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Previously

Previously. Optimization Probability Review Inventory Models Markov Decision Processes. Agenda. Queues. T time in system T q waiting time (time in queue) N #customers in system N q #customers in queue. W = E[T] W q = E[T q ] L = E[N] L q = E[N q ]

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Previously

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  1. Previously • Optimization • Probability Review • Inventory Models • Markov Decision Processes

  2. Agenda • Queues

  3. T time in system Tq waiting time (time in queue) N #customers in system Nq #customers in queue W = E[T] Wq= E[Tq] L = E[N] Lq= E[Nq]  fraction of time servers are busy (utilization) departures arrivals queue servers system Performance Measures

  4. Plain-Vanilla Queue • 1 queue, • 1 class of customers • identical servers • time between customer arrivals is independent • mean rate of arrivals, rate of serviceconstant

  5. What Are We Ignoring? • Rush-hour effects • Priority classes • Balking, Reneging, Jockeying • Batching • Multi-step processes • Queue capacity

  6. Parameters •  mean arrival rate of customers (per unit time) • c servers • µ mean service rate (per unit time)

  7. Some Relations •  = /(cµ) utilization • c = /µ average # of busy servers • W = Wq + 1/µ • L = Lq + c • Little’s Law:Lq =  Wq and L =  W

  8. Lq  So What is Lq? • Depends on details. M/M/1 queue (exponential arrival times, exponential processing times, 1 server)

  9. Qualitatively • Dependence on  1 means Lq • Increased variability (arrival / service times) • Lq increases • Pooling queues • Lq decreases

  10. number of servers: 1, 2, … distribution of the time between arrivals distribution of the processing time Queue Notation M/M/1 D/M/1 M/G/3 … M / M / 1 M = ‘Markov’ exponential distribution D = ‘Deterministic’ constant G = ‘General’ other

  11. Back to Lq… • Lq=E[Nq] • M/G/1 • 2 = variance of the service time • M/D/1 • M/M/1

  12. M/M/1 • N+1 has distribution Geometric(1-) • Var[Nq] = (1+-2) 2/(1-)2 • StDev[Nq] > E[Nq] /  • Pooling queues decreases Lq Ex. (p501) 3 clinics with 1 nurse each (M/M/1) =4/hr, µ=1/13min =87%, Lq=5.6, Wq=1.4hr Consolidated (M/M/3) =12/hr, µ=60/13hr, =87% Lq=4.9, Wq=0.4

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