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Variance of Queueing Output Processes: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

This paper explores the variance of queueing output processes in finite capacity birth-death queues. It provides a main theorem and discusses the calculation of the asymptotic variance rate. The paper also includes illustrations and animations for statisticians.

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Variance of Queueing Output Processes: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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  1. On the Variance of Queueing Output Processes • Yoni Nazarathy • Gideon Weiss • University of Haifa With Illustrations and Animations for “Non-Queueists” (Statisticians) Haifa Statistics Seminar February 20, 2007

  2. Outline • Background • A Queueing Phenomenon: BRAVO • Main Theorem • More on BRAVO • Current, parallel and future work Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  3. Some Background on Queues Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  4. A Bit On Queueing and Queueing Output Processes • A Single Server Queue: Server Buffer … State: 2 3 4 5 0 1 6 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  5. The Classic Theorem on M/M/1 Outputs:Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). A Bit On Queueing and Queueing Output Processes • A Single Server Queue: Server Buffer … State: 2 3 4 5 0 1 6 M/M/1 Queue: • Poisson Arrivals: • Exponential Service times: • State Process is a birth-death CTMC OutputProcess: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  6. The M/M/1/K Queue m “Carried load” Server FiniteBuffer • Buffer size: • Poisson arrivals: • Independent exponential service times: • 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, 2008

  7. 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 (overflows) 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, 2008

  8. 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, 2008

  9. Asymptotic Variance Rate of Outputs: What values do we expect for ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  10. Asymptotic Variance Rate of Outputs: What values do we expect for ? Work in progress by Ward Whitt Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  11. Asymptotic Variance Rate of Outputs: What values do we expect for ? Similar to Poisson: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  12. Asymptotic Variance Rate of Outputs: What values do we expect for ? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  13. 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, 2008

  14. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  15. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  16. Asymptotic Variance of M/M/1/K: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  17. Calculating • Using MAPs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  18. Represented as a MAP (Markovian Arrival Process) (Neuts, Lucantoni et. al.) Transitions with events Transitions without events Generator Birth-Death Process Asymptotic Variance Rate Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  19. 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, 2008

  20. Main Theorem Paper submitted to Queueing Systems Journal, Jan, 2008:The Asymptotic Variance Rate of the Output Process of Finite Capacity Birth-Death Queues. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  21. Scope: Finite, irreducible, stationary,birth-death CTMC that represents a queue: Main Theorem: (Asymptotic Variance Rate of Output Process) Part (i): Part (ii): Calculation of : If: and Then: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  22. Proof Outline Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  23. 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, 2008

  24. Idea of Proof of part (i): 1) Lemma: Look at M(t) instead of D(t). 2) Proposition: The “Fully Counting” MAP of M(t) has an associated MMPP with same variance. 2) Results of Ward Whitt: An 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, 2008

  25. Proposition (relating Fully Counting MAPs to MMPPs) Example: Fully Counting MAP MMPP (Markov Modulated Poisson Process) The Proposition: rate 1Poisson Process rate 1Poisson Process rate 1Poisson Process rate 1Poisson Process rate 4Poisson Process rate 4Poisson Process rate 4Poisson Process rate 4Poisson Process rate 3Poisson Process rate 3Poisson Process rate 3Poisson Process Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  26. More OnBRAVO Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  27. K-1 K 0 1 Some intuition for M/M/1/K: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  28. 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, 2008

  29. 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, 2008

  30. 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, 2008

  31. Other Phenomena at Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  32. Asymptotic Correlation Between Outputs and Overflows M/M/1/K For Large K: M Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  33. The y-intercept of the Linear Asymptote of Var(D(t)) M/M/1/K Proposition: If , then: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  34. The variance function in the short range Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  35. Yet another singularity The “kick-in” time for the BRAVO effect Departures from M/M/1/K with Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  36. How we got here…and where are we going? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  37. A Novel Queueing Network: Push-Pull System (Weiss, Kopzon 2002,2006) Server 1 Server 2 PUSH PULL PROBABLYNOT WITH THESE POLICIES!!! Low variance of the output processes? PULL PUSH Require: “Inherently Unstable” “Inherently Stable” For Both Cases,Positive Recurrent Policies Exist Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  38. Some Queue Size Realizations: BURSTY OUTPUTS BURSTY OUTPUTS BURSTY OUTPUTS Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  39. Work in progress with regards to the Push-Pull system: Server 1 Server 2 PUSH PULL • Can we calculate ? • Is asymptotic variance rate really the right measure of burstines? • Which policies are “good” in terms of burstiness? PULL PUSH Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  40. Future work (or current work by colleagues): • View BRAVO through a Heavy Traffic Perspective, using heavy traffic limits and scaling. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  41. “Fresh” in Progress work by Ward Whitt: Question: What about the null recurrent M/M/1( ) ? Some Guessing: Iglehart and Whitt 1970: Standard independent Brownian motions. 2008 (1 week in progress by Whitt): To be continued… Uniform Integrability SimulationResults Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  42. ThankYou Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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