resource sharing with subexponential distributions
DESCRIPTION
Resource Sharing with Subexponential Distributions. Predrag Jelenković and Petar Momčilvoić Columbia University. Outline. Introduction Processor Sharing Queues Subexponential Distributions Heavy Tailed Distributions Motivation Background Main Result Simulation Results Conclusion. P. - PowerPoint PPT PresentationTRANSCRIPT
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