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Previously Optimization Probability Review Inventory Models Markov Decision Processes
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Agenda Queues
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Performance Measures T time in system T q waiting time (time in queue) N #customers in system N q #customers in queue system arrivals departures queue servers W = E[T] W q = E[T q ] L = E[N] L q = E[N q ] fraction of time servers are busy (utilization)
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Plain-Vanilla Queue 1 queue, 1 class of customers identical servers time between customer arrivals is independent mean rate of arrivals, rate of service constant
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What Are We Ignoring? Rush-hour effects Priority classes Balking, Reneging, Jockeying Batching Multi-step processes Queue capacity
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Parameters mean arrival rate of customers (per unit time) c servers µ mean service rate (per unit time)
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Some Relations = /( cµ) utilization c = / µ average # of busy servers W = W q + 1/ µ L = L q + c Little’s Law : L q = W q andL = W
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So What is L q ? Depends on details. LqLq M/M/1 queue ( exponential arrival times, exponential processing times, 1 server)
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Qualitatively Dependence on 1 means L q Increased variability (arrival / service times) –L q increases Pooling queues –L q decreases
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Queue Notation M / M / 1 M = ‘Markov’ exponential distribution D = ‘Deterministic’constant G = ‘General’ other distribution of the time between arrivals distribution of the processing time number of servers: 1, 2, … M/M/1 D/M/1 M/G/3 …
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Back to L q … L q =E[N q ] M/G/1 – 2 = variance of the service time –M/D/1 –M/M/1
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M/M/1 N+1 has distribution Geometric(1- ) –Var[N q ] = (1+ - 2 ) 2 /(1- ) 2 –StDev[N q ] > E[N q ] / Pooling queues decreases L q Ex. (p501) 3 clinics with 1 nurse each (M/M/1) =4/hr, µ =1/13min =87%, L q =5.6, W q =1.4hr Consolidated (M/M/3) =12/hr, µ =60/13hr, =87% L q =4.9, W q =0.4
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