Question

PROBLEM 1: Suppose that all car owners fill up when their tanks are exactly half full. At the present time, an average of 7.5 customers per hour arrives at a single-pump gas station. It takes an average of 4 minutes to service a car. Assume that inter-arrival times and service times are both exponential. (a) What is the arrival rate? The service rate? The utilization? (b) Hand-compute the average number of cars at the station. (c) Hand-compute the average time (in minutes) a car spends at the station. (a) Use EXCEL Macro to verify your answer to parts ( a ) and (c).

Answer 1

a. Customers joining per hour = 7.5 / hr

=> Arrival rate($\lambda$)=7.5/hr

service time per customer = 4 minutes = 1/15 hr

Service rate($\mu$)= 1/(1/15) = 15/hr

utilization(assuming a single server)($\rho$) = $\lambda$ /$\mu$ = 7.5/15 = 0.5

b. No of customers(for exponential service and inter arrival times, assuming single server) =

p2/(1-P)

= 0.5^2 / 1-0.5 = 0.5

Average no of people waiting in line = 0.5

c. Average Time spent = No of Customers / $\lambda$

= 0.5/7.5 hr

= 60/15 minutes = 4 minutes