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

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” L q just as if normal M/M/s queue

7. Queueing Resources M/M/s –Online http://www.usm.maine.edu/math/JPQ/ 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/ http://www.business.ualberta.ca/aingolfsson/QTP/ G/G/s + Networks –Online http://staff.um.edu.mt/jskl1/simweb 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 guarantees emergency response, missed flights M/M/1 case –N+1 ~ Geometric(1-  ) Otherwise, –ORMM add-in “state probabilities” P(N=k)

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