Parameter Estimation Problems in Queueing and Related Stochastic Models

1. Parameter Estimation Problemsin Queueing and RelatedStochastic Models Yoni Nazarathy School of Mathematics and Physics,The University of Queensland. Australian Statistical Conference, Adelaide, July 11, 2012

2. Talk Goal • A taste of queueing theory • Parameter estimation problems in queues • Departure processes in queueing networks • Estimation through customer streams

3. Queues • Customers: • Communication packets • Production lots • Customers at the ticket box, doctor or similar • Servers: • Routers • Production machines • Tellers, etc… This morning: KayleyNazarathy aged 4, waited 55 minutes for a vaccination in QLD, she was reported by her mother as starting to be loud after 25 minutes saying, “when is it my turn, when is it my turn,….”

4. Queueing Theory • Overview: • Quantifies waiting / congestion phenomena • Mostly stochastic • More than 10,000 papers, more than 100 books • Types of research results: • Phenomena • Performance evaluation: Formulas, computational techniques, asymptotic behavior… • Design and control • Inference and estimation: • Less than 100 serious papers. 1st: “The Statistical Analysis of Congestion”, D. R. Cox, 1955 • Bib: “Parameter and State Estimation in Queues and Related Stochastic Models: A Bibliography.” Y. N. and Philip K. Pollett, on-line

5. The Single Server Queue

6. The Single Server Queue Server Buffer Number in System: … 2 3 4 5 0 1 6 Number in system at time t

7. The Single Server Queue Server Buffer Number in System: … 2 3 4 5 0 1 6 Number in system at time t Arrivals times Inter-arrival times Service requirements The sequence Determines evolution of Q(t)

8. Waiting Times The waiting time of customer n

9. Performance Measures A “key” performance measure: Often assume that the sequence is stochastic and stationary Load If, , there are often limiting distributions: Little’s result: If , The core of queueing theory deals with the distributions of W and/or Q under some assumptions on Typically take i.id. with generic RVs denoted by A, S

10. M/M/1, M/G/1, GI/G/1

11. Notation for Queues • A/S/N/K • A is the arrival process • S represents the service time distributions • N is the number of servers • K is the buffer capacity (default is infinity) M/M/1, M/G/1, GI/G/1 Assumption types on A and S: • M Poisson or exponential or memory-less • G General • GI Renewal process arrivals

12. Mean Stationary Waiting Time

13. Inference Interlude • Understanding the “congestion level” of a given situation implies finding the distribution of W • Queueing theory tells us the distribution of W, based on the distributions of A and S • To quantify the congestion level based on data we are faced with two basic general options: • Perform inference for W directly (do not use queueing theory) • Perform inference for A and S and then use queueing theory Cox 1955: “Such a prediction (i.e. using queueing theory) is of little value when we are merely interested in describing a particular situation, since it is usually no more difficult to measure (i)-(iv) (i.e. W) than to measure arrival or service times (i.e. A and S). However our practical interest is usually in the effect of modifications designed to reduce congestion, and it is often difficult or impossible to find experimentally whether proposed changes are worth while.”

14. Illustration: Queueing Model for Load on the Swinburne Super Computer (Tuan Dinh, Lachlan Andrew, Y.N.) Workload during first half of 2011

15. Interconnecting Queues:Queueing Networks

16. The Basic Model: Open Jackson NetworksJackson 1957, Goodman & Massey 1984 Problem Data: Assume: open, no “dead” nodes Traffic Equations (Stable Case): Product Form “Miracle”: If ,

17. Customer Streams

18. Variance of Outputs Asymptotic Variance Simple Examples: * Stationary stable M/M/1, D(t) is PoissonProcess( ): * Stationary M/M/1/1 with , D(t) is RenewalProcess(Erlang(2, )): Notes: * In general, for renewal process with : * The output process of most queueing systems is NOT renewal

19. Balancing ReducesAsymptotic Variance of Outputs • Theorem (Y. N. , Weiss 2008): For the M/M/1/K queue with : Theorem (Al Hanbali, Mandjes, Y. N. , Whitt 2010):For the GI/G/1 queue with ,under further conditions: • Numerically tested Conjecture (Y. N. , 2011):For the GI/G/1/K queue with : Insight about the asymptotic variance is crucial for inference of customer streams

20. One of the proofs tools: Markov Arrival Processes (MAPs) Transitions with events Transitions without events Generator Birth-Death Process Asymptotic Variance Rate

21. Inference for MAPs 2000 Survey by Tobias Ryden, “Statistical estimation for Markov-modulated Poisson processes and Markovian arrival processes” • Typical methods: • MLE using EM (expectation maximization). The CTMC state is a “hidden variable” • Moments methods (typically for structured MAPS) Proposition (stated loosely) (Y.N., Gideon Weiss): Many MAPs (those that are MMPPs) have equivalent processes that count all transitions of a CTMC (fully counting MAPs). The equivalence is in terms of the mean and variance function. On-going work (with Sophie Hautphenne): Efficient methods (improving on EM for MAPs) for processes generated by all transitions of a CTMC. The idea: “fully counting MAPs” are easier than general MAPS and may approximate customer streams for network decomposition

22. Towards a Survey of Queuing Inference and Estimation Problems

23. Dichotomy for A/S/N/K Models • Bib: “Parameter and State Estimation in Queues and Related Stochastic Models: A Bibliography.” Y. N. and Philip K. Pollett, on-line

24. Closing Remarks • Some processes are well modeled using queueuing models • Using a white or gray box analysis for such systems is often better than a black box • Estimation and inference in queues is NOT yet a highly developed field • As is with other statistical models, there is not yet a definitive answer for model selection