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Resource Sharing with Subexponential Distributions

Explore the relationship between service and waiting times in processor sharing queues with subexponential distributions. Learn how heavy-tailed distributions impact resource sharing. Discover practical implications and simulation results. This work presents new insights and connections for analyzing web traffic dynamics.

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Resource Sharing with Subexponential Distributions

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

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

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

  4. 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.

  5. 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

  6. 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.

  7. Introduction Subexponential Distributions contd. • Examples.

  8. 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.

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

  10. 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.

  11. 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

  12. 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

  13. 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.

  14. 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 servicerequirements of arrivals.

  15. 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)

  16. Main Results

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

  18. Simulation Results

  19. Simulation Results contd.

  20. 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.

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

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