A one-car taxi company receives an average of 18 calls per day. The receptionist takes down details of the requested journey and relays them to the driver by radio. Each passenger's journey takes 25 minutes on average. The working day is 12 hours long.

(i) What constitutes the queue in this particular situation?

(ii) How long, on average, will a client have to wait between placing a pickup call and the arrival of the taxi?

(iii) During this waiting time, how many other clients on average will be picked up by the taxi? (Think about the number of people already waiting in queue when a customer arrives.)

(iv) List all the necessary assumptions for your calculations in (ii) and (iii). For each assumption, explain briefly whether you think they may or may not be realistic in this particular example.

(v) What is the probability that the time between two successive calls will exceed one hour?

(vi) What would your answer to (ii) be if the mean of the service time remains the same but the standard deviation is 15 minutes? What type of queuing system is this?

(vii) Based on observation, the more taxis a company operates, the more calls it receives. If the company operated a second car, what rate of calls would result in the same average waiting time as with one car? Express your answer in terms of the average interval between calls.

(viii) The annual costs of operating the company are as follows:

Office and receptionist (for either 1 or 2 cars) £30,000

Fixed costs per car (depreciation, tax etc.) £5,000

Drivers' wages £20,000

The variable costs per car (petrol etc.) are £18/day, when operating full-time.

For the one-car case described in (i)-(iv), and for the two-car case in (vii), calculate the average charge per client necessary to break even. Assume 300 working days per year.