Opeele Chinkafa Ltd provides Ghanaian businesses and dealerships with high quality technical equipment. They operate tyre service centres, offering modern equipment and comfortable customer waiting zones for fast, reliable, and comfortable service. Due to the nature of their services, they have varied customers bringing their tyres for replacement. Currently, there is just a single server in their Abosokai centre, making the place extremely busy. Generally, $24$ customers bring their tyres for replacement every $2$ hours whilst $4$ customers check out every $15$ minutes after service, which is on a first$-$come, first$-$served basis.

i$.$ Determine the average time a customer must wait to be checked out.

ii$.$ What is the likelihood that $10$ customers are waiting in the system?

iii. What is the probability that an arriving customer will have to wait for service?

b$)$ Assume Opeele Chinkafa Ltd decides to advertise for a new mechanic to handle all tyre changes for customers ordering a new set of tyres. Two mechanics applied for the job. One mechanic $($new$)$ with limited experience, can be hired for $$14$ per hour, and can service an average of twenty customers per hour. The other mechanic with considerable experience, can service a customer every $2$ minutes, but must be paid $$20$ per hour. Assume that customers arrive at Opeele Chinkafa$$s garage at the rate of one customer every $30$ minutes.

i$.$ In each mechanic$$s case, determine the average number of customers in the queuing system,

the average waiting time, and the average time in the queuing system.

ii$.$ If the company assigns a cost of $$30$ per hour to every average number of customers in the

queuing system, and applying a cost$-$effective strategy, which mechanic should be hired given the total operating cost per hour?

iii. Following from a$($i$),$ Opeele Chinkafa Ltd estimates that each minute$$s reduction in customer waiting time would increase the company$$s total revenue by $$10$ per hour. Which mechanic would generate more net profit for the company?