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 ${?}^{\text{i}\text{n}\text{a}\text{n}\text{a}\text{g}\text{e}\text{r}}$$$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 $$\frac{200}{?}$ week, the amortized cost of building the new counters will be $$\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$)$