A motorway service station

At a motorway service station, cars approach the fuel pumps on a long single-lane service road and queue at a point for the next available fuel pump. There are 12 pumps. On filling their car, drivers take it to the paying kiosk. On completion of payment, drivers may then exit the service station with optional stops for shop, toilets, air and water, and refreshment outlets.

Vehicles arrive at the service station at the rate of 135 per hour and spend an average of 4 minutes at the pump before driving to a pay kiosk. The average service time at the pay kiosk is 20 seconds. Arrivals at the service station are random with a Poisson distribution. All service rates have a negative exponential distribution.

i. Calculate:

a. the pump facility utilisation.

(1 mark)

b. the probability of no cars at the pumps. (Calculate this value; do not use tables.)

(4 marks)

c. the average time from joining the queue to leaving the pumps.

(3 marks)

d. the probability that there is at least 1 car but not more than 5 cars queuing for the pumps at any one time.
the probability that there are at least 2 cars in the queue for the kiosk at any given time.

(4 marks)

what false assumption has been made as far as the queue for the kiosk is concerned.

(3 marks)

Briefly discuss:
How might the site be redesigned to reduce the effect of this false assumption?

(3 marks)

If no redesign is made, how

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