Please help with details Thanks Cars arrive at the drive-through of a bank at a

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Please help with details

Thanks

Cars arrive at the drive-through of a bank at a rate of 0.15 car per minute There is only one service window and the mean service time is 2.5 minutes Assume that this process is an M/M/1 queue Let N be the number of cars when the system is in steady-state. Find the service rate lambda_s in number of cars per minute. Find the arrival/service ratio r as an exact decimal number. Find the expected value and standard deviation of the number of cars in the system in steady-state. Give the standard deviation to 4 decimal places. Find the long-run proportion of time that the queue is idle. Find the probability that two or more cars are in the system in steady-state. On average, how many minutes does a car spend in the drive-through?

Explanation / Answer

a) Service Rate , s = 1/2.5 = 0.4 cars per minute

b) Arrival by service ratio = 0.15/0.4 = 0.375

c) Long run proportion of time that the queue is idle = 1- service rate = 1 - 0.375 = 0.625

d) Probability of two or more cars = 14.06% = 1 - (probability of No car + probability of 1 car) = 1 - (0.625 + 0.2344)

Probability of n Cars in system is calculated by the following formula, Pn = (1-)*n, where is the ratio - arrival rate/service rate = 0.375, and n is the number of cars

e) Average time spent by car in the drive-through = 1/(0.4-0.15) = 4 minutes , [calculated by the formula 1/(-), where is service rate, and is arrival rate ]

=

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