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EMIS 7370

PROJECT PRESENTATION. “QUEUEING THEORY”. EMIS 7370. Yavuz SAHIN. QUEUEING THEORY. Nobody likes waiting But, reducing waiting times needs investment Queueing theory analyzes this situation It is applicable to communication networks Computer systems Supermarkets hospitals. Server 1.

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EMIS 7370

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  1. PROJECT PRESENTATION “QUEUEING THEORY” EMIS 7370 Yavuz SAHIN

  2. QUEUEING THEORY • Nobody likes waiting • But, reducing waiting times needs investment • Queueing theory analyzes this situation • It is applicable to • communication networks • Computer systems • Supermarkets • hospitals

  3. Server 1 Server 2 Server 3 Queue (waiting line) Server n

  4. SOME COMMON DISCIPLINES • FIFO: (First in, First out) • LIFO: (Last in, First out) • Random Service: the customers in the queue are served in random order • Round Robin: every customer gets a time slice. If her service is not completed, she will re-enter the queue • Priority Disciplines: every customer has a (static or dynamic) priority

  5. Kendall Notation • used for a short characterization of queueing systems • A/B/m/N-S • A : the distribution of the inter arrival time • B : the distribution of the service times • m : the number of servers • N : the maximum size of the waiting line in the finite case • S : the service discipline used

  6. M (Markov) : it is memoryless • D (Deterministic):all values from a deterministic “distribution” are constant • Erlang-k: • Hk (Hyper-k): • G (General): general distribution

  7. Markovian Systems • M/M/1 Queue: • the simplest queueing system to analyze • Arrival and service time are negative exponentially distributed (Poisson) • Mean number of customers • Mean response time • The probability of that the system have k or more customers

  8. Markovian Systems • M/M/m-Queue: • m serversand the waiting line is infinitely long • The mean number of customers • Probability formula (Erlang C)

  9. Markovian Systems • M/M/1/K -Queue: • has exponential interarrival time and service time distributions • FIFO, single server so system can hold K customers at the same time. • When a new customer comes it is blocked • pure birth-death system • suited to approximate “real systems”

  10. Markovian Systems • M/M/1/K -Queue: • The mean number of customers in the system: • N Graph for K=10

  11. Comparison of Markovian Systems • Comparison is made that each system has the probability for a new customer • The best one is M/M/m, the worst one is M/M/1/K according to mean delay times...

  12. Queueing Netwoks • Open networks:receive customers from an external source and send them to an external destination. • have a fixed population that moves between the queues but never leaves the system

  13. Summary • Queueing theory is very useful to analyse serving time or blockage rate for a lot of systems • It has a lot of combination to explain most situation based on number of servers, availability of blockage (overflow) or number of customers • It determines how much investment is needed to have better Quality of Service. Also we can see if investment is justified.

  14. References • http://www.dcs.ed.ac.uk/home/jeh/Simjava/queueing/ • http://www.win.tue.nl/~iadan/queueing.pdf • http://www.eventhelix.com/realtimemantra/CongestionControl/queueing_theory.htm • http://www.eventhelix.com/realtimemantra/CongestionControl/m_m_1_queue.htm • A Short Introduction to Queueing Theory, A.Willig, TUB, 1999 • An introduction into Queueing Theory, M. Krondorf, TUD, 2006 • Kuyruk Teorisi, Y.SUCU, M. SIMSEKLER, Trakya University, 2003

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