Question

An average of 10 cars per hour reaches an ATM with a single server that provides service without leaving the car. Suppose the time of average service for each client is 4 minutes, and how long the times between arrival and service times are exponential. Answer the following questions:
a) What is the notation for this problem?
b) What is the probability that the cashier is idle?
c) What is the average number of cars in the cashier's queue?
d) What is the average amount of time a customer spends in the parking lot of the bank (including service time)?
e) How many customers will the cashier serve on average per hour?

Answer 1

a)

The problem is for single server model with infinite queue length. The notation for this problem is M/M/1

b)

Arrival rate = 10 cars per hour = (10/60) cars per minute = (1/6) cars per minute

Service rate, = 1 per 4 minute = (1/4) per minute

Utilization rate, p = / = (1/6) / (1/4) = 2/3

Probability that the cashier is idle = 1 - p = 1 - 2/3 = 1/3

c)

average number of cars in the cashier's queue = p / ( - )

= (2/3) * (1/6) / (1/4 - 1/6)

= (1/9) / (1/12)

= 12/9 = 4/3 minutes = 1.33 cars

d)

average amount of time a customer spends in the parking lot of the bank (including service time)

= 1 / ( - )

= 1 / ((1/4) - (1/6))

= 12 minutes

e)

Number of customers will the cashier serve on average per hour = / ( - )

= (1/6) / (1/4 - 1/6)

= (1/6) / (1/12)

= 2