12 Fischer Masi Gross Shortle Tr2005

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The Telecommunications Review 2005 79

One-Parameter Pareto, Two-Parameter Pareto, Three-

Parameter Pareto: Is There a Modeling Difference?

Martin J. Fischer, Ph.D.

Denise M. Bevilacqua Masi, Ph.D.

Donald Gross, Ph.D.

John F. Shortle, Ph.D.

The Pareto distribution was first formulated in the late 1800s by the Italian economist Vilfredo Pareto. He

presented the argument that in all countries and times, the distribution of income and wealth could be

described by the formula log(N) = log(A) + mlog(x), where N is the proportion of income earners who receive

incomes higher than x, andA andm are constants. Over the years, Paretos Law has held up in empirical

studies. The Pareto distribution has recently been used as a model for file sizes on the Internet, insurance

losses, financial behavior of the stock market, and in telecommunications systems. It has various forms; here,

we consider a one-parameter form and a two-parameter form. Thus, we question if using one form of the

Pareto gives different results than using another form. In this paper, we numerically address this question bystudying queueing systems with either Pareto arrivals or service times. The two Pareto forms are studied in

detail: Case 1, both Pareto forms have equal means and variances; and Case 2, both Pareto forms have equal

mean and shape parameters. For both cases, our numerical results, substantiated by simulation studies, show

that using the two-parameter Pareto results in lower congestion than the comparable one-parameter Pareto.

IntroductionPareto distributions play an important role in queueing

models of Internet traffic and financial insurance

claims. The Pareto distribution is a power-tailed dis-

tribution which is a special case of a heavy-tailed dis-tribution. Heavy-tailed distributions have tails that go

to zero more slowly than exponential. A cumulativedistribution function, F(x), has a power tail if thereexist positive constants c and a such that

for )(1)( xFxF =

lim[ ( )] .ax

x F x c

=

That is, the tail decays geometrically in the limit.

Sometimes these distributions are also said to be fat-tailed, heavy-tailed, or long-tailed. But, the latter

terms are used to describe the larger class of distribu-

tions in which the tail probabilities satisfy

(x)Feax

x

=

lim

for every a > 0. That is, their survival functions go to0 more slowly than any exponential, but not necessar-

ily as slowly as a power-tailed function. A power-

tailed distribution is also a heavy-tailed distribution,

but not necessarily the reverse.

Application in Queueing ModelingWith the growth of the Internet and the World Wide

Web (WWW), heavy-tailed distributions have played

an important role in characterizing many of the traffic

invariants. [1, 2, 3] In addition, the application ofthese distributions has also been seen in the financial

and insurance communities. [4, 5, 6, 7]Figure 1 shows the distribution (complementary

cumulative distribution function [CCDF]) of file

transmission times on the Web. The tail of the distri-

bution is approximately linear on a log-log scale. This

corresponds to a CCDF which decays as a power law.For contrast, we have also plotted the CCDF of an

exponential distribution, which is often used to model

the holding times of voice traffic. While the voice

holding time distribution drops off quickly to zero, the

file transmission time distribution decays linearly inthis scale. From a queueing point of view, this says

that every once in a while there is a request for an ex-

tremely large size file that occupies the outgoing linkto the Web for an extraordinary length of time.

Power tails are also observed in the distribution of

insurance claims. Figure 2 shows the 30 most costly

insurance losses, worldwide, from 1970 to 1995. [8]We have fitted the distribution with both a power tail

and an exponential tail. Visually, the power tail is a

much better fit. Again, the implication is that there is


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Heavy-Tailed

Traffic: Files on

the Web

Light-Tailed

Traffic (e.g.,

Voice)

0

-0.5

-1

-1.5

-2-2.5

-3

-3.5

-4

-4.5

-5

-1.5 -1 -0.5 0 0.5 1.5 2 2.5 3 3.5

Log10(Pr(time>x))

Log10 (Transmission Time in Seconds)

Heavy-Tailed

Traffic: Files on

the Web

Light-Tailed

Traffic (e.g.,

Voice)

0

-0.5

-1

-1.5

-2-2.5

-3

-3.5

-4

-4.5

-5

-1.5 -1 -0.5 0 0.5 1.5 2 2.5 3 3.5

Log10(Pr(time>x))

Log10 (Transmission Time in Seconds)

Figure 1. Log-Log Plot of the Web [1] and Voice Transmission Time

0

5

10

15

20

25

30

$0 $4,000 $8,000 $12,000 $16,000

Rank

Loss (Millions of $US, 1992 Prices)

Exponential

Fit

Pareto

Fit Hurricane

Andrew,

1992

Northridge

Earthquake

(CA),

1994

0

5

10

15

20

25

30

$0 $4,000 $8,000 $12,000 $16,000

Rank

Loss (Millions of $US, 1992 Prices)

Exponential

Fit

Pareto

Fit Hurricane

Andrew,

1992

Northridge

Earthquake

(CA),

1994

Figure 2. The 30 Most Costly Insurance Losses, 1970 to 1995 [7]

a non-trivial probability of an extremely large insur-

ance loss. From a queueing perspective, one can show

that the probability of ruin for an insurance company

with initial cash reserve u is the same as the steady-

state queue wait probability P(Wq > u) for a G/G/1queue. The service distribution G corresponds to the

distribution of insurance losses (power-tailed in this

case) and the arrival distribution G corresponds to thearrival process of claims.

In this paper, we seek to determine how the

choice of arrival or service distributionspecifically,

how different forms of the Pareto distributionaffectsqueueing performance. Suppose one is using the

M/G/1 queue to model the previous applications. The

Pollaczek-Khintchine (P-K) formula [9] implies that

the expected delay is the same for all service distribu-tions with equal mean and variance. But, the delay

quantiles vary with the form of the service distribu-

tion. We compared the heavy-tailed LogNormal,

Pareto (single parameter), and Weibull (with shape

parameter less than 1) service distributions with equal

mean and variance. [10] The Pareto yielded thesmallest quantiles and the Weibull the largest for wait

in queue. Thus, the selection of the service distribu-

tion can make a difference in modeling the congestionof the systems of interest.

Paxson and Floyd state that heavy-tailed distribu-

tions (in particular the Pareto) can serve as models for

packet interarrival times. [2] We compare the use ofthe Pareto, LogNormal, and Weibull with the same

mean and variance as arrival distributions for loss and

delay queueing systems. [11, 12] Based on our nu-

merical investigations, we found that the blockingprobability for loss systems and the expected queueing


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delay for delay systems were again ordered with the

Pareto arrival distribution resulting in smallest meas-

ures of performance and the Weibull the largest.

Our analysis to date has shown that modelingresults can vary when one uses a Pareto, LogNormal,

or Weibull for arrival or service distributions in M/G/1

or G/M/1 queueing systems. In these analyses, we

kept the mean and variance of the distribution thesame. Here, we extend our M/G/1 and G/M/1

comparisons to various forms of the Pareto.

Forms of the Pareto DistributionThere are several forms of the Pareto distribution and

we study here how the particular form of the Pareto

might influence the results of the particular queueingmodel in question. The distribution was named after

Italian economist Vilfredo Pareto (18481923). In

Cours DEconomie Politique Professe a lUniversite

de Lausanne, Volumes I, II, and III, 18961897,

Pareto presented the argument that in all countries andtimes the distribution of income and wealth could be

described by the formula (called Paretos Law)

log(N) = log(A) + mlog(x),

where N is the proportion of income earners who re-

ceive incomes higher than x, and A and m are con-stants. Paretos Law is equivalent to the probability

statement Pr{X>x} = 1- F(x) =Axm . Note that for this

to be a valid probability distribution, m must be nega-tive since the CDF F(x), must go to 1 as x goes to in-

finity, hence Axm must go to zero. Over the years,

Paretos Law has held up in empirical studies.

There are many forms of the Pareto distribution asshown in Table 1, where F(x) is the CDF and Fc(x)

denotes the complement. Some have three parameters

(shape, scale, and location (shift); others have just two

parameters; and one form of the Pareto has only a sin-gle (scale) parameter. The two and single parameter

versions can be obtained from the three-parameter

version (the most general) by setting some parametersto specific values as shown in Table 1.

All forms above are valid Pareto distributions.

We note that Form Reference Number 2c seems to be

the most popular one in use (many texts use this form,as does the popular distribution fitting package, Ex-

pertFit). This is also the form that directly fitsParetos citation given above, where A = and m =-, and is used in many papers dealing with the appli-cations of heavy-tailed distributions to Internet traffic[1, 2, 13], and to insurance claims processing. [4, 14]

In our previous work, [3, 15, 16] we have generally

used the third form, which is easy to work with due toits simplicity, and allows the random variable to start

at zero, instead of a minimum threshold. The question

we ponder here is does the form used influence the

results of a queueing model. We attempt to answer

this question by comparing waiting times for variousforms of the Pareto in P/M/1 and M/P/1 queueing

models. Actually, the two forms we compare are the

popular Form Reference Number 2c with a minimum

x value of, to the single parameter case Form Refer-ence Number 3 where the minimum x value is 0. We

look at two cases, one matching the first two moments

of the two forms of the distribution, and the othermatching the shape parameter and the mean. Forthese situations, matching the other forms shown in

Table 1 reduce to either Form Reference Number 2c or

Form Reference Number 3, so it suffices to compare

only these two forms, i.e., one Form Reference Num-ber 2c with a shift parameter, and the other FormReference Number 3 without. Table 2 shows a nu-

merical example of the matches.Note that when matching mean and variance (line

1) in the numerical example above, the shift parameterof Form Reference Number 2c, i.e., the minimumvalue of the random variable, is .256 (more than half

the mean value), while in Form Reference Number 3,

the minimum value of the random variable is 0. When

matching alpha and the mean (line 2), the minimum

value for Form Reference Number 2c is .323, almostthree-fourths of the mean value, while again, the

minimum value for Form Reference Number 3 is 0.

Much of the available analytic results in queueing

theory rely on the input distributions (interarrival andservice times) having closed form expressions of their

Laplace transforms. The Laplace transform of the

Pareto (regardless of the particular form) does not.Thus, we employ a technique which we call the Trans-

form Approximation Method (TAM), and its associ-

ated numerical procedure called the TAM Recursion

Method (TRM), to generate the queue waiting times

for models with either Pareto arrivals (P/M/1) orPareto service (M/P/1) distributions. [3, 15, 17] We

briefly summarize the TAM and TRM procedures in

the Appendix to this paper. The next section discussesthe Pareto distributions compared in this study.

The Pareto Distributions Compared in

the StudyIn our investigation, we compare two of the forms ofthe Pareto distribution listed in Table 1; namely, oneof the two parameter cases, Form Reference Number

2c, and the single parameter case, Form Reference

Number 3. As mentioned above, Form Reference

Number 2c is the most used in the literature and unlike

Form Reference Number 3, has a minimum threshold


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Number of

Parameters

Parameter

RestrictionsF

c= 1-F(x)

Form Reference

Number

Three , > 0, 0

+x

(x ) 1

Set = 0

+x (x 0) 2a

Set =1

+ 1

1

x(x ) 2bTwo

Set =

x

(x > 0) 2c

One Set = 0, =1

+ 11

x(x 0) 3

Table 1. Various Forms of the Pareto Distribution, With Parameters = Shape,

= Scale, and = Location

One-Parameter: Form Reference Number 3 Two-Parameter: Form Reference Number 2cCase

Alpha Mean Variance CV Alpha Gamma Mean Variance CV

Case 1:Matched

Mean

and

Variance

3.1 0.47619 0.639043 1.678744 2.1639 0.256 0.47595 0.638715 1.679159

Case 2:

MatchedAlpha

and

Mean

3.1 0.47619 0.639043 1.678744 3.1 0.32258 0.47619 0.066497 0.54153

Table 2. Numerical Example of Matching Two Forms of Pareto

value of the random variable greater than zero. Ta-

ble 3 shows the complementary CDF and the means

and variances of the two cases. We note that for themean to exist, the shape parameter must be greaterthan 1, and for the variance to exist, must be greaterthan 2.

Based on Table 1 (shown earlier) and the twocases, one matching the first two moments of the two

forms of the distribution (Case 1) and the other

matching the shape parameter and the mean (Case

2), we see that if one matched the first two moments,then the one-parameter Pareto has three moments

(since = 3.1); but the two-parameter only has twomoments (since is just over 2). Therefore, if onewere considering an M/P/1 queueing system, the ex-

pected queue waiting time would be equal for both

forms of the Pareto, but the second moment of the

two-parameter Pareto would not exist. [18] This is the

first indication that the choice of the form of thePareto does result in different congestion measures,

even when the first two moments are equal. In this

example, the differences are significant in that the

second moment of the waiting time exists in one caseand does not exist in another.

Let us look at Case 1, where the means and vari-

ances are equal. Since we are equating means and

variances, we must have > 2 and 2 > 2. If is

known then we have 5.02 ))1

1(2(1

+= and

)1(

1

2

2

=

. As a function of , 2 is monotoni-cally increasing starting at 2 and bounded above by

2.414. One can also solve for as a function of2,

which results in .21

2222

+

= For 2 < 2 2.414, is less than 0. Thus, for the case where we have equal


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Form Reference

NumberFc(x) Mean Variance Conditions

3

+ 1x1 1/(-1) 2/[(-2)(-1)]-[1/(-1)]2 x>0, >2

2c 2x

2/(2-1) 2 2/(2-2)-[2 /( 2-1)]2 x> , 2>2Table 3. The Pareto Distributions Compared

means and variances, we will always have > 2 and2.414 > 2 > 2. This implies that the one-parameterPareto could have more than just the first two

moments but the two-parameter Pareto will only have

its first two moments.For Case 2, we are considering equal means and

shape parameters; we have > 1 and =1/ . So is amonotonically decreasing function of and is alwaysless than 1.

The P/M/1 ModelFor P/M/1, the usual approach for obtaining the

stationary delay-time distributions and system-size

probabilities requires solving a root-finding probleminvolving the Laplace-Stieltjes Transform (LST),

A*(s), of the interarrival-time distribution function.

The appropriate form of the problem (often called the

fundamental equation of the branching processes) is tosolve for z in

)]z1([*Az =

where 1/ is the expected service time. [9] The load,, equals /, where is the customer arrival rate, andfor the problem to have a non-trivial solution, one

must have < 1. The unique root of the fundamentalequation of the branching process, say r0 in (0,1), then

becomes the parameter of a geometric distribution for

steady-state system sizes at the embedded arrival

points. These geometric probabilities are then com-

bined with convolutions of the exponential servicedistribution to derive the stationary line-delay distri-

bution. For the case of Pareto arrivals, a closed form

forA*(s) does not exist. We use TAM forA*(s) and

then use Newtons method to solve for r0.

Once the root is found, the complete CDF of thequeue or system waiting time is easily determined. [9]

It has the same functional form as the M/M/1 queue

except with r0 replacing . The expected queue wait-ing time, Wq, is given by

)0

r1(

0r

qW

=

.

In actuality, the equivalent load on the system is

the root, r0, and not . For the case of bursty arrivals,one can show r0 isgreater than .

Here we look at using the one-parameter or two-

parameter Pareto as the customer interarrival distribu-tion. We considered two cases; in Case 1, the two

forms of the Pareto have equal mean and variances,

and in Case 2, they have equal means and shape pa-

rameters. Our analysis focuses on solving for the rootof fundamental equation of the branching process.

First we look at Case 1; in those comparison was fixed at 0.8. For this case, we have seen that theshape parameter () of the one-parameter Pareto isgreater than 2, and the shape parameter of the two-

parameter Pareto (2) is contained in the interval (2,2.414). Thus, the two-parameter Pareto does not pos-sess moments higher than its second (discussed ear-

lier).

In Figure 3, we compared the expected queuewaiting time for the one-parameter and two-parameter

Pareto, as well as with Poisson arrivals with the same

mean (but, here, the standard deviation of the Poisson

equals the mean so that the variance differs from those

of the Pareto cases where both mean and varianceswere matched). The most important thing we see is

that using the two-parameter Pareto results in lower

expected queue waiting times than does the one-pa-rameter Pareto. However, what is more surprising is

the fact that it is lower than with Poisson arrivals.

This implies that the root of the fundamental branch-

ing equation is less than when using the two-pa-rameter Pareto for the arrival distribution.

Figure 4 shows the root of the fundamental equa-tion of the branching process. For Poisson arrivals,

the root is ; but for the one-parameter Pareto, the rootis greater than and for the two-parameter Pareto the

root is less than . This result is quite significant. Letus look into it a bit more.In Figures 3 and 4, for each , the corresponding

2 and is determined so that the resulting means andvariance are equal. Table 4 presents the results used in

Figures 3 and 4, and we see that the coefficient of

variation (CV = the standard deviation divided by themean) is greater than 1 for both the one-parameter and

two-parameter Pareto. One would expect that this


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Expected Queue Waiting Time

0

2

4

6

8

10

2 2.5 3 3.5 4 4.5 5 5.5 6

Wq

1Parm 2Parm Poisson

Expected Queue Waiting Time

0

2

4

6

8

10

2 2.5 3 3.5 4 4.5 5 5.5 6

Wq

1Parm 2Parm Poisson

Figure 3. Expected Queue Waiting Time Comparisons (Case 1)

Root

0.70

0.75

0.80

0.85

0.90

0.95

2 3 4 5 6

1Parm 2Parm Poisson

r

0

Root

0.70

0.75

0.80

0.85

0.90

0.95

2 3 4 5 6

1Parm 2Parm Poisson

r

0r

0

Figure 4. Root of the Fundamental Equation of the Branching

Process (Case 1)

would be reflected in bursty arrivals; that is, arrivals

that occur in clusters and so experience longer delays

than seen by the smoother Poisson arrivals and have a

root greater than . For the one-parameter Pareto, thiswas true (as shown in Figure 4); but for the two-pa-

rameter Pareto, this was not true. That is, for the two-

parameter Pareto, we had a coefficient of variationgreater than 1, but the root and expected queue waiting

time less than 1 would get with Poisson arrivals

(where the CV = 1).This would tend to say that there is a significant

difference in the peakedness factor (PF) of the offered

load. [19] The PF is defined as the standard deviationof the offered load divided by the mean. The mean

and variance of the offered load is found from the

probability distribution of the number of customers

present (or equivalently the number of busy servers) at

a random point in time in the P/M// system. [19]

The mean equals , but the variance depends on theform of the Pareto. More specifically, to find the dis-

tribution of the number of busy servers in P/M//,

one first uses the results given in Introduction to the

Theory of Queues [20] to find the number of custom-

ers an arrival sees in a P/M/S/S with S = . This is theprobability, Pj, of an arrival seeing j customers present

in a P/M// system. The probability of j customerspresent at a random point in time, Qj, is given by j Qj =

Pj-1 for j = 1, 2 and Q0 can be found by normali-zation. [20] The mean and variance of the offered

load is the mean and variance of the probability distri-bution Qj. This procedure was used to generate the PF

columns as shown inTable 4. We again use the TAM

approximation for the Pareto arrivals. The mean andvariance in Table 4 is the actual mean and variance of

the Pareto.

Bursty arrivals have a PF greater than 1. We see

the one-parameter Pareto is bursty, but the two-pa-

rameter Pareto is not. Bursty arrival processes see

congestion worse than Poisson (r0 > ) and non-burstysee congestion less than Poisson (r0 < ). In this


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2 Mean Variance CV PF: One-Parameter ParetoPF: Two-

Parameter Pareto

2.1 2.024 0.460 0.909 17.355 4.583 1.298 0.987

2.5 2.095 0.349 0.667 2.222 2.236 1.264 0.976

2.9 2.145 0.281 0.526 0.893 1.795 1.236 0.971

3.1 2.164 0.256 0.476 0.639 1.679 1.226 0.969

3.5 2.195 0.218 0.400 0.373 1.528 1.209 0.965

3.9 2.219 0.189 0.345 0.244 1.433 1.198 0.965

4.1 2.230 0.178 0.323 0.203 1.397 1.192 0.962

4.5 2.247 0.159 0.286 0.147 1.342 1.185 0.960

4.9 2.262 0.143 0.256 0.111 1.300 1.177 0.959

5.1 2.268 0.136 0.244 0.098 1.283 1.174 0.959

5.5 2.279 0.125 0.222 0.078 1.254 1.171 0.957

5.9 2.289 0.115 0.204 0.063 1.230 1.165 0.955

Table 4. The Offered Load and PF Comparison (Case 1)

example, the CV of both forms of the Pareto is greaterthan 1, but the two-parameter Pareto is not bursty, i.e.,

its PF is < 1, while the PF of the one-parameter Pareto

is >1.One of the possible reasons is because the two-

parameter Pareto arrivals are always greater than ,whereas the one-parameter Pareto can have customers

arriving before . In the case of the two-parameterPareto, this has a tendency to clear out the queue.Figure 5 plots the CDF of the one-parameter Pareto

evaluated at for the , pairs shown in Table 4. We

see there is over a 0.45 probability that the one-parameter Pareto will have an arrival in the interval (0,

); whereas that probability is 0 in the case of a two-parameter Pareto.

As , we have 2 21+ and 0.One can numerically investigate this convergence and

see that it is slow. As gets large, the one-parameterPareto is converging to an exponential distribution, as

can be seen by looking at its CDF and taking the limit.

That is, we have F(x) = 1- (1+x)-, and as gets large,it is straight forward to show that F(x) 1- e-x 1-e-(-1) x. Correspondingly, the two-parameter Pareto is

converging to a distribution that has only two

moments and has a concentration closer and closer tozero; but is not deterministic because of lack of

moments greater than two.

For Case 2, equal means and shape parameters,

the story is different. In this case, we have

= 2and = 1/ 2. able 5 gives the root of thefundamental equation of the branching process for

Case 2, with > 2 and getting larger and = 0.8. Asthe shape parameter gets large, we see that the coeffi-

cient of variation for the one-parameter Pareto is ap-proaching one from above, and in the two-parameter

case it is going to zero. Thus, as the shape parameter

gets large, the one-parameter Pareto is again converg-ing to an exponential distribution and the two-pa-

rameter Pareto is converging to a deterministic distri-

bution. This observation is further illustrated in Figure

6. In that figure, the CDF is plotted for = 10.5. Theone-parameter Pareto CDF is compared to an expo-

nential distribution with parameter equal to 9.5; that is,

one with the same mean. We see the two CDFs are

very close.For the two-parameter Pareto, as 2 gets large, wehave = 1/2 going to zero and the corresponding CVis going to zero. For = 10.5 we see the two-

parameter CDF is approaching a deterministicdistribution with mean equal to .105. For Case 2, we

have the two-parameter CDF, 2a

2)

x

1(1)x(F

= for

x >1/2 and = 0 for x


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Probability One-Parameter Pareto Less Than

0.4600

0.4800

0.5000

0.5200

0.5400

0.5600

2 2.5 3 3.5 4 4.5 5 5.5 6

F(

)

Probability One-Parameter Pareto Less Than

0.4600

0.4800

0.5000

0.5200

0.5400

0.5600

2 2.5 3 3.5 4 4.5 5 5.5 6

F(

)

Figure 5. CDF of One-Parameter Pareto Evaluated at (Case 1)One-Parameter Pareto Two-Parameter Pareto

Mean Variance CV r0 2 Mean Variance CV r02.5 0.667 2.222 2.236 0.8894 2.5 0.400 0.667 0.356 0.894 0.7195

3.5 0.400 0.373 1.528 0.8582 3.5 0.286 0.400 0.030 0.436 0.6682

4.5 0.286 0.147 1.342 0.8443 4.5 0.222 0.286 0.007 0.298 0.6549

5.5 0.222 0.078 1.254 0.8349 5.5 0.182 0.222 0.003 0.228 0.6427

6.5 0.182 0.048 1.202 0.8258 6.5 0.154 0.182 0.001 0.185 0.6325

7.5 0.154 0.032 1.168 0.8234 7.5 0.133 0.154 0.001 0.156 0.6380

8.5 0.133 0.023 1.144 0.8231 8.5 0.118 0.133 0.000 0.135 0.6340

9.5 0.118 0.018 1.125 0.8175 9.5 0.105 0.118 0.000 0.118 0.6344

10.5 0.105 0.014 1.111 0.8146 10.5 0.095 0.105 0.000 0.106 0.6319

Table 5. Root of the Fundamental Equation of the Branching Process (Case 2)

CDF Comparisons

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2

x

F(x)

F(x):1parm F(x):exp(alpha-1) F(x):2parm F(x):D

CDF Comparisons

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2

x

F(x)


CDF Comparisons

0.0

0.2

0.4

0.6

0.8

1.0

0 0.1 0.2

x

F(x)


Figure 6. One-Parameter Pareto and Two-Parameter Pareto

CDF as Shape Parameter Gets Large (Case 2, = 10.5)


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have a PF greater than 1? To examine this issue, we

look at Case 2 with 1 < = 2 < 2; that is, their meansexist, but their second moments do not. Thus, their

coefficients of variation are not strictly defined;

however, if we assume the variances are infinite, then

the associated CV > 1 for both forms of the Pareto.

In Table 6, we examine Case 2 with set to 0.8and, for all but the last row, both the one-parameterPareto and the two-parameter Pareto do not have a

variance. For those cases, we see that the expected

queue waiting time was greater than if the arrivalprocess were Poisson. Thus, the two-parameter Pareto

has peaked or bursty arrivals when 1 < 2 < 2. Wealso see that for this example, the one-parameter

Pareto is significantly more peaked than the

corresponding two-parameter Pareto.

The last row of Table 6 presents the situation

where the one- and two-parameter Paretos do havevariances. We see the one-parameter Pareto

maintaining its PF being greater than 1, but the

expected queue waiting time for the two-parameterPareto is less than that of Poisson arrivals; indicating

a PF less than 1.

An important point shown in Table 6 is when 1 0 and so tended to clear out the queue.When used as the service time distribution in M/P/1, it

would say that the service time is always greater than > 0 and, hence, one would think the two-parameter

Pareto would introduce more congestion. Figure 7 and

Table 8 show this is not true. Since this result iscounter to what we expected, we verified our findings

using a simulation.For Cases 1a and 1b, the CVs were 4.583 and

1.679, respectively. We thought it would be

worthwhile to look at a situation where the CV waslarge, say CV = 10, to see if our numerical conclusions

still held. Again, = 0.8 and because of the largeCV, the expected queue waiting time was 198.02.Table 9 compares the quantiles obtained using TRM

and the simulation. Again we see very close

agreement. In addition, the two-parameter Pareto once

again results in less quantile congestion than the one-

parameter Pareto, even in the case of a large expectedqueue waiting time.

In summary for M/P/1, for the case where eachform of the Pareto has equal means and variance or

just equal mean and shape parameters, using the two-

parameter Pareto will result in smaller quantile

congestion. This result is not intuitive, but has been

validated against simulation results.

ConclusionsIn this paper, we numerically investigated whether

using a one- and two-parameter Pareto makes a differ-

ence in the results obtained from congestion models.Two cases were numerically investigated. In Case 1,

each form of the Pareto had equal means and vari-ances; in Case 2, each form of the Pareto had equal

means and shape parameters. We used the TAM to

find the root of the fundamental branching process in

P/M/1 and the TRM to numerically find the CDF of

the queue waiting time in M/P/1.

For both queueing systems and each case, we nu-merically found the performance measure given by the

two-parameter Pareto was less than the performance

measure using the one-parameter Pareto. For theP/M/1 queue, this result made sense as the two-pa-

rameter Pareto guaranteed no arrivals during a certain

period of time, thereby clearing out the queue. For theM/P/1, this result was not intuitive, but was substanti-

ated using simulation results. For both queues, further

investigation into the reasons for this ordering are

planned. One of the possible reasons for the ordering

is that it appears the two-parameter Pareto has a fattertail than the one-parameter Pareto.

For P/M/1 and for Case 1, the coefficient of

variation (standard deviation divided by the mean) wasgreater than one for both forms, but for the two-pa-

rameter Pareto, the PF was less than 1. The PF factor

is defined as the standard deviation of the number of

customers in a P/M// divided by the mean at a ran-

dom point in time. This result was again counter-in-tuitive, as one would expect that if a distribution had

the coefficient of variation greater than 1, then its as-

sociated PF would also be greater than 1. This result

was also verified using simulation. The net effect ofthis was that using a two-parameter Pareto in P/M/1

could yield performance measures that were less than

those obtained with comparable Poisson arrivals.


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Wq(t) Quantiles

One-Parameter Pareto Two-Parameter Pareto 2 E[Wq] Method0.5 0.8 0.9 0.95 0.5 0.8 0.9 0.95

SIM 4.67 24.84 52.03 97.11 1.83 7.82 15.29 27.330.8162 2.02 2.005 0.4913 198.02

TRM 4.68 24.79 51.66 95.99 1.82 7.81 15.26 27.21

Table 9. Quantile Comparison of TRM and Simulations for CV = 10

All of these results highlight the importance ofselecting the distribution most appropriate to the

application or data being studied, as the queueing

measures can be quite different.

AcknowledgementsThis work was partially supported by the NationalScience Foundation Grant DMII-0140232:

Development of Procedures to Analyze Queueing

Models with Heavy-Tailed Interarrival and ServiceTimes. Drs. Fischer and Masi would also like to thank

Mitretek Systems for their support of this work.

Appendix: Transform Approximation

Method (TAM) and TAM Recursion

Method (TRM)One problem with using heavy-tailed distributions is

that they do not have closed-form Laplace transforms.

This makes numerical techniques involving heavy-tailed distributions more challenging. Such techniques

generally require nesting two numerical procedures:

(a)numerically approximating the Laplace transform

of the heavy-tailed distribution; and (b)numerically

inverting the Laplace transform of the distribution ofinterest. There are multiple ways to do both (a) and

(b). In this paper, we use the TAM [17] to do (a). We

use this method for its generality and ease ofimplementation. Other methods would also work

(e.g., methods to approximate Laplace transforms of

heavy-tailed distributions. [21, 22] The results of this

paper do not depend significantly on the underlyingnumerical methods. To invert the distribution of

interest (b), we use a recursion method for the M/G/1

queue that is based on TAM called TRM. [23] Again,

other inversion methods would also work, [24] but weuse this one for ease of implementation. For

completeness, we briefly summarize these twomethods.

Transform Approximation Method

Given a CDFFwith Laplace-Stieltjes transform

=

0

sx* )x(dFe)s(B

and pointsx1, ,xN, the TAM approximation is:

=

N

1i

sxi

* .ep)s(B i

where

1N,,3,2i,2

)x(F)x(Fp 1i1ii =

= + K .

The idea is to assign to xi half of the probability

between the points to the left and right ofxi. There are

exceptions at the boundaries where the leftoverprobability near zero and infinity must be counted so

all weights add to 1:

2

)x(F)x(F1p,

2

)x(F)x(Fp N1NN

211

+=

+=

The approximation points xi are arbitrary. One

way to choose them is as follows: Choose xi so that

Fc(xi) = qi, for some constant 0 < q < 1. The idea is to

choose points that rapidly get far out in the tail of thedistribution. For the one-parameter Pareto (Case 3),

this becomes

1qx /ii = .

For the two-parameter Pareto (Case 2c), this becomes

xi= qi / .

TAM Recursion Method

The recursion method for the M/G/1 queue [21] issummarized as follows: Let T >0 be some small

number and let Fn = F(nT). Fn can be approximated

through the following recursion:

T1

1F0

, )T1/(FcTFF1N

0jjjn1nn

= ,

where cn is the sum of the pi such that n = Round(xi /

T), xi and pi are parameters from the TAM

approximation, and c0 is assumed to be 0.


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Notes and References1. Crovella, M. E., M. S. Taqqu, and A. Bestavros,

Heavy-Tailed Probability Distributions in the

World Wide Web, A Practical Guide to Heavy

Tails, R. J. Adler, R. E. Feldman, and M. S.Taqqu, Editors, Birkhauser, Boston, MA, pp. 3

27, 1998.2. Paxson, V. and S. Floyd, Wide Area Traffic:

The Failure of Poisson Modeling, IEEE/ACM

Transactions on Networking, Volume 3, pp. 226

244, June 1995.

3. Harris, C. M., P. H. Brill, and M. J. Fischer,Internet-Type Queues with Power-Tailed Interar-

rival Times and Computational Methods for Their

Analysis, INFORMS Journal on Computing,

Volume 12, pp. 261271, 2000.4. Juneja, S., P. Shahabuddin, and A. Chandra,

Simulating Heavy-Tailed Processes Using De-

layed Hazard Rate Twisting, Proceedings of the

1999 Winter Simulation Conference, P. A. Far-rington, H. B. Nembhard, D. T. Sturrock, and

G. W. Evans, Editors, Phoenix, AZ, pp. 420427,

1999.

5. Smith, R. L., Statistics of Extremes, With Appli-cations in Environment, Insurance, and Finance,

Chapter 1, Extreme Values in Finance, Telecom-

munications and the Environment, B. Finkenstadt

and H. Rootzen, Editors, Chapman and Hall/CRCPress, London, pp. 178, 2003.

6. Mayhew, S., Security Price Dynamics and Simu-lation in Financial Engineering, Proceedings ofthe 2002 Winter Simulation Conference, E. Yce-

san, C. H. Chen, J. L. Snowdon, and J. M. Char-nes, Editors, San Diego, CA, pp. 15681574,

2002.7. Embrechts, P., S. Resnick, and G. Samorodnitsky,

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Tool, North American Actuarial Journal, Vol-

ume 3, pp. 3041, 1999.8. Embrechts, P., C. Kluppelberg, and T. Mikosch,

Modeling Extremal Events, Springer, New York,

1997.9. Gross, D. and C. M. Harris, Fundamentals of

Queueing Theory, Third Edition, John Wiley and

Sons, New York, 1998.

10. Masi, D. M. B., M. J. Fischer, D. Gross, and J. F.Shortle, Using Quantile Estimates in SimulatingInternet Queues with Heavy-Tailed Service

Times, Proceedings of 5th World Multi-Confer-

ence on Systemics, Cybernetics, and Informatics,Orlando, FL, pp. 414419, July 2225, 2001.

11. Fischer, M. J., D. M. B. Masi, D. Gross, and J. F.Shortle, Loss Systems With Heavy-Tailed Arri-

vals, The Telecommunications Review, Volume

15, Mitretek Systems, 2004.

12. Fischer, M. J., D. M. B. Masi, P. H. Brill,D. Gross, and J. F. Shortle, Using the CorrectHeavy-Tailed Arrival Distribution in Modeling

Congestion Systems, The 11th International

Conference on Telecommunication Systems Man-

agement, Naval Postgraduate School, Monterey,CA, October 25, 2003.

13. Fowler, T. B., A Short Tutorial on Fractals andInternet Traffic, The Telecommunications Re-view, Volume 10, Mitretek Systems, 1999.

14. Asmussen, S. and K. Binswanger, Simulation ofRuin Probabilities for Subexponential Claims,

ASTIN Bulletin, Volume 27, pp. 297318, 1997.15. Shortle, J. F., P. H. Brill, M. J. Fischer, D. Gross,

and D. M. B. Masi, An Algorithm to Compute

the Waiting Time Distribution for the M/G/1

Queue, INFORMS Journal on Computing, Vol-ume 16(2), pp. 152161, 2004.

16. Gross, D., J. F. Shortle, M. J. Fischer, and D. M.B. Masi, Difficulties in Simulating Queues withPareto Service, Proceedings of the 2002 Winter

Simulation Conference, E. Ycesan, C. H. Chen,

J. L. Snowdon, and J. M. Charnes, Editors, San

Diego, CA, pp. 407415, 2002.

17. Fischer, M. J., D. M. B. Masi, P. H. Brill,D. Gross, and J. F. Shortle, Development of

Procedures to Analyze Queueing Models with

Heavy-Tailed Interarrival and Service TimeA

Status Report, 2003 NSF Design, Service, andManufacturing Grantees and Research

Conference Proceedings, R. G. Reddy, Editor,

Birmingham, AL, January 69, 2003.18. Cohen, J. W., The Single Server Queue, Revised,

North-Holland, Amsterdam, 1982.

19. Cooper, R. B., Introduction to Queueing Theory,Third Edition, CEEPress, Washington, DC, 1990.

20. Takacs, L.,Introduction to the Theory of Queues,Oxford University Press, New York, 1961.

21. Feldmann, A. and W. Whitt, Fitting Mixtures ofExponential to Long-Tail Distributions to Ana-lyze Network Performance Models,Performance

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24. Abate, J., G. L. Choudhury, and W. Whitt, AnIntroduction to Numerical Transform Inversion

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About the Authors

Martin J. Fischer, Ph.D., is a senior

fellow at Mitretek. His experienceincludes network design and perform-ance analysis in telecommunications.

He has published over 30 articles in

refereed journals. He received his doc-torate degree in operations research

from Southern Methodist University.Dr. Fischer may be contacted at

[email protected].

Denise M. Bevilacqua Masi, Ph.D., is asenior principal engineer at Mitretek.Her experience and research interestsinclude queueing theory and simulation

applied to telecommunications net-works. She received her doctorate de-

gree in information technology and

engineering at George Mason Univer-sity. Dr. Masi may be contacted at

[email protected].

Donald Gross, Ph.D., is a researchprofessor in the Department of Systems

Engineering and Operations Researchat George Mason University and pro-

fessor emeritus of Operations Researchat George Washington University. He

is the co-author of the well-known book,

Fundamentals of Queueing Theory. Hehas authored numerous publications in the field of queueingtheory, and is past president of INFORMS. He has received

the INFORMS Kimball Medal for Service to the OperationsResearch Profession. Dr. Gross may be contacted [email protected].

John F. Shortle, Ph.D., is an assistantprofessor of Systems Engineering at

George Mason University. His experi-ence includes developing stochastic,

queueing, and simulation models tooptimize networks and operations.

His research interests include simula-tion and queueing applications in tele-

communications and air transporta-tion. He received his doctorate degree in operationsresearch from UC Berkeley. Dr. Shortle may be contactedat [email protected].

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