Question

Many of a bank’s customers use its automatic teller machine to transact business after normal banking hours. During the early evening hours in the summer months, customers arrive at a certain location at the rate of one every other minute. This can be modeled using a Poisson distribution. Each customer spends an average of 94 seconds completing his or her transactions. Transaction time is exponentially distributed.

a. Determine the average time customers spend at the machine, including waiting in line and completing transactions. (Do not round intermediate calculations. Round your answer to the nearest whole number.)

b. Determine the probability that a customer will not have to wait upon arriving at the automatic teller machine. (Round your answer to 2 decimal places.)

c. Determine the average number of customers waiting to use the machine. (Round your answer to 2 decimal places.)

Answer 1

Arrival rate , A = 30 customers per hour ( 1 every other minute)

Service rate , S = 3600 / 94 = ~38 customers per hour ( 1 hour = 3600 seconds)

a) Utilization , p = A / S = 30/38 = 79%

Customers waiting , I = p² / (1-p) =  0.79 * 0.79 / (1-0.79) = 0.79*0.79 / 0.21 = 2.97 customers

Waiting time = I / A = 2.97 / 30 = 0.099 hours = 5.94 minutes

Total time in system = Service time + Watiting time = 5.94 + 94/60 mins = 7.50 minutes

b) Probability of no wait = 1 - A/S = 1 - 0.79 = 0.21 = 21%

c) Number of customers in system = Customer waiting + Utilization = 2.97 + 0.79 = 3.76 or 4 customers