Question $2$: Consider the simulation model you did for Joines Chapter $2($The ice cream store$).$ At time zero, zero customers are in the system and the servers are idle. Suppose, the management is interested in the steady$-$state behavior of the store during an operating day with $8$ hours in terms of the following performance measure: average time a customer stays in the store.

Suppose $Y_{r,j}($each element in the following table$)$ denotes the average time customers spend in the store in replication $r$ from time $0$ up to the end of $j^{th}$ batch of $20$ minutes. For example, $Y_{5,4}$ is the average time customers spend in the store in the first $420=80$ minutes in the $5^{th}$ replication.

Suppose you have run the model with $R=5$ replications and $n=10$ batches, and have collected the following data:

$\backslash$table$[[$Replication$,$Observations$],[2,0\mathrm{.}120,0\mathrm{.}145,0\mathrm{.}164,0\mathrm{.}175,0\mathrm{.}175,0\mathrm{.}169,0\mathrm{.}164,0\mathrm{.}159,0\mathrm{.}156,0\mathrm{.}153],[3,0\mathrm{.}128,0\mathrm{.}127,0\mathrm{.}126,0\mathrm{.}128,0\mathrm{.}132,0\mathrm{.}131,0\mathrm{.}132,0\mathrm{.}133,0\mathrm{.}139,0\mathrm{.}149],[5,0\mathrm{.}132,0\mathrm{.}150,0\mathrm{.}156,0\mathrm{.}159,0\mathrm{.}159,0\mathrm{.}158,0\mathrm{.}155,0\mathrm{.}152,0\mathrm{.}151,0\mathrm{.}150]]$

a$.$ Calculate the Ensemble averages.

b$.$ Plot the Ensemble averages versus time in hours ranging from $0$ to $200$ minutes.

c$.$ Using the Ensemble averages method, what do you suggest as $d($the first $d$ observations should be deleted for steady states calculation$)?$ Why? How long should be the initialization period?

d$.$ Find a $95\%$ confidence interval for the average time in system. Report your answer as lower and upper bounds with three decimal places.