At a border inspection station, vehicles arrive at the rate of 10 per hour in a

Question

At a border inspection station, vehicles arrive at the rate of 10 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 14 per hour in an exponentially distributed fashion. a. What is the average length of the waiting line? (Round your answer to 2 decimal places.) Average length customers b. What is the average total time it takes for a vehicle to get through the system? (Round your answer to 2 decimal places.) Average time c. What is the utilization of the inspector? (Round your answer to 1 decimal place.) minutes Utilization 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.) Probability

Explanation / Answer

Arrival rate (A) = 10

Service rate (S) = 14

a) Average length (Lq) = A^2/(S*(S-A)) = 10^2/(14*(14-10)) = 1.79 customers

b) Total time = (Lq/A + 1/S)*60 = (1.79/10 + 1/14)*60 = 15.03 minutes

c) Utilization = A/S = 10/14*100 = 71.4%

d) Probability of three of more vehicle = (A/S)^3 = (10/14)^3*100 = 36.4

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