MS $3053-$ Problem Set #$11$

Due Date: May $1,2024$

#$1$

For this question, we return to Pat's, one of the Philadelphia cheesesteak restaurants mentioned in Problem Set #$8$ about market share analysis. Cheesesteak service at Pat's is limited to a single walk$-$up window: as a customer, you get into a single$-$file line and, upon reaching the service window, you order your sandwich, pay for it$,$ receive it$,$ and leave $[]$ Suppose, on average, that $42$ customers arrive per hour, while the average service time per customer is $48$ seconds. Assume that the single$-$channel $(\frac{M}{M}\frac{?}{1})$ queucing model is appropriate for this scenario. $($Include at least three decimal places in your answers.$)$

$($a$)$ Indicate the average arrival rate $$ in customers per minute.

$($b$)$ Indicate the average service rate $$ in customers per minute.

$($c$)$ What is the probability that the service window completes a customer's order within...

$($i$)...30$ seconds?

$($ii$)...45$ seconds?

$($iii$)...90$ seconds?

$($d$)$ What is the probability that the number of arrivals in a onc$-$minute period is$...$