Suppose that the output process from a queueing simulation is $L(t),0tT,$ the total number in queue at time $t.$ A continuous$-$time output process can be converted into the sort of discretetime process $Y_1,Y_2,$dots described in this chapter by first forming $k=\frac{T}{m}$ batch means of size $m$ time units:

$Y_j=\frac{1}{m}{\displaystyle \underset{(j-1)m}{\overset{jm}{}}}L(t)dt$

for $j=1,2,$dots,$k.$ Ensemble averages of these batch means can be plotted to check for initialcondition 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 $\frac{M}{M}\frac{?}{1}$ queue with $=1$ and $=1\mathrm{.}25$ 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.