# Answered Essay: At a border inspection station, vehicles arrive at the rate of 12 per hour in a Poisson distribution. For

At a border inspection station, vehicles arrive at the rate of 12 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 13 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             minutes

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

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             %

a) Average length of the waiting line:::::$\gamma 2/\mu (\mu -\gamma )$

=(12)2 / 13(13-12)

=144 /13 (1)

=144/13

=11.07 customer

$\gamma$= Arrival time

$\mu$= Service time

___________________________________________________________________________________

b)Average total time = 1 /$\mu -\gamma$

=1/13-12

=1/1

=1 minutes

________________________________________________

c) utilization of the inspector = p $\gamma / \mu$

=12/13

=92.31%

________________________________________________

d)P=(n>4)

=($\gamma / \mu$)5

=(12/13)5

=(0.923)5

=0.669

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