Problem $3$: Fast$-$Food Restaurant Design $[40$ points$]$

A large fast$-$food restaurant chain operator is analyzing a new restaurant opening project. Based on

data from existing locations, it expects that the new potential restaurant would see an average

demand of $25$ customers per hour. All customers would need to first order their food at one of $3$

planned counters, at which point $80\%$ of these customers are expected to choose to eat in as

opposed to take$-$away. On average, eat$-$in customers spend $30$ minutes in the restaurant$$s seating

area once they are given their food at the counter.

$3\mathrm{.}1$ Assume unlimited seating capacity for eat in customers. Under these assumptions,

what will be the average number of eat in customers in the seating area? $[8$ points$]$

$3\mathrm{.}2$ What would be the maximum total volume of incoming customers under which the

average number of eat in customers could be accommodated with a seating area of $14$

seats $?[6$ points$]$

$3\mathrm{.}3$ Consider now the customer lines forming in front of the $3$ ordering counters.

Assume the customer arrival process has the baseline average demand rate of $25$

customers per hour, with a coefficient of variation of $1\mathrm{.}2$ for the inter$-$arrival times.

Assume also that the service time for each customer at any one of the ordering counters

has an average of $6$ minutes and a coefficient of variation of $1\mathrm{.}1.$

a$.$ What is the average fraction of time when staff at the ordering counters are idle? $[6$

points$]$

b$.$ Assume that arriving customers separate randomly into $3$ separate queues in front of

each ordering counter, and that moving from one queue to the next is never possible.

What is the average waiting time of a customer before ordering? $[10$ points$]$

c$.$ Assume now that arriving customers join a single queue forming in front of the $3$

ordering counter, and that the customer first in line joins the first available counter.

What is the average waiting time of a customer before ordering? $[5$ points$]$

d$.$ Provide a qualitative explanation for the difference of average customer waiting times

found in your answers to the two previous questions. $[5$ points$]$