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A bit on Queueing Theory: M/M/1, M/G/1, GI/G/1

A bit on Queueing Theory: M/M/1, M/G/1, GI/G/1. Yoni Nazarathy * EURANDOM, Eindhoven University of Technology, The Netherlands. (As of Dec 1: Swinburne University of Technology, Melbourne). Swinburne University Seminar, Melbourne, July 29, 2010.

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A bit on Queueing Theory: M/M/1, M/G/1, GI/G/1

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  1. A bit on Queueing Theory:M/M/1, M/G/1, GI/G/1 Yoni Nazarathy* EURANDOM, Eindhoven University of Technology,The Netherlands.(As of Dec 1: Swinburne University of Technology, Melbourne) Swinburne University Seminar, Melbourne, July 29, 2010. *Supported by NWO-VIDI Grant 639.072.072 of Erjen Lefeber

  2. Outline • The term: queueing theory • The single server queue • M/M/1, M/G/1, GI/G/1 • Mean waiting time formulas • Derivation of the M/M/1 result • A glimpse at my queueing research

  3. The Term: Queueing Theory

  4. Queues • Customers: • Communication packets • Production lots • Customers at the ticket box • Servers: • Routers • Production machines • Tellers • Queueing theory: • Quantifies waiting/congestion phenomena • Abstract models of reality • Mostly stochastic • Outputs: • Performance evaluation (formulas, numbers, graphs) • Design and control (decision: what to do)

  5. Queueing Research • 1909: Erlang – telephone lines • Dedicated journal: Queueing Systems • Other key journals: • Stochastic Models • Applied Probability Journals (JAP/Advances) • Annals of Applied Probability • OR, ANOR, ORL, EJOR… • About 5 other applied probability journals • Books: Around 200 Teaching/Research • Active researchers: ~500 • Researchers that “speak the language”: ~2000 • Related terms: “Applied Probability”, “Stochastic Modeling”

  6. Queueing Theory Applied in Practice • Here and there… • Practice motivates many new queueing problems • BUT: Queueing results not so often applied • Accurate data sometimes hard to obtain • Models are often too simple for very complex realities • Simulation can do much more… • …but say much less • Insight gained from queueing theory is important

  7. The Single Server Queue

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

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

  10. The waiting time of customer n

  11. Performance Measures Assume the sequence is stochastic and stationary Load Stable when Some important performance measures: Little’s result: We can quantify L (or W) under some further assumptions on

  12. M/M/1, M/G/1, GI/G/1

  13. Notation for Queues • A/B/N/K • A is the arrival process • B is the service times • N Is the number of servers • K is the buffer capacity (default is infinity) M/M/1, M/G/1, GI/G/1 Assumptions on : • M Poisson or exponential or memory-less • G General • GI Renewal process arrivals

  14. Results for Mean Waiting Time

  15. Mean Waiting Time

  16. Derivation of the M/M/1 Result

  17. A Markov Jump Process Due to M/M (Exponential), at time t, Q(t) describes the state of the process 2 0 1 Stationary distribution Utilization

  18. The Stationary Distribution 2 0 1 Solution: Performance measures:

  19. My Research

  20. During PhD • Control and stability of Queueing Networks • Queueing Output Processes

  21. During Post-doc • BRAVO Effect(Seminar tomorrow at Melbourne University) • Sojourn Time Tail Asymptotics • Methods of Control Theory Applied to Queues • Stability of Queueing Networks • Asymptotic scaling of stochastic systems • Optical Packet Switching Applications

  22. In future… • Research area: Model selection and statistics of queueing networks (from data) • Engineering applications • More on previous subjects • Power supply networks

  23. Thanks for Listening and See you Dec 1, 2010

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