Resource Sharing with Subexponential Distributions

Predrag Jelenković and Petar MomčilvoićColumbia University

Outline

Introduction Processor Sharing Queues Subexponential Distributions Heavy Tailed Distributions

Motivation Background

Main Result Simulation Results Conclusion

Used in modeling of resource sharing applications.

IntroductionProcessor Sharing Queues.

J1

J5J4

J3

J2

J6

J7

J8

PJ9JAJB . . .

IntroductionProcessor Sharing Queues contd. Each user will get a quantum of server time. This quantum depends on the processor

(resource) sharing schemes used. The order of service will also depend of the

resource sharing scheme used. Extensively used in modeling of computer

and communication systems.

IntroductionProcessor Sharing Queues contd. Properties

Customers arrive according to a Poisson process The service time is independent and identically distributed

random variables. The number of customers L in the queue depends only on

E[B].

The stationary remaining service time of customers are i.i.d random variables having a distribution of

IntroductionSubexponential Distributions

A subexponential distribution has the property to decay slower than any exponential distribution, i.e., its cumulative distribution function F(t) satisfies

Examples of Subexponential distributions are Lognormal and Weibull distributions.

0 a allfor tas )](1[ atetF

Introduction Subexponential Distributions contd. Examples.

Subexponential Distributions

0.0001

0.001

0.01

0.1

1

10 20 30 40 50 60 70 80 90 100

x

P[X

>x

]

Weibull Distribution Lognormal Distribution

IntroductionHeavy tail distributions A distribution is said to be heavy tailed if

This means that regardless of the distribution for small values of the random variable, if the asymptotic shape of the distribution is hyperbolic, it is heavy-tailed.

20 , xas ~][ xxXP

Motivation

This work was mainly motivated by the finding that, server access patterns and file sizes have a moderately heavy tails.

MotivationRecent and related work. Asymptotic behavior of M/G/1 PS queues

with polynomial like tails was covered by A.P. Zwart et al.

Zwart et al investigated PS queues with multiple classes of arrivals with different polynomial tails.

Main Result

Provides a relationship between the waiting time and the service distribution of a customer for subexponentially distributed service times.

It has been shown that If B IR and [[B]] < , >1 then as x

Main Result contd

Waiting time of a customer depends on three factors. (for a PS queue)

1. Work load of the customer

2. Workloads of the customers already present in the system

3. Workloads of the customers that arrive during the service

Main Result contd.

Let Bi and Vi represents the Service time and the waiting time of customer arriving at time Ti, then the waiting time of the customer who arrives at time T0=0 is given by

Where,

Waiting time caused by customers already in the system Waiting time caused by new

arrivals.

Main Result contd.

Remarks Asymptotically long waiting time of a customer cannot be

caused by customers present in the system upon arrival, i.e. the effect of is limited.

The waiting time can be become large only if it actually has a long waiting time, and not because of large service requirements of arrivals.

Main result

For a M/G/1 PS queue with service time distributions that belongs to a class of Subexponential distributions with tails heavier than will exhibit the following property as

This result only applies to service distributions with tails heavier than e-sqrt(x)

xe

])1([~][ ..

1])1([

][

xBPxVPei

xBP

xVP

x

Main ResultsHeavy Tail Distributions

1E-56

1E-53

1E-50

1E-47

1E-44

1E-41

1E-38

1E-35

1E-32

1E-29

1E-26

1E-23

1E-20

1E-17

1E-14

1E-11

1E-08

1E-05

0.01

10

X

P[X

>x

]

EX Weibull Lognormal Exponential(Exp(-x^0.6))

xe

xe 6.0

Simulation results

Parameters Service time distribution According to the results of this paper , the waiting time

distribution should be,

2.0120][ xexBP

15.0 15.0

2.0)102()( xexP

Simulation Results

Simulation Results

0.001

0.01

0.1

1

5 11 17 23 29 35 40 46 52 58 64 70 76

x

P[V

>x

]

Simulation results Numerical results

Simulation Results contd.Comparison of Different Service Time distributions

0.01

0.1

1

x

P[v

>x]

Simulation E^sqrt(x) Numerical Result E^sqrt(0.85x)

Simulation Results exp^sqrt(100x) Numerica Results exp^sqrt(85x)

Conclusion

Proposes a Asymptotic relationship between the service time distribution and the waiting time for a M/G/1 processor sharing queue with subexponential waiting time distributions.

The result is useful in the analysis of web traffic.

Intermediately regularly varying distributions A nonnegative r.v. X is called intermediately regularly

varying if