Answer Happy

2. Suppose that the output process from a queueing simulation is L(t), O SES T, the total number in queue at time t. A continuous-time output process can be converted into the sort of discrete-time process Y1, Y2... described in this chapter by first forming k =T/m batch means of size m time units: 1 jm v; - Som) ) Y = L(t) dt т for j = 1, 2,...,k. Ensemble averages of these batch means can be plotted to check for initial- condition bias. a. Show algebraically that the batch mean over 2m time units can be obtained by averaging two adjacent batch means over m time units. (Hint: This implies that we can start with batch means over rather small time intervals m and build up batch means over longer intervals without reanalyzing all of the data.] b. Simulate an M/M/ 1 queue with 1 = 1 and u = 1.25 for a run length of 4000 time units, computing batch means of the total number in queue with batch size m = 4 time units. Make replications and use a mean plot to determine an appropriate number of batches to delete. Convert this number of batches to delete into a deletion time.