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Queuing theory

operations research ..waiting lines

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Queuing theory

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  1. QUEUES:INTRODUCTION LT COL JITHESH

  2. Murphy’s Law on Queues • If anything can go wrong, it will. • If you change queues, the one you have left will start to move faster than the one you are in now. • Your queue always goes the slowest. • Whatever queue you join, no matter how short it looks, it will always take the longest for you to get served.

  3. Queuing Theory • Waiting!!! • The aim of all investigations in queueing theory is to get the main performance measures of the system which are the probabilistic properties ( distribution function, density function, mean, variance ) of the following random variables: • number of customers in the system, • number of waiting customers, • utilization of the server/s, • response time of a customer, • waiting time of a customer, • idle time of the server, • busy time of a server • Agnes KrarupErlang: 1909 • http://web2.uwindsor.ca/math/hlynka/queue.html • Kendal Notation

  4. Performance measures of Queuing System • Interarrival time • Service time • Capacity of System • Structure of service • Number of servers • Service discipline • FIFO - First In First Out: who comes earlier leaves earlier • LIFO - Last Come First Out: who comes later leaves earlier • RS - Random Service: the customer is selected randomly • Priority • Queue length • Number of customers in the system • Waiting time • Response time

  5. Kendal Notation A / B / m / K / n/ D Where A: distribution function of the interarrival times, B: distribution function of the service times, m: number of servers, K: capacity of the system, the maximum number of customers in the system including the one being serviced, n: population size, number of sources of customers, D: service discipline. (David George Kendal)

  6. Markovian Process • Andrey Markov, 1906 • Stochastic Process • is a sequence of events in which the outcome at any stage depends on some probability. • Markov Process • is a stochastic process with the following properties: • The number of possible outcomes or states is finite. • The outcome at any stage depends only on the outcome of the previous stage. • The probabilities are constant over time. • Memoryless • M Exponentially distributed random variables

  7. Poisson Distribution • Denise Poisson • The Poisson distribution is a discrete probability distribution for the counts of events that occur randomly in a given interval of time (or space). • The number of cases of a disease in different towns • The number of mutations in set sized regions of a chromosome • The number of dolphin pod sightings along a flight path through a region • The number of particles emitted by a radioactive source in a given time • The number of births per hour during a given day

  8. Questions!!!!!!!!!!!

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