Question

At a border inspection station, vehicles arrive at the rate of 8 per hour in a Poisson distribution. For simplicity in this problem, assume that there is only one lane and one inspector, who can inspect vehicles at the rate of 15 per hour in an exponentially distributed fashion.

a. What is the average length of the waiting line? (Round your answer to 2 decimal places.)

b. What is the average total time it takes for a vehicle to get through the system? (Round your answer to 2 decimal places.)

c. What is the utilization of the inspector? (Round your answer to 1 decimal place.)

d. What is the probability that when you arrive there will be three or more vehicles ahead of you? (Round your answer to 1 decimal place.)

Answer 1

Service Rate (Mu) = 15 per hour

Arrival Rate (Lambda) = 8 per hour

a. Average Length of waiting line (Lq)

b. Average total time it takes for a vehicle go through system (W)

c. Utilization of Inspector

Utilization % = 53.3%

d.

Probability that less than/ three vehicles in the line (P) = P0 + P1 + P2

P0 = (1-lambda/Mu) = 1 - 8/15 = 0.4666

P1 = (Lambda/Mu)^1*P0 = (8/15)*0.4666 = 0.2489

P2 = (Lambda/Mu)^2*P0 = (8/15)^2*0.4666 = 0.1327

P = 0.4666 + 0.2489 + 0.1327 = 0.8482

Probability that three or more vehicles ahead of you(P') = 1 - P = 1 -0.8482 = 0.1518 ~ 15.2%