Question

Cars arrive at an automatic car wash system every 10 minutes on average. The cars inter-arrival times are exponentially distributed. Washing time for each is 6 minutes per car and is purely deterministic (i.e., the waiting line system is M/D/c). Assuming that the car wash has a single bay to serve the cars, what is the average number of cars waiting in line (L.)?

Answer 1

the average number of cars waiting in line L
`q` is 0.45

Explanation:

Given the data in the question;

Cars arrive at an automatic car wash system every 10 minutes on average.

Car arrival rate λ = 1 per 10 min = [ 1/10 × 60 ]per hrs = 6 cars per hour

Washing time for each is 6 minutes per car

Car service rate μ = 6min per car = [ 1/6 × 60 ] per hrs = 10 cars per hour

so

P = λ/μ = 6 / 10 = 0.6

Using the length of queue in M/D/1 system since there is only one service bay;

L
`q` =
`(1)/(2)`[ P² / ( 1 - P ) ]

so we substitute

L
`q` =
`(1)/(2)`[ (0.6)² / ( 1 - 0.6 ) ]

L
`q` =
`(1)/(2)`[ 0.36 / 0.4 ]

L
`q` =
`(1)/(2)`[ 0.9 ]

L
`q` = 0.45

Therefore, the average number of cars waiting in line L
`q` is 0.45