Resource Sharing with Subexponential Distributions Predrag Jelenković and Petar Momčilvoić Columbia University

Outline Introduction Motivation Main Result Simulation Results Processor Sharing Queues Subexponential Distributions Heavy Tailed Distributions Motivation Background Main Result Simulation Results Conclusion

Introduction Processor Sharing Queues. Used in modeling of resource sharing applications. J5 J4 J6 P . . . JB JA J9 J3 J7 J2 J8 J1

Introduction Processor 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.

Introduction Processor 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

Introduction Subexponential 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.

Introduction Subexponential Distributions contd. Examples.

Introduction Heavy 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.

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

Motivation Recent 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) Work load of the customer Workloads of the customers already present in the system 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)

Main Results

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

Simulation Results

Simulation Results contd.

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