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

2. Agenda Hwk Additional Topics Simulation

3. Additional Topics?

4. 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

5. What can we calculate? W and L Utilization Distribution of L (for M/M/s) How often T q >c (for M/M/s) Networks (for M/M/s) Staffing necessary. Cost trade-offs.

6. What can we not do? Distribution of T q Networks (for G/G/s) Rush-hour effects Priority classes Balking, Reneging, Jockeying Batching Queue capacity

7. Simulation (Ch 15) Interested in quantity X (it is random) Run simulation to get realizations of X: –X 1, X 2, X 3, …, X n Evaluate output: –look at averageE[X] ≈ AVERAGE(X 1,…,X n ) –standard deviation  [X] ≈ STDEV(X 1,…,X n ) –distribution of realizations

8. Examples What is X? Time from check-in to boarding Rush-hour effects Probability waiting time > 1 hr

9. Agenda Confidence intervals –for output evaluation Generating realizations

10. Independent Case Suppose X 1, X 2, X 3, …, X n are independent, –a n = AVERAGE(X 1,…,X n ) –s n = STDEV(X 1,…,X n ) Central limit theorem: –a n - E[X] ≈ normal distribution –mean 0, standard deviation s n / n 1/2 Confidence interval for E[X] –P( E[X] < a n +y) ≈ P( N(0,s n 2 /n) < y) –with probability p, E[X] not in [a n -y, a n +y] –y = - NORMINV (p/2, 0, s n / n 1/2 )

11. Example X = # customers in line at lunch place at noon Data X 1,…,X 20 from last month (n=20) a n =5.5, s n =2 Want 90% confidence interval for E[X]: –y = -NORMINV(5%,0, 2/√20) ≈ 0.7 –E[X] in [4.8,6.2] with 90% probability

12. Rare Event Estimate the probability that all the ambulances are busy when a call arrives X = 1 if ambulance not available when call arrives, 0 if available Suppose n=1000, a n =7/1000 s n = 0.08 90% confidence interval is [0.3%,1.1%] What n is needed for 10% error bound? –10% error  y=0.1% (factor 4 smaller)  16x samples needed  n = 16000