Question

1. From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the
   1. average number of cars inline.
   2. average time a car waits before it is washed.
   3. average time a car spends in the service system.
   4. utilization rate of the car wash.
   5. probability that no cars are in the system.

Please be detailed and show your work.

Answer 1

SOLUTION:

Arrival rate = 10 cars Per hour

Service rate = One car at every 5 minutes

Service rate = 12 cars per hour

Number of servers (s) = 1

So

(A)

Average number of cars in line

Average no of cars in line(Lq) = 4.166666667

(B)

Average time a car waits before it is washed      
Average waiting time in queue(Wq) = 0.416666667   hours
Average waiting time in queue(Wq) =(0.8333/((0.1667)(12)))=0.4165 hours= 25   Mins

(C)

Average time a car spends in the service system      
Average time in service system(W) = 0.5 hours  
Average time in service system(W) = 30 Mins

(D)

Utilization rate of the car wash          
Average utilization of service system = 0.833333333 = 83.33 Percent
(E) Probability that no cars are in the system          
Probability of no car in the system(P(0)) = 0.166666667