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Previously

Previously. Optimization Probability Review Inventory Models Markov Decision Processes. Agenda. Hwk Projects Additional Topics Finish queues Start simulation. Projects. 10% of grade Comparing optimization algorithms Diet problem Vehicle routing Safe-Ride Limos

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Previously

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

  2. Agenda • Hwk • Projects • Additional Topics • Finish queues • Start simulation

  3. Projects • 10% of grade • Comparing optimization algorithms • Diet problem • Vehicle routing • Safe-Ride • Limos • Airplane ticket pricing • Over time • Different fare classes / demands

  4. Additional Topics? • Case studies • Pricing options • Utility theory (ch 9-10) • Game theory (ch 16)

  5. service rate µ departures arrivals rate  queue servers c system Queues • M/M/s (arrivals / service / # servers)M=exponential dist., G=general • W = E[T], Wq = E[Tq] waiting time in system (queue) • L = E[N], Lq = E[Nq] #customers in system (queue) •  = /(cµ) utilization (fraction of time servers are busy)

  6. Networks of Queues (14.10) • Look at flow rates • Outflow =  when  < 1 • What is the distribution between arrivals? • Not independent, formulas fail. • Special case: all queues are M/M/s “Jackson Network” Lq just as if normal M/M/s queue

  7. Queueing Resources • M/M/s • Online http://www.usm.maine.edu/math/JPQ/ • Lpc(rho,c) function from textbook (fails on excel 2007,2008) • G/G/s • QTP (fails on mac excel) http://www.business.ualberta.ca/aingolfsson/QTP/ • G/G/s + Networks • Online http://staff.um.edu.mt/jskl1/simweb • ORMM book queue.xla at http://www.me.utexas.edu/~jensen/ORMM/frontpage/jensen.lib

  8. Distribution of Queue Length • Why care? • service guaranteesemergency response, missed flights • M/M/1 case • N+1 ~ Geometric(1-) • Otherwise, • ORMM add-in “state probabilities” P(N=k)

  9. ER Example (p508) Surgery c=3 µ=2/hr 12/hr 2/hr 1/6 5.3/hr 1/3 3.3/hr 5/6 Diagnosis c=4 µ=4/hr 10/hr 2/3 Other units

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