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All trucks traveling on Interstate 40 between Albuquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 0.42 trucks/min, and the station can weigh, on the average, 0.46 trucks/min. Determine the following:

a. Average number of trucks waiting in the queue, Lq
b. Average waiting time in the queue for each truck, Wq
c. Average time spent in the system by each truck, W
d. Probability that arriving truck has to wait for service, PW
e. Probability that one truck is waiting in the queue and one truck is being served, P2
f. What is the probability that more than two trucks are waiting in the queue (i.e. more than three in the system)?

Sagot :

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Answer:

9.587 ; 22.83 ; 25 minutes

Step-by-step explanation:

Given that:

Arrival rate (λ) = 0.42 trucks/ min

Service rate (μ) = 0.46 trucks / min

Trucks arrive at the weigh station at a rate of 0.42 trucks/min, and the station can weigh, on the average, 0.46 trucks/min. Determine the following:

a. Average number of trucks waiting in the queue, Lq

Lq = λ² / μ(μ - λ)

Lq = 0.42² / 0.46(0.46 - 0.42)

Lq = 0.1764 / 0.0184

= 9.5869565

= 9.587

b. Average waiting time in the queue for each truck, Wq

Wq = λ / μ(μ - λ)

Wq = 0.42 / 0.46(0.46 - 0.42) = 22.83 minutes

c. Average time spent in the system by each truck, W

W = 1 / (μ - λ)

W = 1 / (0.46 - 0.42)

W = 1 / 0.04

W = 25 minutes

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