Q$1.($Total $70$ points$)$ The Fast Shop Drive$-$In Market has one checkout counter where

one employee operates the cash register. The combination of the cash register and the

operator is the check out channel in this waiting line system; the customers who line up at

the counter to pay for their selections form the waiting line. Customers arrive at a rate of

one every $2\mathrm{.}5$ minutes according to a Poisson distribution, and service times are

exponentially distributed, with a mean rate of $30$ customers per hour.

a$)$ The market manager wants to determine the following for this waiting line system:

points$)$

I. Probability of no customers in the system

II$.$ Average number of customers in the total system

III. Average time spent in the system per customer

IV$.$ Average time spent waiting in line

V$.$ Probability that there are more than $1$ customer waiting in line

b$)$ Because of the nature of the store, customers purchase a few items and expect quick

service. Given customers' expectations, the manager believes that the waiting time is

unacceptable. The manager wants to test two alternatives for reducing customer waiting

time: I$)$ adding extra employees to pack up the purchases $($and hence increasing the

service rate$),$ II$)$ adding more checkout counters with all the counters to serve the same

waiting line. With the help of the market's national office's marketing research group, the

manager has determined that each customer's waiting cost in the system is $$75$ per week

to the store. The regular salary of the cashier is $$\frac{200}{?}$ week.

Alternative I: Adding extra employee

Adding extra employees will cost the market manager $$100$ per week for each employee

added $($since these extra employees will not be cashiers but only packers, so the rate is

less$).$ However, each extra employee will increase the service rate by $10$ customers per

hour. Note that there will be a single cash counter.

Alternative II: Adding more checkout counters with a single line serving all counters

In this case the market can decide to build new checkout counters with each counter

being served by a separate cashier $($there will be no helpers$).$ The customer at the top of

the waiting line goes to whichever counter is free. While each cashier will cost $$200/$

week, the amortized cost of building the new counters will be $$\frac{1000}{?}$ year for each new

counter added.

Taking both the capacity and waiting costs into account, which alternative is the best for

the market $($assume $52$ weeks$/$year$)?$ What will be the average waiting time in the system

and in the line for the customers if the market is operating at its optimal configuration?

points$)$ Good handwriting please