Q$\mathrm{.}3.$ Smith is a worker working for an oil change facility. Each car requires on average $15$ minuies of service by Smith. Cars visits the oil change facility at a rate of $3$ cars per hour. Assuming that the arrivals follows a Poisson distribution, whereas the service time is exponentially distributed. Currently Smith is paid $$8$ per hour. If the owner estimates that the cost of an idle car i$.$e$.$ waiting for service is $$2$ per hour. $(5$ marks$)$

a$.$ Determine the proportion of time Smith is busy?

b$.$ What is the average number of cars waiting for service?

c$.$ How long on average will each car wait till it is being served?

d$.$ Assume that the owner is thinking about hiring another worker. Assume the second worker payment and service rate is exactly similar to the first one. If the workshop is open for $8$ hours per day. Construct a daily cost analysis and determine whether he should hire a second worker or not?