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Yoni Nazarathy Gideon Weiss University of Haifa

The Asymptotic Variance of the Output Process of Finite Capacity Queues. Yoni Nazarathy Gideon Weiss University of Haifa. Meeting of the euro working group on stochastic modeling. Koc university, Istanbul. June 23-25, 2008. Typical Queueing Performance Measures. Sojourn Times. Queue Sizes.

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Yoni Nazarathy Gideon Weiss University of Haifa

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  1. The Asymptotic Variance of theOutput Process of Finite Capacity Queues • Yoni Nazarathy • Gideon Weiss • University of Haifa Meeting of the euro working group on stochastic modeling.Koc university, Istanbul. June 23-25, 2008

  2. Typical Queueing Performance Measures Sojourn Times Queue Sizes Server Idleness Lost Job Rates Variability of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  3. Variability of Outputs Queueing Networks Setting Sometimes related to down-stream queue sizes Manufacturing Setting Aim for little variability over [0,T] Asymptotic Variance Rate of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  4. Previous work – Asymptotic Variance Rate of Outputs Baris Tan, Asymptotic variance rate of the output in production lines with finite buffers, Annals of Operations Research, 2000. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  5. In this work… Analyze for simple finite queueing systems e.g. M/M/1/K Results: • A surprising phenomenon • Simple formula for • Extensions Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  6. The M/M/1/K Queue m FiniteBuffer NOTE: output process D(t) is non-renewal. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  7. What values do we expect for ? Keep and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  8. What values do we expect for ? Keep and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  9. What values do we expect for ? Keep and fixed. Similar to Poisson: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  10. What values do we expect for ? Keep and fixed. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  11. What values do we expect for ? Keep and fixed. Balancing Reduces Asymptotic Variance of Outputs Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  12. Output from M/M/1/K Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  14. 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

  15. 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

  16. Main Theorem Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  18. Explicit Formula for M/M/1/K Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  19. Idea of Proof Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  20. Counts of Transitions Births Deaths Asymptotic Variance Rate of M(t): , Observe: MAP of M(t) is “Fully Counting” – all transitions result in counts of events. Lemma: Proof Outline 1) Use above Lemma: Look at M(t) instead of D(t). 2) Use Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance. 3) Use Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP. Book: 2001 - Stochastic Process Limits,. Paper: 1992 - Asymptotic Formulas for Markov Processes… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  22. Some intuition for M/M/1/K … K 0 1 K – 1 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  23. Intuition for M/M/1/K doesn’t carry over to M/M/c/K But BRAVO does M/M/1/40 c=20 c=30 K=20 K=30 M/M/10/10 M/M/40/40 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  24. BRAVO also occurs in GI/G/1/K MAP used for PH/PH/1/40 with Erlang and Hyper-Exp distributions Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  25. The “2/3 property” • GI/G/1/K • SCV of arrival = SCV of service Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  26. M/M/1+ Impatient Customers - Simulation Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  27. Thank You Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  28. Extensions Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  30. Traffic Processes • Counts of point processes: • - Arrivals during • - Entrances • - Outputs • - Lost jobs M/M/1/K Poisson Renewal Renewal Renewal Non-Renewal Renewal Non-Renewal Poisson Poisson Poisson Poisson Book: Traffic Processes in Queueing Networks, Disney, Kiessler 1987. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  31. Push-Pull Queueing Network(Weiss, Kopzon 2002,2006) Server 2 Server 1 • Require: • Stable Queues PUSH PULL PULL PUSH Positive Recurrent Policies Exist!!! Asymptotic Variance Rate of the output processes? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

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

  34. The y-intercept of the Linear Asymptote M/M/1/K Proposition: For , Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  36. Lemma: Proof: Q.E.D Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  37. Fully Counting MAP and associated MMPP Example: Transitions with events Transitions without events Fully Counting MAP MMPP(Markov Modulated Poisson Process) Proposition rate 2 rate 2 rate 2 rate 2 rate 4 rate 4 rate 4 rate 4Poisson Process rate 3 rate 3 rate 3 Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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