1. On the Variance of Output Counts of Some Queueing Systems Yoni Nazarathy Gideon Weiss SE Club, TU/e April 20, 2008

2. 2 Haifa

3. 3 Overview 1.Introduction and background 2.Results for M/M/1/K 3.Results for Re-entrant lines 4.Possible Future Work

4. 4 A Bit On Queueing Output Processes Buffer Server 01 2345 6 … State: A Single Server Queue:

5. 5 The Classic Theorem on M/M/1 Outputs: Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). Buffer Server 01 2345 6 … State: Output Process: Poisson Arrivals: M/M/1 Queue: Exponential Service times: State Process is a birth-death CTMC A Bit On Queueing Output Processes A Single Server Queue:

6. 6 PLANT OUTPUT Problem Domain: Analysis of Output Processes Desired: 1.High Throughput 2.Low Variability Model as a Queueing System

7. 7 Example 1: Stationary stable M/M/1, D(t) is PoissonProcess( ): Example 2: Stationary M/M/1/1 with. D(t) is RenewalProcess(Erlang(2, )): Variability of Outputs Asymptotic Variance Rate of Outputs For Renewal Processes: Plant

8. 8 Taken from Baris Tan, ANOR, 2000. Previous Work: Numerical

9. 9 Summary of our Results Queueing System Without LossesFinite Capacity Birth Death Queue Push Pull Queueing NetworkInfinite Supply Re-Entrant Line

10. 10 Overview 1.Introduction and background 2.Results for M/M/1/K 3.Results for Re-entrant lines 4.Possible Future Work

11. 11 The M/M/1/K Queue  Finite Buffer NOTE: output process D(t) is non-renewal. Stationary Distribution:

12. 12 What values do we expect for ? Keep and fixed.

13. 13 What values do we expect for ? Keep and fixed.

14. 14 Similar to Poisson: What values do we expect for ? Keep and fixed.

15. 15 What values do we expect for ? Keep and fixed.

16. 16 B alancing R educes A symptotic V ariance of O utputs What values do we expect for ? Keep and fixed.

17. 17 BRAVO Effect

18. 18 Theorem Part (i) Part (ii) Scope: Finite, irreducible, stationary, birth-death CTMC that represents a queue. and If Then Calculation of (Asymptotic Variance Rate of Output Process)

19. 19 Explicit Formula in case of M/M/1/K

20. 20 01 K K-1 Some (partial) intuition for M/M/1/K

21. 21 Overview 1.Introduction and background 2.Results for M/M/1/K 3.Results for Re-entrant lines 4.Possible Future Work

22. 22 Infinite Supply Re-entrant Line 4 2 1 3 5 6 7 8 10 9

23. 23 Stability Result for Re-entrant Line (Guo, Zhang, 2008 – Pre-print) QueuesResiduals is Markov with state space Theorem (Guo Zhang): X(t) is positive (Harris) recurrent. Proof follows framework of Jim Dai (1995) 2 Things to Prove: 1.Stability of fluid limit model 2.Compact sets are petite Positive Harris Recurrence: There exists, Note: We have similar result for Push-Pull Network.

24. 24 for Re-entrant lines Remember for renewal Process: Proof Method: Find diffusion limit of: It is Brownian Motion

25. 25 “Renewal Like” 4 2 1 3 5 6 7 8 10 9 1 6 8

26. 26 Overview 1.Introduction and background 2.Results for M/M/1/K 3.Results for Re-entrant lines 4.Possible Future Work

27. 27 Naive Estimation of : There is bias due to intercept: Remember: Busy Cycle Duration Number Customers Served Use “Regenerative Simulation”: Alternative: Future Work: Smith (50’s), Brown Solomon (1975) ???

28. 28 Thank You