.1 0.3225806 0.47619 0.066498 0.5415
In this section, we investigate the performance of the P/M/1 queue using different forms of the
Pareto distribution. For notation, let be the mean service rate; let be the mean arrival rate; let
*
be the traffic intensity (with < 1). Let A ( s)
be the Laplace-Stieltjes transform (LST)
of the inter-arrival time distribution.
The usual approach to obtaining the stationary queue-wait distribution is to solve the following
equation involving A * (s) (often called the fundamental equation of the branching process):
*
z = A [ µ ( 1−
z)]
.
Let 0 ( 0,
1)
be the unique root of this equation. Then, the expected queue waiting time W q and
the queue waiting time cumulative distribution function (CDF) have the same functional
form as for the M/M/1 queue, except with r0 replacing (e.g., [8]):
∈ r
Wq (t)
r0
−µ
( 1−r0
) t
Wq = , Wq
( t)
= 1−
r0e
(t ≥ 0).
µ 1−
r )
( 0
For the case of Pareto arrivals, a closed form for A * (s) does not exist, so we use an approximation
technique called the Transform Approximation Method (TAM, [10], [16], [17]). That is, we use
TAM to approximate A * (s) and then use Newton’s method to solve for r0.
We now investigate the effects of using the 1- and 2-parameter Pareto forms (1 and 2c) for the
inter-arrival distribution. We first look at Case I, where we match the mean and variance between
the two forms. Figure 1 compares the expected queue waiting time of three G/M/1 queues, where
G is either Pareto-form (1), Pareto-form (2c), or an exponential. The mean of the exponential is
matched to the mean of the Pareto distributions. (However, the variance of the exponential differs
from those of the Pareto cases; in particular, the CV of the exponential is 1 while the CV’s for the
Pareto cases are larger than 1.) We vary the shape parameter in form (1), choose matching
parameters for form (2c), and choose a service rate so that is fixed at 0.8.
4

The 2-parameter Pareto results in lower expected queue waiting times than the 1-parameter Pareto.
This is somewhat intuitive since the 2-parameter Pareto has a guarantee that inter-arrival times are
greater than γ, while for the 1-parameter Pareto case, inter-arrival times can be near zero. Also, the
2-parameter Pareto has a fatter tail (smaller shape parameter) so there is a greater possibility for
very large inter-arrival times which would also tend to empty out the system.
8
7
6
5
Wq 4
3
2
1
0
2 3 4 5 6
α
Figure 1. Expected queue waiting time (Case I).
1-Parm.
2-Parm.
Poisson
Somewhat surprising is the fact that the waiting-time for the 2-parameter Pareto is lower than for
Poisson arrivals. This result is significant, and we look into it further. Table 4 presents the
parameter values used for both forms of the Pareto. In all cases, the CV (for each form) is greater
than 1. One would expect that this would be reflected in arrivals that occur in clusters (sometimes
referred to as bursty arrivals) and so experience longer delays than seen by the smoother Poisson
arrivals. For the 1-parameter Pareto this was true, but for the 2-parameter Pareto this was not true
(Figure 1).
This implies there is a significant difference between the peakedness factors (PF) for the two
systems. The PF is defined as the variance of the offered load divided by the mean of the offered
load, where the offered load is the number of servers busy at a point in time in an equivalent
infinite-server system – here, a P/M/∞ queue (see, e.g., [3] for a discussion of offered load and
peakedness factor). For Poisson arrivals, the PF is 1.We refer to a system with a PF greater than 1
as “bursty.”
Table 4 shows the PF for the P/M/1 queue, using the 1- and 2-parameter forms. 2 For all cases
shown in the table, the 1-parameter Pareto is bursty (PF > 1), but the 2-parameter Pareto is not
2 To calculate the PF values in the table (e.g., [19]), let Pj be the probability that an arrival sees j customers present in a
P/M/∞ queue. Let Qj be the probability that j customers are present at a random point in time. Then, j Qj = Pj-1, for j
= 1, 2 …, and Q0 can be found by normalization. The mean and variance of the offered load are the mean and variance
of the probability distribution Qj. In this calculation, we again use the TAM approximation for the LST of the Pareto
inter-arrival distribution.
5

ursty (PF < 1). Bursty arrival processes see congestion worse that Poisson and non-bursty
processes see congestion less than Poisson.
Table 4. Offered load and peakedness factor (PF) comparison (Case I).
Form Form(2c) Form(2c)
PF: PF:
2
CV Form (1) Form (2c)
2.1 2.024 0.460 4.583 1.298 0.987
2.5 2.095 0.349 2.236 1.264 0.976
2.9 2.145 0.281 1.795 1.236 0.971
3.1 2.164 0.256 1.679 1.226 0.969
3.5 2.195 0.218 1.528 1.209 0.965
3.9 2.219 0.189 1.433 1.198 0.965
4.1 2.230 0.178 1.397 1.192 0.962
4.5 2.247 0.159 1.342 1.185 0.960
4.9 2.262 0.143 1.300 1.177 0.959
5.1 2.268 0.136 1.283 1.174 0.959
5.5 2.279 0.125 1.254 1.171 0.957
5.9 2.289 0.115 1.230 1.165 0.955
The ordering of the mean queue waiting times can also be seen from the root r0 of the fundamental
equation of the branching process. Figure 2 shows r0 for both P/M/1 queues and the M/M/1 queue.
For Poisson arrivals, the root is in all cases; for the 1-parameter Pareto, the root is greater than
and for the 2-parameter Pareto the root is less than . Since W q is monotonic in r0 for 0 < r0 < 1,
arrival processes with r0 > see congestion worse than Poisson and arrival processes with r0 <
see congestion less than Poisson. Thus, although the CV of both forms of the Pareto is greater than
one, the 2-parameter Pareto is not bursty (and r0 < , while the 1-parameter is bursty (and r0 > ).
r 0
1.0
0.9
0.9
0.8
0.8
0.7
2 3 4 5 6
α
1-Parm.
2-Parm.
Poisson
Figure 2. Root of the fundamental equation of the branching process (Case I).
One of the possible reasons is that the 2-parameter Pareto has a minimal value of , which tends to
empty out the queue. In contrast, the 1-parameter Pareto can have customers arriving before
Figure 3 shows the CDF of the 1-parameter Pareto evaluated at for the ( , pairs shown in
Table 4. In all cases, there is at least about 0.47 probability that the 1-parameter Pareto has an
arrival in the interval (0, ); whereas that probability is 0 in the case of the 2-parameter Pareto.
6

F(γ )
0.56
0.54
0.52
0.50
0.48
0.46
2 3 4 5
Figure 3. CDF of 1-parameter Pareto evaluated at . (Case I)
We now consider Case II, where we match the means and shape parameters of the two forms of the
Pareto ( and = 1/ Since we are not matching the second moment, we can consider > 1
(instead of > 2). Table 5 shows sample performance measures for the P/M/1 queue for both
forms of the Pareto distribution for a range of > 1. For all cases, we choose the exponential
service rate to be such that = 0.8.
Table 5. Sample performance measures of P/M/1 queue, Case II.
Parameters CV r0 Wq
2 Mean (1) (2c) (1) (2c) (1) (2c) M/M/1
1.3 0.769 3.333 -- -- 0.9945 0.9834 479.55 157.91 10.67
1.5 0.667 2.000 -- -- 0.9730 0.9188 57.57 18.10 6.40
1.7 0.588 1.429 -- -- 0.9494 0.8546 21.43 6.72 4.57
1.9 0.526 1.111 -- -- 0.9294 0.8049 11.70 3.67 3.56
2.1 0.476 0.909 4.58 2.18 0.9132 0.7676 7.65 2.40 2.91
2.5 0.400 0.667 2.24 0.89 0.8894 0.7195 4.29 1.37 2.13
4.5 0.222 0.286 1.34 0.30 0.8443 0.6549 1.24 0.43 0.91
6.5 0.154 0.182 1.20 0.10 0.8258 0.6325 0.69 0.25 0.58
8.5 0.118 0.133 1.14 0.14 0.8231 0.6340 0.50 0.19 0.43
10.5 0.095 0.105 1.11 0.11 0.8146 0.6319 0.37 0.15 0.34
As before, the root r0 for the 1-parameter Pareto is greater than 0.8 in all cases. Hence, the
expected queue wait for the 1-parameter case is larger than for the matching M/M/1 queue. On the
other hand, the root r0 for the 2-parameter Pareto is greater than 0.8 for small values of and less
than 0.8 for large values of (the fact that the cut-over point is near 2 is coincidental). In general,
higher CV’s correspond to higher values of r0. For the 2-parameter Pareto, it is possible to have
non-bursty arrivals (r0 < ), even when the CV of the Pareto is greater than 1 (e.g., = 2.1 in the
table).
The results given so far assume that = 0.8. We now establish more general results. First, we
establish general properties of the 1-parameter Pareto (form 1) and its associated P/M/1 queue:
The CV of the 1-parameter Pareto is always greater than 1. This can be seen from Table 2
and observing that the variance is strictly greater than the square of the mean.
α
6
7

The root r 0 of the fundamental branching equation for the 1-parameter P/M/1 queue is
always greater than / , where 1/ is the mean of the Pareto distribution. In other
words, the 1-parameter P/M/1 queue always has a higher expected queue-wait than a
matching M/M/1 queue. This result follows from the theorem below which shows that the
Laplace-Stieltjes transform (LST) of the 1-parameter Pareto is larger than the LST of a
matching exponential.
Thus, the 1-parameter Pareto behaves as expected – it has a CV greater than 1, and its associated
P/M/1 queue experiences greater congestion than Poisson arrivals.
Theorem 1. Let f (t)
be the probability density function (PDF) of an exponential distribution with
mean 1 / λ . Let g(t) be the PDF of a 1-parameter Pareto distribution, also with mean 1 / λ (that is,
α = λ + 1).
Then, for any real s ≥ 0 ,
Proof. See Appendix.
∞
∫
0
e
−st
g
∞
−st
( t)
dt ≥∫ e f ( t)
dt
0
The 2-parameter Pareto has somewhat different properties:
Its CV is greater than 1 when 1 < α 2 < 1+
2 (when 1 2 2 ≤ < α , its CV is technically
infinite); its CV is less than 1 when α 2 > 1+
2 . 3
The root r 0 of the fundamental branching equation may or may not be greater than /
depending on the value of
Figure 4 summarizes properties of the 2-parameter Pareto for Cases I and II, for a range of 2 and
. The results are based on numerical computations as described earlier. In Case I, 2 can take
values between 2 and 1+ 2 (hence, the CV of the 2-parameter Pareto is always greater than 1).
The root r 0 of the fundamental branching equation may or may not be greater than , depending
on the value of . When is low enough, the P/M/1 queue experiences less congestion than the
matching M/M/1 queue (that is, r0 < ), even though the CV of the 2-parameter Pareto is greater
than 1. For high values of , the P/M/1 queue experiences more congestion than Poisson arrivals.
Case II exhibits similar behavior, except that 2 can take any value greater than 1. When
1 < α 2 < 1+
2 , the CV of the 2-parameter Pareto is greater than 1. In this region, the P/M/1
queue experiences less congestion than Poisson arrivals, for low enough values of ; it experiences
greater congestion than Poisson arrivals for high enough values of . There is also a third region,
α 2 > 1+
2 , where the CV of the 2-parameter Pareto is less than 1 and the P/M/1 queue
experiences less congestion than Poisson arrivals.
In summary, using different forms of the Pareto to characterize the inter-arrival distribution can
yield significantly different performance results. In all cases, we have observed that the 1parameter
Pareto yields higher congestion than the matching 2-parameter Pareto. Further, the 1parameter
Pareto has peaked (or bursty) arrivals for all values of > 1 (proved directly in
Theorem 1), and thus yields a larger expected queue wait than a matching M/M/1 queue. This is
2
3 α
CV > 1 2γ
⎛ α ⎞ 2γ
⇔ − ⎜ ⎟
α − 2 ⎝α
−1⎠
2
2
2
⎛ α 2γ
⎞
> ⎜ ⎟
⎝α
−1⎠
2
2
2
⇔ −α
+ α + 1 > 0 ⇔ 1− 2 < α 2 < 1+
2 .
2
2 2
.
8

intuitively expected, since the CV of the 1-parameter Pareto is greater than 1 for all values of .
On the other hand, the 2-parameter Pareto does not always have peaked arrivals (specifically,
when is low enough), and can therefore yield a smaller expected queue wait than a matching
M/M/1 queue. Surprisingly, this can occur when the CV of the 2-parameter Pareto is bigger than 1.
r 0 >
CV > 1
1
0.8
0.6
0.4
0.2
4. The M/P/1 Model
Case I Case II
r 0 <
CV > 1
0
2 2.1 2.2 2.3 2.4
2
1
0.8
0.6
0.4
0.2
0
r 0 >
CV > 1
r 0 <
CV > 1
1.25 1.5 1.75 2 2.25 2.5 2.75 3
2
r 0 <
CV < 1
Figure 4. Properties of the P/M/1 queue, using a 2-parameter Pareto.
In this section, we examine how different forms of the Pareto distribution affect the queue waiting
time in an M/P/1 queue. We consider both the mean queue wait W q and the queue wait CDF
Wq (t)
. We assume that < 1 and that the service distribution has a finite second moment. The first
condition implies that Wq (t)
exists (e.g., [2]), and both conditions imply that Wq
is finite, by the
Pollaczek-Khintchine (PK) formula (e.g., [8]):
2 2
λ / µ + λsσ
W q =
.
2(
1−
ρ)
For Pareto distributions, the second moment is finite when the shape parameter ( or is greater
than 2. Table 6 gives specific examples we use for comparison in this section.
Table 6. Parameters Used in M/P/1 Examples
Matching Form (1) Form (2c) Form (1) Form (2c)
Example Case 2 Wq Wq
A I 2.1 2.024 0.46000 0.88 40.00 40.00
B I 3.1 2.164 0.25614 1.68 3.64 3.64
C II 2.1 2.100 0.47619 0.88 40.00 10.48
D II 3.1 3.100 0.32258 1.68 3.64 1.23
We first consider examples where we match the mean and shape parameter of Pareto forms (that
is, Case II). For Case II, it is straight-forward to check that the variance of the 1-parameter form is
strictly greater than the variance of the 2-parameter form (see Table 2, when 2 = and = 1 / ).
9

This implies, via the PK formula, that the expected queue-wait is strictly greater for the 1parameter
case (since the first moments are equal). Examples C and D in the table illustrate this.
We now consider examples where we match the first two moments of the Pareto distributions (that
is, Case I). In general, when the first two moments are equal, the expected queue waiting times Wq
are the same for both Pareto distributions, since the PK formula is a function of the first two
moments of the service distribution. Examples A and B illustrate this.
Although the mean waiting times are equal, the distributions Wq (t)
are not. Figure 5 shows Wq (t)
for both forms of the Pareto using Example B. For low values of t, Wq (t)
is smaller in the case of
the 1-parameter Pareto. But, for high enough values of t, this ordering reverses. This implies that
the 2-parameter Pareto yields a higher likelihood that a customer experiences a high waiting time
and also a higher likelihood that a customer experiences a low waiting time. In other words,
although the expected values are the same for the two distributions, the 2-parameter case tends to
yield more extreme values (both high and low) for the waiting time.
1.0
0.8
0.6
Wq(t)
0.4
0.2
0.0
0 10 20 30 40
t
1.000
0.995
0.990
0.985
1 Parm.
2 Parm.
0.980
10 20 30 40 50
Figure 5. Queue waiting time CDF comparisons for 1- and 2-parameter Pareto.
This behavior can be explained as follows: For the M/G/1 queue with heavy-tailed service
distribution G (e.g., [18]),
c ρ c
Wq
( t)
~ Ge
( t)
,
1−
ρ
where G (t)
is the equilibrium distribution of the service distribution
c
e
∞
∫
t
c
c
G ( t)
= ( 1/
E(
G))
G ( u)
du ,
e
and where a(
t)
~ b(
t)
⇔ lim a(
t)
/ b(
t)
= 1.
For the 1- and 2-parameter Pareto distributions, it is
x→∞
−( α −1)
easy to check by direct integration that G ( t)
~ K1t
for form (1) and for
form (2c), where K1 and K2 are constants. Thus, in each case is power-tailed with shape (or
tail) parameter one degree less than the corresponding service shape parameter. Since 2 < when
c
−( α 2 −1)
e
G ( t)
~ K 2t
c
e
W (t)
c
q
10

2 (shown in Section 2), W (t)
is asymptotically greater for the 2-parameter Pareto case
(yielding a higher probability of a very large waiting time).
c
q
The asymptotic tail expression also illustrates an extreme difference between the waiting time
distributions in Example B. In this example, the shape parameters for the 1- and 2-parameter
distributions are = 3.1 and 2 = 2.164. Therefore, the tail parameters for the respective waiting
times are – 1 = 2.1 and 2 – 1 = 1.164. This implies that the second moment of the queue waiting
time is finite in the 1-parameter case (since – 1 > 2), but infinite in the 2-parameter case (since
2 – 1 < 2; see also [2]). Furthermore, since 2 is always less than 2 under the moment-matching
method of Case I, the second moment of the queue waiting time is always infinite in this case for
the 2-parameter Pareto.
To calculate Wq (t)
in these figures, we have used TAM and its associated numerical procedure
called the TAM Recursion Method (TRM). Specifically, we used TAM to approximate the LST of
the Pareto service time and TRM to numerically invert the LST of the waiting time (see [16], [17]).
Because the method is approximate, we have additionally verified these results with simulation.
(Specifically, for all cases considered, we obtained the 0.50, 0.80, 0.90, 0.95, and 0.99 quantiles
using simulation and they match the results here; see also [9]).
5. Conclusions
In this paper, we have investigated the impact of using different forms of the Pareto distribution on
congestion measures of queueing systems. We specifically considered 1- and 2-parameter forms of
the Pareto distribution. The 2-parameter form appears to be the most common in the literature. It
has the property that its minimum value is greater than some > 0.
For the P/M/1 model, we have observed in all examples that the 1-parameter queue yields worse
congestion than the matching 2-parameter queue. One explanation is that the minimum value of
the inter-arrival time allows the system to clear out. 4 Further, we have found that the two forms of
the Pareto distribution can yield qualitatively different measures of congestion, with respect to
analogous exponential models. This is true for both ways of matching the Pareto distributions. In
particular, the 1-parameter P/M/1 queue yields worse congestion than the analogous M/M/1 model,
while the 2-parameter P/M/1 queue may yield better congestion than the M/M/1 model. The result
is counter-intuitive for the 2-parameter model, since the CV of the Pareto inter-arrival distribution
may be greater than 1.
For the M/P/1 model, under a matching of the first two moments, both models yield the same
expected queue wait, but the 2-parameter model yields a fatter tail for the complementary CDF. In
other words, there is a higher probability of a very long waiting time. In fact, under this momentmatching
scheme, the variance of the queue wait for the 2-parameter case is always infinite, while
the variance for the 1-parameter case is finite when > 3. Under Case II, a matching of the first
moment and the shape parameter, the variance of the 1-parameter Pareto is always larger than the
4 An interesting consequence of the minimum inter-arrival time for the 2-parameter Pareto is the following: A queue
with constant service times (a P/D/1 queue) has no queueing when the fixed service time is less than the shift
parameter , regardless of the traffic load.
11

variance of the 2-parameter Pareto. This yields a larger mean queue waiting time for the 1parameter
case.
All of these results highlight the importance of selecting the distribution most appropriate to the
application or data being studied, and understanding the implications of assumptions underlying
each distribution.
Acknowledgement
This work was partially supported by the National Science Foundation Grant DMII-0140232:
Development of Procedures to Analyze Queueing Models with Heavy-Tailed Interarrival and
Service Times. Drs. Fischer and Masi would also like to thank Mitretek Systems for their support
on this work.
Appendix. Proof of Theorem 1.
Let F(t ) be the CDF of an exponential distribution with mean 1 / λ . Let G(t)
be the CDF of a 1parameter
Pareto distribution, also with mean 1 / λ (that is, α = λ + 1).
Let F (t)
and be
the complementary CDF's of the equilibrium distributions of the exponential and 1-parameter
Pareto, respectively. For the exponential distribution,
c
e G (t)
c
e
∞ ∞
c 1 c
−λu
−λt
Fe
( t)
= ∫F( u)
du = λ∫edu
= e .
E(
F)
For the 1-parameter Pareto,
c 1
Ge
( t)
=
E(
G)
t t
∞ ∞
−α
−(
α −1)
α ( 1+
u)
du = ( 1+
t)
.
c
∫G( u)
du = ( −1)
∫
t t
c
c
−λt
t −λ
−(
α −1)
We show that Ge
( t)
≥ Fe
( t)
(for t ≥ 0 ). This follows since e = ( e ) ≤ ( 1+
t)
, where the
inequality follows since e t
t
≥ 1 + and λ = α −1.
The remainder of this proof is similar to that in
2 c 2
[12]. First, observe that the PDFs can be written: f ( t)
= ( d Fe
( t)
/ dt ) / λ and
2 c 2
g(
t)
= ( d G ( t)
/ dt ) /( α −1)
. Using properties of Laplace transforms:
e
∞
− ∫
∞
2∫ c
e
0
0
∞
2 −st
c ∫e
Fe
c
t)
dt − s dFe
/ dt
c
t=
0 − Fe
( 0)
=
∞
−st
∫e
f ( t)
0
0
c
c
G ( t)
≥ Fe
( t)
c
e / dt t=
0
c
= dFe
/ dt t=
0
st
−st
c
c
( α − 1)
e g(
t)
dt = s e Ge
( t)
dt − s dGe
/ dt t=
0 − G
≥ s
( 0)
( λ dt .
c
c
The inequality follows since e , dG , and G e ( 0)
= Fe
( 0)
. Since
α = λ + 1,
the result is proved.
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