An Equivalent Random Method with hyper-exponential service

Performance Evaluation 57 (2004) 409–422
An Equivalent Random Method with hyper-exponential service
John F. Shortle
Systems Engineering and Operations Research, George Mason University, 4400 University Drive,
MS 4A6, Fairfax, VA 22030, USA
Received 17 March 2003; received in revised form 18 February 2004
Available online 27 April 2004
Abstract
The Equivalent Random Method (ERM) has been widely used to predict blocking probabilities at overflow service stations.
The method assumes that service times follow an exponential distribution. While this may be a reasonable assumption for
voice traffic, it is not a good assumption for dial-up Internet traffic, where service times typically have a coefficient of
variation (standard deviation/mean) greater than 1. In this paper, we give a modified ERM for two-term hyper-exponential
service distributions. The method is based on an efficient algorithm to estimate the peakedness of the overflow process of
an M/H2/S/S queue. Finally, we investigate the accuracy of the modified ERM using simulation and also compare systems
with hyper-exponential service to systems with heavy-tailed service.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Equivalent Random Method; Internet traffic; Overflow queues; Queues with blocking
1. Introduction
The purpose of this paper is to investigate the accuracy of the Equivalent Random Method (ERM)
when service times are not exponential. We are particularly interested in hyper-exponential service times,
which are typical of dial-up Internet connections (e.g., [12]).
The Equivalent Random Method (ERM) is a method to size the number of servers (or trunks) along both
primary and overflow routes in a circuit-switched network. It was formulated by Wilkinson in 1956 [15]
(the same kind of method was also proposed at the same time by Bretschneider [2]) and later extended
by many others. The method has been used extensively since the 1970s to design the public-switched
telephone network (PSTN).
One drawback is that the ERM assumes call lengths have a common exponential distribution. While
this may be a reasonable assumption for voice traffic, it is not a good assumption for dial-up Internet
traffic, where service times typically have a coefficient of variation (standard deviation/mean) greater
than 1. The exponential assumption is also not valid when the network carries a mixture of traffic types
(like voice and data), each with a different mean holding time.
E-mail address: jshortle@gmu.edu (J.F. Shortle).
0166-5316/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.peva.2004.02.003

410 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
In response to these and other problems in modern telecommunications networks, several generalizations
of the ERM have been given. For example, Borst et al. [1] consider an application to wireless
communications where calls at the secondary server can be transferred back to the primary server. Fischer
et al. [5] examine a system where primary traffic is also offered to the secondary server, and they compute
the blocking level for both streams. Fredericks [7] and Labetoulle [9] consider mixtures of traffic
and estimate blocking probabilities for both streams. Schehrer [14] considers an application to ISDN
traffic where the mean holding times at the primary and secondary stations are different. Machihara [11]
considers hyper-exponential service.
This paper gives a modified ERM which works when the service distribution is a two-term hyperexponential
distribution. We develop the method by extending a peakedness calculation given by Machihara
[11]. While Machihara’s approximation works only for certain offered loads, the modified approximation
works for all loads. Using this peakedness calculation, we can construct a modified ERM for
hyper-exponential distributions. We also conduct simulation experiments to compare the hyper-exponential
service model with heavy-tailed service models.
Section 2 gives an overview of the classical ERM. Section 3 gives an efficient algorithm to approximate
overflow characteristics when the service distribution is a two-term hyper-exponential. Section 4 gives
a modified ERM based on this approximation. Section 5 uses simulation to evaluate the performance
of the modified ERM and to compare hyper-exponential service distributions with heavy-tailed service
distributions.
2. The Equivalent Random Method
This section reviews the ERM (see also [16]) and defines notation. The ERM can be used to efficiently
approximate blocking probabilities on primary and overflow trunk groups.
Fig. 1 shows a typical configuration for overflow traffic. Each call arrives at one of n primary trunk
groups, where trunk group i has Si trunks. If all primary trunks are busy, the call overflows to a common,
secondary trunk group, which has K trunks. If all secondary trunks are busy, the call is blocked. A problem
is to find the number of secondary trunks K such that the blocking probability at these trunks is b (say,
b = 0.01).
To do this, we typically characterize all traffic streams with two quantities: load a and peakedness
z. 1 Load is the mean number of trunks in service when the stream is offered to an infinite group of
trunks. Let V be the variance of the number of trunks in service when offered to an infinite group
of trunks. Then z ≡ V/a is the peakedness of this input stream. A stream with Poisson arrivals has
peakedness z = 1. Overflow streams typically have peakedness z > 1, since overflow calls tend
to come in bursts. In Fig. 1, the arriving stream to trunk group i has load and peakedness (Ei, 1),
representing Poisson arrivals. The overflow stream from trunk group i has load and peakedness
(ai,zi).
If service times are exponential, we can quickly calculate ai and zi as follows (e.g., [16]):
ai = EiB(Ei,Si), (1)
1 While these two parameters are not sufficient to determine blocking probabilities for a general arrival stream [8], these
parameters appear to characterize overflow streams quite well for exponential service.

J.F. Shortle / Performance Evaluation 57 (2004) 409–422 411
Arriving Traffic Overflow Traffic
(E 1 , 1)
(E 2 , 1)
(E n , 1)
S 1 Trunks
S2 Trunks
.
S n Trunks
(a 2 , z 2 )
(a 1, z 1)
(a n , z n )
K Trunks
Fig. 1. A typical overflow configuration: calls which are blocked at the primary trunks overflow to a common set of secondary
trunks.
(E, 1)
Similar System
S Trunks
(a, z)
K Trunks
Fig. 2. An overflow configuration stochastically similar to Fig. 1.
Ei
zi = 1 − ai +
, (2)
Si − Ei + ai + 1
where B(·, ·) is the Erlang-B blocking probability. (The Erlang-B blocking probability applies to general
service distributions and only requires Poisson arrivals. Thus, Eq. (1) holds for arbitrary service distributions,
but Eq. (2) only holds for exponential service.) If the n arrival streams are independent, then the
load and peakedness of the combined overflow stream are
n
a = ai, z = 1
n
ziai. (3)
a
i=1
i=1
The ERM seeks to find a simpler network in the form of Fig. 2 which is stochastically similar to the
original network in Fig. 1. Specifically, the ERM finds an offered load E and a trunk group size S such
that load and peakedness (a, z) of the overflow stream in Fig. 2 are close to the load and peakedness of the
combined overflow stream in Fig. 1 (Eq. (3)). Given such an E and S, the probability that a call offered
to the secondary trunk group is blocked is
B(E, K + S)
. (4)
B(E, S)
We seek to generalize this process for hyper-exponential service.
3. Peakedness calculation with hyper-exponential service
A necessary step in the ERM is computation of the peakedness of the overflow traffic streams. In
this section, we give an efficient algorithm to compute this when the service distribution is a two-term

412 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
hyper-exponential. In other words, we seek an analogous expression for zi, Eq. (2), for hyper-exponential
service. The method is based on work by Machihara [10,11] who derives an analytical expression for zi
(though this expression can be numerically challenging to evaluate). For completeness, we summarize
Machihara’s analytical result below.
Consider an M/H2/S/S queue with hyper-exponential service distribution G:
G c (x) = 1 − G(x) = k1 e −µ1t
+ k2 e −µ2t
, (5)
where k1 + k2 = 1. Let µ = (k1/µ1 + k2/µ2) −1 be the service rate. Let λ be the Poisson arrival rate, let
E = λ/µ be the offered load, and let a = EB(E, S) be the load of the overflow process. The peakedness
of the overflow process is [10]
⎛
2
z = 1 − a + ⎝1 + 2µi
⎞
2 kj ⎠ k2 i µ
· qS(µi)(IS+1 − S(µi)) −1 e T , (6)
i=1
ki
µi + µj
j=1,j=i
where IS+1 is the (S + 1) × (S + 1) identity matrix, e is the 1 × (S + 1) vector (1, 1,... ,1), q is the
1 × (S + 1) probability vector that solves qS(0) = q and qeT = 1, and S(s) is an (S + 1) × (S + 1)
matrix computed recursively as follows:
λ
0(s) = ,
s + λ
n(s) = In+1 − 1
λ Qn(s)An,n−1n−1(s)An−1,n
−1 λQn(s),
where Qn(s) is a diagonal (n + 1) × (n + 1) matrix
⎡
1
⎤
⎢ s + nµ1 + λ
⎢
Qn(s) = ⎢
⎣
1
s + (n − 1)µ1 + µ2 + λ
. ..
1
s + nµ2 + λ
⎥ ,
⎥
⎦
An,n−1 is a constant (n + 1) × n matrix
⎡
nµ1
⎢ µ2 (n − 1)µ1
⎢ 2µ2
⎢
(n − 2)µ1
An,n−1 = ⎢
. ..
⎢
⎣
and An−1,n is a constant n × (n + 1) matrix
⎡
k1 k2
⎤
⎢
An−1,n = ⎣
. .. . ..
⎥
⎦ .
k1 k2
µi
. ..
(n − 1)µ2 µ1
nµ2
⎤
⎥ ,
⎥
⎦

80
Servers (S)
0
0
J.F. Shortle / Performance Evaluation 57 (2004) 409–422 413
Blocking < 0.01%
Blocking = 0.01%
Offered Load (E)
Fig. 3. A contour plot of (zhyper − zexp)/zexp. The figure only shows values for which blocking at the primary station is greater
than 0.01%.
Fig. 3 shows the difference in peakedness z between Eq. (6) (when the service distribution is hyperexponential)
and Eq. (2) (when the service distribution is exponential). Specifically, the figure is a contour
plot of
z = zhyper − zexp
,
zexp
where zhyper is computed using Eq. (6) and zexp is computed using Eq. (2). The hyper-exponential distribution
(Eq. (5)) has parameters
1
1
= 0.28597512, = 6.60204571, k1 = 0.88695109, (7)
µ1
µ2
giving a mean of 1 and a standard deviation of 3. The exponential distribution has the same mean of 1.
The figure also shows the contour line of 0.01% blocking (that is, where B(E, S) = 0.01%). We did not
evaluate z for blocking levels less than 0.01%, since the overflow process is less relevant for such low
blocking probabilities.
We make several observations from the figure. First, the peakedness with hyper-exponential service is
always less than the peakedness with exponential service, since z < 0. Second, the gap between zhyper
and zexp increases with increasing S. Fixing the blocking level at 0.01%, we still see that the difference
between zhyper and zexp increases moving away from the origin.
One problem with the exact calculation is that it is slow when the number of servers is large. For
example, on a 1 GHz PC, the time to calculate z with S = 80 servers is about 1.5 min. While this is
not impossibly slow, we typically need to evaluate z at much larger values of S (several hundred); in
addition, the modified ERM, which we give later, requires multiple evaluations of z. Thus, we seek a
quicker method. Machihara [11] has given a quick approximation; however, it only applies for certain
combinations of E and S. In this section, we give a modified approximation which works for all values.
Machihara’s approximation is based on computing a lower bound zL and an upper bound zU for z.
These bounds come from the following queueing systems:
• Upper bound (zU): M/M/S/S → /M/∞.
• Exact (z): M/H2/S/S → /H2/∞.
• Lower bound (zL): M/M/S/S → /H2/∞.
70
-25%
-20%
-15%
-10%

414 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
More specifically, the exact value z is the peakedness of the overflow process of an M/H2/S/S queue
offered to an infinite trunk group (with hyper-exponential service at the infinite trunk group). The upper
bound zU is the peakedness of the same overflow process, but service times at both the primary and
overflow stations are exponential. This is the classical overflow peakedness, so
E
zU = 1 − a +
,
S − E + a + 1
(8)
which is Eq. (2) and with a = EB(E, S). Empirically, zU is an upper bound for z from Fig. 3 and [11].
The lower bound zL is the peakedness of the overflow process when customers at the primary station
have exponential service, but customers at the overflow station have hyper-exponential service. Eckberg
[4] shows that this is a lower bound. Machihara [11] gives that
⎛
2
zL = 1 − a + ⎝1 + 2µi
⎞
2 kj ⎠ k2 i µ ˜gS(µi)
,
1 −˜gS(µi)
(9)
where
i=1
ki
µi + µj
j=1,j=i
λ
˜gn(s) =
s + λ + nµ(1 −˜gn−1(s)) , n = 1, 2,... ,S, ˜g0(s) = λ
s + λ .
Both zL and zU can be calculated quickly. Now, there is some ρ, 0≤ ρ ≤ 1, such that
z = ρzU + (1 − ρ)zL.
More generally, we can regard ρ as a function of E and S and rewrite the above as
ρ(E, S) = z(E, S) − zL(E, S)
zU(E, S) − zL(E, S) .
When S is not too large, we can calculate ρ(E, S) exactly (since we can calculate z(E, S) exactly using
Eq. (6)). Fig. 4 shows a contour plot of ρ(E, S). Contours of ρ are roughly straight lines in (E, S) space.
(Machihara [11] found that ρ is approximately 2/3, a constant. In fact, this is only true for certain values
of E and S. InFig. 4, ρ(E, S) = 2/3 only when the blocking level B(E, S) is about 5%.)
Fig. 4. A contour plot of ρ(E, S). The figure only shows values for which blocking at the primary station is greater than 0.01%.
µi

J.F. Shortle / Performance Evaluation 57 (2004) 409–422 415
To approximate z(E, S) when S is large, we can exactly calculate z(E, S) for small values of S and
then extrapolate along these straight lines. The following algorithm does this.
Algorithm 1. Given E and S (offered load and number of servers), approximate the peakedness z(E, S)
of the overflow stream from an M/H2/S/S queue:
1. Let s2 >s1 > 0 be two positive integers, such that we can quickly calculate the exact values z(E, s1)
and z(E, s2). (In our implementation, we choose s1 = 10,s2 = 20. Evaluating z(E, s1) and z(E, s2)
takes less than a second each on a 1 GHz PC.)
2. Find E1,E2 such that
and
S − s1 = s2 − s1
(E − E1), (10)
E2 − E1
ρ(E1,s1) = ρ(E2,s2). (11)
The first equation implies that (E1,s1), (E2,s2), and (E, S) are co-linear. The second equation implies
that (E1,s1) and (E2,s2) lie on the same contour of ρ. To solve both equations, we use a nested binary
search:
(a) Do a binary search on E2 to solve Eq. (10), where E1 is found by calling (b) as a subroutine.
(b) Given E2, do a binary search on E1 to solve Eq. (11). 2
3. Let ρ ∗ = ρ(E1,s1) = ρ(E2,s2) (using E1 and E2 from the previous step). From the straight-line
assumption, we assume that ρ(E, S) ≈ ρ ∗ .
4. Return:
z = ρ ∗ zU(E, S) + (1 − ρ ∗ )zL(E, S),
where zU(E, S) is given in Eq. (8) and zL(E, S) is given in Eq. (9).
Since the contours in Fig. 4 are not completely straight, we can improve the approximation for ρ∗ in
Step 3 using one extra point, by adding the following step to the algorithm.
3 ′ . Let s3 be a positive integer greater than s2 and s1 (we choose s3 = 40); let E3 be such that (E3,s3) is
co-linear with (E1,s1) and (E2,s2). Compute ρ(E3,s3) exactly. Then adjust ρ∗ as follows:
ρ ∗ ← ρ ∗ + (ρ(E3,s3) − ρ ∗ E − E2
) .
E3 − E2
This is linearly extrapolating ρ(E, S) along the line through (E1,s1), (E2,s2), and (E3,s3), based on
the difference between ρ(E3,s3) and ρ(E2,s2).
2 To start the binary search in (b), we choose the initial lower bound for E1 to be L = E2/2. If ρ(L, s1) >ρ(E2,s2), then this
is not a valid lower bound. We divide L by 2 and repeat until ρ(L, s1) ≤ ρ(E2,s2). Similarly, to find an initial upper bound, we
start with U = 2E2 and double until ρ(U, s1) ≥ ρ(E2,s2). Then, we run the binary search until the bounds differ by less than
0.01. The binary search for (a) is similar. Let E ∗ be such that B(E ∗ ,s2) = B(E, S) (we expect (E2,s2) to have about the same
blocking level as (E, S)). The initial lower and upper bounds for E2 are L = E ∗ /2 and U = 2E ∗ , where we may need to double
or halve the values to ensure the binary search converges. Then, we run the binary search until the bounds differ by less than
0.01.

416 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
Table 1
Performance of Algorithms 1 and 1 ′
E S Blocking (%) Percent error
Exact peakd. Algorithm 1 (%) Algorithm 1 ′ (%) Machihara (%) Exp. (%)
84.06 100 1.0 3.927 0.62 0.92 3.79 24.49
131.58 150 1.0 4.832 0.85 1.00 3.03 24.77
179.74 200 1.0 5.610 0.90 0.55 2.41 24.74
120.64 100 20.0 3.496 −0.11 −0.13 −8.41 7.87
182.98 150 20.0 3.783 −0.13 0.01 −9.58 6.78
245.38 200 20.0 3.971 −0.07 0.06 −10.38 5.97
Time to calculate (s)
Exact peakd. Algorithm 1 Algorithm 1 ′ Machihara Exp.
84.06 100 1.0 150 18 0.2 ≈0 ≈0
131.58 150 1.0 711 18 0.2 ≈0 ≈0
179.74 200 1.0 2222 18 0.2 ≈0 ≈0
120.64 100 20.0 143 19 0.1 ≈0 ≈0
182.98 150 20.0 710 19 0.1 ≈0 ≈0
245.38 200 20.0 2231 19 0.1 ≈0 ≈0
Table 1 shows the performance of the algorithm for several large values of E and S. For these cases,
the algorithm has a relative error of less than 1%. It is also much faster than the exact calculation (over
100 times faster when S = 200). The table also shows Machihara’s approximation. Although it is very
quick, it is not consistently accurate. The last column shows the difference between using an exponential
approximation to the hyper-exponential (matching the mean and finding z using Eq. (2)) and the exact
value.
It is also possible to speed up Algorithm 1 if the service distribution does not change from problem
to problem. In this case, we can pre-compute the values for ρ(E, S) for S = s1,s2,s3. This gives the
following modification.
Algorithm 1 ′ . Pre-compute ρ(E, S) for S = s1,s2,s3 and E = 1, 2,... ,250 (or some other large
number). Use Algorithm 1 as described above, but use the pre-computed lookup table for all evaluations
of ρ(E, S). When E is not an integer, interpolate between the pre-computed values.
Table 1 shows that the modification is slightly less accurate than the original, but about 100 times faster
(the table ignores the pre-computation time).
4. Modified Equivalent Random Method
The following algorithm generalizes the Equivalent Random Method for hyper-exponential service.
Algorithm 2 (ERM with hyper-exponential service). Given: n, Si, and Ei (i = 1, 2,...n), where n is
the number of primary trunk groups, Si the number of trunks in group i, and Ei the offered load to group

J.F. Shortle / Performance Evaluation 57 (2004) 409–422 417
i. Determine an equivalent single trunk group system (E, S), where E is the offered load and S is the
number of trunks, such that the overflow load and peakedness from this trunk group are the same as the
combined overflow load and peakedness from the original n trunk groups. Assume that all service times
follow a common two-term hyper-exponential distribution:
1. For i = 1, 2,... ,n, compute the overflow load and peakedness:
ai = EiB(Ei,Si), zi = z(Ei,Si),
where z(Ei,Si) can be evaluated exactly using Eq. (6) or approximately using Algorithm 1, depending
on the size of Si.
2. Compute the load a and peakedness z of the combined overflow stream using Eq. (3).
3. Find (E, S) such that
z(E, S) ≈ z, EB(E, S) ≈ a,
where z(E, S) is computed using Algorithm 1:
(a) For an initial guess for (E, S), use the Rapp approximation (e.g., [13;16, p. 354]). The Rapp approximation
solves Step 3 in the classical ERM, when service times are exponential. Specifically,
let:
Ê = az + 3z(z − 1),
Ê(a + z)
S =
− a − 1 . (12)
a + z − 1
Eq. (12) is obtained by solving Eq. (2) for S and truncating to get an integer. To keep equality in
Eq. (2) (since S was truncated), let
(z − 1 + a)(S + a + 1)
E = . (13)
a + z
This approximation works well in the classical ERM so long as S ≤ E (e.g., [16]). However, we
need to do a search on S in the hyper-exponential case.
(b) Decrement S by one unit. Recalculate E using Eq. (13). Evaluate z(E, S) using Algorithm 1.
(c) Repeat the previous step until z(E, S) = z (or until two consecutive estimates for z(E, S) lie on
either side of z).
4. Return (E, S), the single trunk group system approximately equivalent to the original system.
Then, we estimate the blocking probability on K overflow trunks as
B(E, K + S)
.
B(E, S)
Fig. 5 shows a sample application of Algorithm 2. The purpose is to compare the performance of a system
with hyper-exponential service to a similar system with exponential service. Specifically, the figure is
a contour plot of b = (bhyper − bexp)/bexp, where bhyper is the blocking probability at an overflow
trunk group, where all service times are hyper-exponential (this is computed using Algorithm 2); bexp

418 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
Fig. 5. A contour plot of (bhyper − bexp)/bexp—that is, a comparison of the blocking probabilities using hyper-exponential service
and exponential service.
is the blocking probability at the same overflow trunk group, where all service times are exponential,
with matching mean (this is computed using the standard ERM as in [16, Chapter 7], using the Rapp
approximation).
The figure specifically considers the following example: there are n = 10 primary trunk groups. Each
trunk group has S servers (the y-axis) and the offered load to each trunk group is E (the x-axis). For a given
(E, S) pair, we choose the number K of overflow trunks to be such that bexp is about 1% (specifically,
as high as possible but less than 1%). The figure only shows results when the Rapp approximation is
accurate (the upper left hand corner is omitted, where blocking probabilities are very low). We use the
hyper-exponential distribution parameters in Eq. (7).
The figure shows that as the number of primary servers S increases, the difference between bhyper and
bexp also increases. In other words, for systems with lower blocking, the hyper-exponential distribution
makes more of a difference in system performance, compared with the classical model.
5. Heavy-tailed distributions
In this section, we compare systems with hyper-exponential service to those with heavy-tailed service.
Heavy-tailed distributions arise frequently in modern telecommunications networks—particularly,
those associated with Internet traffic. For example, file sizes on the World Wide Web [3], dial-up Internet
call lengths [12], and TCP connection times have all been modeled using heavy-tailed distributions
(for a survey, see [6]). These distributions have tails which decay more slowly than any exponential
function (e ax P(X>x)→∞,a > 0). Three common heavy-tailed distributions are the Weibull (with a
shape parameter less than 1), the lognormal, and the Pareto.
To compare heavy-tailed distributions with hyper-exponential distributions, we match the first three
moments using the parameters µ1, µ2, and k1 in Eq. (5). (It is not always possible to create this match,
since the moment-matching equations may give values for k1 which lie outside of [0, 1].) We consider
the following distributions:

2.6
2.5
2.4
2.3
2.2
2.1
2
J.F. Shortle / Performance Evaluation 57 (2004) 409–422 419
Weibull Lognormal Pareto
A. Hyper Apx (Theory)
B. Hyper Apx (Sim)
C. Original Dist (Sim)
Fig. 6. Comparison of peakedness of overflow process of M/G/S/S and M/H2/S/S queues, where H2 is a hyper-exponential
fittoG. The 95% confidence intervals (not shown) are all within ±0.02 of the mean values shown.
• Weibull
G c (x) = exp
x
α
− , α = 0.4113, β = 0.3244.
β
• Lognormal
2
−( ln x − µ)
g(x) = exp
/(x √ 2πσ2 ), µ =−1.1513, σ = 1.5174.
• Pareto
2σ 2
G c (x) = (1 + x/k) −α , α = 2.25, k = 1.25.
All distributions have the same first two moments: E(X) = 1 and E(X 2 ) = 10 (which gives a
coefficient of variation of 3—typical of dial-up Internet call lengths [12]). The third moments E(X 3 ) are:
312.5 (Weibull), 1000 (lognormal), and ∞ (Pareto). Table 2 gives the two-term hyper-exponential fit to
these distributions. (It is not technically possible to create a moment-matched fittothis Pareto distribution,
since it has an infinite third moment. Instead, we can create a pseudo-fit by assuming the third moment
is some large number—here, 1,000,000.)
Fig. 6 compares the peakedness of the overflow traffic from these service distributions. Specifically,
the figure shows the peakedness z(E, S) (where E = 15 and S = 20) calculated three different ways for
each distribution:
A. Using the hyper-exponential fitinTable 2 and computing the overflow peakedness (from an M/H2/S/S
queue) exactly using Eq. (6).
Table 2
Two-term hyper-exponential fit to three distributions
1/µ1 1/µ2 k1
Weibull 0.60649922 11.16516406 0.962731956
Lognormal 0.89615921 39.52050746 0.997311520
Pareto 0.99990399 41664.41676 0.999999998

420 J.F. Shortle / Performance Evaluation 57 (2004) 409–422
5.5%
5.0%
4.5%
4.0%
3.5%
3.0%
Weibull Lognormal Pareto Exponential
Classical ERM
Prediction
A. Hyper Apx (ERM)
B. Hyper Apx (Sim)
C. Original Dist (Sim)
Fig. 7. Comparison of blocking probabilities on overflow trunks.
B. Same as above, but estimating the overflow peakedness using simulation.
C. Using the original distribution and estimating the overflow peakedness (from an M/G/S/S queue)
using simulation.
The simulation involved 10 replications of about 1.5 million arrivals each, with a warm-up period of
about 15,000 arrivals each.
Comparing A and B in Fig. 6 shows that the exact and simulated values for an M/H2/S/S queue are
almost indistinguishable. In other words, simulation validates Eq. (6) from [10]. Comparing B and C,
the Weibull and its hyper-exponential fit yield nearly the same overflow peakedness. However, there is
substantial deviation for the lognormal and Pareto. This is most likely because the Weibull distribution
is the “least heavy-tailed” of the three.
Fig. 7 compares the blocking probabilities from these service distributions. We consider the following
example: there are n = 10 primary trunk groups. Each trunk group has S = 40 servers; the offered load to
each trunk group is E = 33. (This gives a 3.49% blocking probability on each trunk group.) The number
of common overflow trunks is K = 23. Fig. 7 shows the blocking probability at these overflow trunks
calculated three ways (similar to Fig. 6): (A) using the hyper-exponential fit and calculating the blocking
probability using the modified ERM in Section 4, (B) using the hyper-exponential fit, but estimating the
blocking probability using simulation, and (C) using the original service distribution and simulation.
(For exponential service, the hyper-exponential “fit” in A and B is just the exponential distribution itself.
Thus, case A corresponds to the blocking probability predicted by the classical ERM.) For each case, the
simulation involved about 49.5 million total arrivals with a warm-up period of about 100,000 arrivals.
In the figure, we observe the following: (1) The modified ERM gives a better prediction of blocking than
the original ERM. In other words, if we regard case C as “exact,” then A is closer to C than the original
ERM estimate (dashed line: case A, exponential service). (2) For all distributions, the modified ERM (case
A) under-estimates the blocking probability observed in the analogous hyper-exponential simulation (case
B). (3) The trend for all cases (A–C) is that higher peakedness yields higher blocking. That is, the Weibull
distribution (more specifically, its hyper-exponential fit) has the least overflow peakedness among the
distributions, followed by the lognormal, the Pareto, and then the exponential. The blocking probabilities
in all cases increase from left to right in the figure. In particular, exponential service yields the worst
blocking among the distributions.

J.F. Shortle / Performance Evaluation 57 (2004) 409–422 421
One possible explanation, comparing the Weibull and exponential cases, is the following: the Weibull
distribution is more volatile than the exponential distribution (with the same mean). That is, relatively
speaking, the Weibull case has many very small service times and, additionally, a few very large service
times. Hence, when all servers are busy, there is a high chance that a server will become free, since many
of the service times will finish soon. Hence, the overflow process is less bursty in the Weibull case than
in the exponential case. (This argument is not as strong comparing the three heavy-tailed distributions
with each other, since they all have the same first and second moments. This is perhaps why the Pareto
case has higher blocking than the Weibull case, even though the Pareto has a larger third moment.)
6. Conclusions
We developed an efficient algorithm to estimate the peakedness of an overflow process of an M/H2/S/S
queue (hyper-exponential service) when S is large. Numerical experiments indicated that the method is
accurate at least up to 200 servers. We then developed a modified ERM based on this approximation. In
all numerical experiments (Figs. 3 and 5), hyper-exponential service changed the system metrics more for
larger systems (more servers). Somewhat surprising was that hyper-exponential distributions exhibited
less peakedness and blocking than corresponding queues with exponential service. Generally, we expect
service with higher variance to yield worse quality-of-service metrics.
We also conducted simulation experiments to compare the hyper-exponential service model with
heavy-tailed service models. While the modified ERM was a better predictor than the classical ERM
for these experiments, the modified ERM does not appear to be a precise predictor for these heavy-tailed
systems. One exception is the peakedness calculation where the hyper-exponential and Weibull cases
matched very closely.
Acknowledgements
This work was partially funded by the ORAU Ralph E. Powe Junior Faculty Enhancement Award. The
author would like to express his thanks to ORAU for their support and to the anonymous reviewers for
their helpful comments.
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John F. Shortle was born in Santa Barbara, CA. He received a B.S. in mathematics from Harvey Mudd
College in 1992. He received a Ph.D. and M.S. in operations research from UC Berkeley in 1996. He
worked for 3 years at US WEST Advanced Technologies developing stochastic, queueing, and simulation
models to optimize networks and operations. In 2000, he won the INFORMS Daniel H. Wagner Prize for
excellence in Operations Research Practice. He is currently an assistant professor of Systems Engineering
at George Mason University. His research interests include simulation and queueing applications in
telecommunications and air transportation.