At an automatic car wash, cars arrive randomly at a rate of 7 cars every 30 minutes. The car wash takes exactly 4 minutes (this is constant). On average, what would be the length of the queue line?a) 3.86 b) 7.83 c) 4.35 d) 6.53 e) 5.92

This is a constant service time modeling challenge. In order to solve this question, you have to set a time period and this should be the same for all variables. In this case, our time variable will be 1 hour. The random rate of cars per hour coming to car wash is 14 since 7 in half an hour would correspond to 14 in one hour. Also, the washing rate is constant and it will be 15 cars washed in an hour if 1 car is washed in 4 minutes. Since our time constant is the same now we can solve this question. One important note here is the cars arrive randomly, so it would be a mistake to think like there will be no queque at the end of one hour. This is calculated via the following equation:

lambda= number of cars coming to car wash in one hour

nu= service rate which must be constant in my time span (1 hour)

cars in the queue= [tex]\frac{lambda^{2} }{2*nu*(nu-lambda)}[/tex]

Thus replacing the numbers lambda=14 and nu=15

[tex]\frac{15^2}{2*14*(15-14)}[/tex] is 6.53 average cars in the cue