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On the Variance of Output Counts of Some Queueing Systems

On the Variance of Output Counts of Some Queueing Systems. Yoni Nazarathy Gideon Weiss. SE Club, TU/e April 20, 2008. Haifa. Overview. Introduction and background Results for M/M/1/K Results for Re-entrant lines Possible Future Work. A Bit On Queueing Output Processes.

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On the Variance of Output Counts of Some Queueing Systems

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

  2. Haifa

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

  4. A Bit On Queueing Output Processes A Single Server Queue: Server Buffer … State: 2 3 4 5 0 1 6

  5. The Classic Theorem on M/M/1 Outputs:Burkes Theorem (50’s): Output process of stationary version is Poisson ( ). A Bit On 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:

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

  7. Variability of Outputs Asymptotic Variance Rate of Outputs For Renewal Processes: Plant 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, )):

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

  9. Summary of our Results Queueing System Without Losses Finite Capacity Birth Death Queue Push Pull Queueing Network Infinite Supply Re-Entrant Line

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

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

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

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

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

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

  16. What values do we expect for ? Keep and fixed. Balancing Reduces Asymptotic Variance of Outputs

  17. BRAVO Effect

  18. 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

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

  20. K-1 K 0 1 Some (partial) intuition for M/M/1/K

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

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

  23. Stability Resultfor Re-entrant Line (Guo, Zhang, 2008 – Pre-print) Queues Residuals 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: • Stability of fluid limit model • Compact sets are petite Note: We have similar result for Push-Pull Network. Positive Harris Recurrence: There exists,

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

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

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

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

  28. Thank You

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