2. Suppose that the output process from a queueing simulation is L(t), 0 s ts T, the total number in queue at time t. A continuous-time output process can be converted into the sort of discrete-time process Y, Y2,... described in this chapter by first forming k -T/m batch means of size m time units: L(t) dt forj= 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 MM/ 1 queue with -1 and -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. 2. Suppose that the output process from a queueing simulation is L(t), 0 s ts T, the total number in queue at time t. A continuous-time output process can be converted into the sort of discrete-time process Y, Y2,... described in this chapter by first forming k -T/m batch means of size m time units: L(t) dt forj= 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 MM/ 1 queue with -1 and -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