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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. EURANDOM QPA Seminar April 4, 2008. Outline. Background A Queueing Phenomenon: BRAVO Main Theorem More on BRAVO Some open questions. The M/M/1/K Queue. m.

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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 EURANDOM QPA Seminar April 4, 2008

  2. Outline • Background • A Queueing Phenomenon: BRAVO • Main Theorem • More on BRAVO • Some open questions Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  3. 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. • Assume is stationary. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  4. 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

  5. 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 80’s) • What about ? Asymptotic Variance Rate: Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

  7. What values do we expect for ? Keep and fixed. Work in progress by Ward Whitt Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

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

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

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

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

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

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

  15. 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

  16. 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

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

  18. 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

  19. Proof Outline(of part i) Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  20. Define The Transition Counting Process - Counts the number of transitions in [0,t] Births Deaths Asymptotic Variance Rate of M(t): , MAP of M(t) is “Fully Counting” – all transitions result in counts of events. Lemma: Proof: Q.E.D Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  21. Proof Outline 1) Lemma: Look at M(t) instead of D(t). 2) Proposition: The “Fully Counting” MAP of M(t) has associated MMPP with same variance. 3) Results of Ward Whitt: An explicit expression of asymptotic variance rate of birth-death MMPP. Whitt: Book: 2001 - Stochastic Process Limits,. Paper: 1992 - Asymptotic Formulas for Markov Processes… Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  22. 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

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

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

  25. 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

  26. 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

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

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

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

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

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

  32. Why lookedat asymptotic variance rate? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  33. Push-Pull Queueing Network(Weiss, Kopzon 2002,2006) Server 2 Server 1 • Require: • Stable Queues PUSH PULL PULL PUSH Positive Recurrent Policies Exist!!! PROBABLYNOT WITH THESE POLICIES!!! Low variance of the output processes? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  34. Queue Size Realizations BURSTY OUTPUTS BURSTY OUTPUTS Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  35. Work in progress Server 1 Server 2 PUSH PULL PULL PUSH • Can we calculate ? • Diffusion Approximations of the Outputs. • Is the right measure of burstines? • Which policies are “good” in terms of burstiness? Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  36. Other Questions • Heavy Traffic Scaling, Whitt. Prove the 2/3 Property for GI/G/1/K. • BRAVO - What is going on? • M/M/1 with . • Formulas for asymptotic variance of outputs from other systems. Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

  37. In Progress by Ward Whitt Question: What about the null recurrent M/M/1( ) ? Some Guessing 1970, Iglehart and Whitt Standard independent Brownian motions. 2008, (1 week in progress by Whitt) Uniform Integrability SimulationResults Yoni Nazarathy, Gideon Weiss, University of Haifa, 2008

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

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

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