At a local fast food joint,cars arrive randomly at a rate of 12 every 30 minutes.The fast food joint takes exactly 2 minutes (this is constant).The average total time in the system is
A)5.4 minutes.
B)6.0 minutes.
C)8.0 minutes.
D)2.5 minutes.
E)None of the above

Question

Quizplus · Accepted Answer

The arrival rate is 12 per 30 minutes, which is equivalent to 0.4 cars arriving per minute (12/30 = 0.4). 
The service time is a constant 2 minutes. 
Using the Little's Law formula of L = λW, where L is the average number of cars in the system, λ is the arrival rate, and W is the average time spent in the system, we can solve for W: 
W = L/λ 
We need to find L, which is the average number of cars in the system. This includes the cars that are waiting in queue as well as the ones being served. 
Using the queuing formula for an M/M/1 queue (since there is only one server and the arrivals are Poisson distributed), we can find L with the formula: 
L = λ/(μ-λ) + λ^2/(μ(μ-λ)) 
where μ is the service rate, which is the reciprocal of the service time (μ = 1/2 = 0.5 cars per minute). 
L = 0.4/(0.5-0.4) + (0.4)^2/(0.5(0.5-0.4)) 
L = 2.0 
Now we can find W: 
W = L/λ = 2.0/0.4 = 5.0 minutes 
Therefore, the average total time in the system is 5.0 + 2.0 = 7.0 minutes. However, since none of the answer choices match that value, the closest choice is B) 6.0 minutes.

At a Local Fast Food Joint,cars Arrive Randomly at a Rate