1. Previously Optimization Probability Review Inventory Models Markov Decision Processes

2. Agenda Queues

3. 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)

4. 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

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 = W q + 1/ µ L = L q + c  Little’s Law : L q = W q andL = W

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

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

10. 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 …

11. Back to L q … L q =E[N q ] 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[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