Question

Problem 15-3 (Algorithmic) Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 18 customers per hour or 0.3 customers per minute. In the same bank waiting line system, assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 30 customers per hour, or 0.5 customers per minute. Determine the following operating characteristics for the system:

The probability that no customers are in the system. If required, round your answer to four decimal places. P0 =

The average number of customers waiting. If required, round your answer to four decimal places.

Lq =

The average number of customers in the system. If required, round your answer to the nearest whole number.

L =

the average time a customer spends waiting. If required, round your answer to four decimal places.

Wq = min

The average time a customer spends in the system. If required, round your answer to the nearest whole number.

W = min

The probability that arriving customers will have to wait for service. If required, round your answer to four decimal places.

Pw =

Answer 1

Given

Arrival rate , 18/hr

Departure rate , 30 /hr

Number of servers = 1

By M/M/1 model,

Probability that no customers in the system ,P(0)= 1 -18/30 = 0.4

Average number of customers waiting , L_q = = 18^2/ 30×12= 0.9

Average number of customers in the system , L = = 18/12= 1.5

Average time of a customer spend waiting , W_q = = 18/30*12= 0.05 hr = 3 min

Average time of customer in the system,W= =1/12= 0.0833 hr = 5 min

Probability that arriving customer will have to wait for device,P(w)=1-P(0)=1-0.4=0.6