1. Previously Optimization Probability Review Inventory Models Markov Decision Processes

2. Agenda Queues

3. 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, …

4. W= E[T] time in system W q = E[T q ] waiting time (time in queue) L = E[N] #customers in system L q = E[N q ]#customers in queue  = /( cµ) utilization (fraction of time servers are busy) Setup system arrivals departures queue servers rate service rate µ c

5. Formulas Simple –W = W q + 1/ µ –c  average # of busy servers –L = L q + c  Little’s Law : L q = W q andL = W M/G/1 queue: (  2 = variance of the service time )

6. Qualitatively  1 means L q  L q increases with variability (of arrival / service times) L q decreases when pooling queues (a lot for M/M/1) ( or equivalently adding servers )

7. Simulation What if not M/G/1? (ex. multiple servers) What if qualitative results not enough?

8. Simulation Online M/M/s http://www.usm.maine.edu/math/JPQ/ http://www.usm.maine.edu/math/JPQ/ G/G/s http://staff.um.edu.mt/jskl1/simweb/sq1/sq1.html http://staff.um.edu.mt/jskl1/simweb/sq1/sq1.html Excel add-in (nothing for Excel 2008) From book (for M/M/s, fails for Excel 2007) QTP (fails on mac) http://www.business.ualberta.ca/aingolfsson/QTP/ ORMM book queue.xla at http://www.me.utexas.edu/~jensen/ORMM/frontpage/jensen.lib

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

10. Networks of Queues (14.10) Look at flow rates Outflow = when  < 1 Time 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