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Yoni Nazarathy Gideon Weiss University of Haifa

On the Asymptotic Variance Rate of the Output Process of Finite Capacity Queues. Yoni Nazarathy Gideon Weiss University of Haifa. Queueing Analysis, Control and Games December 20,2007 Technion, Israel. The M/M/1/K Queue. m. Server. Buffer. Poisson arrivals:

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Yoni Nazarathy Gideon Weiss University of Haifa

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  1. On the Asymptotic Variance Rate of the Output Process of Finite Capacity Queues • Yoni Nazarathy • Gideon Weiss • University of Haifa Queueing Analysis, Control and Games December 20,2007 Technion, Israel

  2. The M/M/1/K Queue m Server Buffer • Poisson arrivals: • Independent exponential service times: • Finite buffer size: • Jobs arriving to a full system are a lost. • Number in system, , is represented by a finite state irreducible birth-death CTMC: M Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  3. Traffic Processes M/M/1/K • Counts of point processes: • - The arrivals during • - The entrances into the system during • - The outputs from the system during • - The lost jobs during Poisson Renewal Renewal Non-Renewal Renewal Non-Renewal Renewal Poisson Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987. Poisson Poisson Poisson Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  4. D(t) – The Output process: • Some Attributes: (Disney, Kiessler, Farrell, de Morias 70’s) • Not a renewal process (but a Markov Renewal Process). • Expressions for . • Transition probability kernel of Markov Renewal Process. • A Markovian Arrival Process (MAP) (Neuts 1980’s). • What about ? Asymptotic Variance Rate: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  5. Asymptotic Variance Rate of Outputs: What values do we expect for ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  6. Asymptotic Variance Rate of Outputs: What values do we expect for ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  7. Asymptotic Variance Rate of Outputs: What values do we expect for ? Similar to Poisson: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  8. Asymptotic Variance Rate of Outputs: What values do we expect for ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  9. Asymptotic Variance Rate of Outputs: What values do we expect for ? Balancing Reduces Asymptotic Variance of Outputs M Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  10. Some Results Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  11. Results for M/M/1/K: Other M/M/1/K results: • Asymptotic correlation between outputs and overflows. • Formula for y-intercept of linear asymptote when . Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  12. Calculating • Using MAPs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  13. Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Transitions with events Transitions without events Generator Asymptotic Variance Rate Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  14. For , there is a nice structure to the inverse… Attempting to evaluate directly… But This doesn’t get us far… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  15. Main Theorem Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  16. Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue: Main Theorem: (Asymptotic Variance Rate of Output Process) Part (i): Part (ii): If: Calculation of : and or and Then: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  17. Proof Outline Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  18. Use the Transition Counting Process - Counts the number of transitions in the state space in [0,t] Births Deaths Asymptotic Variance Rate of M(t): Lemma: Proof: Q.E.D Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  19. Idea of Proof of part (i): 1) Look at M(t) instead of D(t). 2) The MAP of M(t) has an associated MMPP with same variance. 2) Results of Ward Whitt allow to obtain explicit expression for the asymptotic variance rate of MMPP with birth-death structure. Whitt: Book: Stochastic Process Limits, 2001. Paper: 1992 –Asymptotic Formulas for Markov Processes… Proof of part (ii), is technical. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  20. More BRAVO Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  21. K-1 K 0 1 Trying to understand what is going on…. M/M/1/K: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  22. Intuition for M/M/1/K doesn’t carry over to M/M/c/K… But BRAVO does… c=30 c=20 M/M/c/40 c=1 K=30 K=20 M/M/40/40 K=10 M/M/K/K Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  23. BRAVO also occurs in GI/G/1/K… MAP is used to evaluate Var Rate for PH/PH/1/40 queue with Erlang and Hyper-Exp Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  24. The “2/3 property” seems to hold for GI/G/1/K!!! and increase K for different CVs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

  25. ThankYou Yoni Nazarathy, Gideon Weiss, University of Haifa, 2007

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