Question

Star Car Wash estimates that dirty cars arrive at the rate of 15 per hour all day and at the wash line, the cars can be cleaned at the rate of one every 4 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:

(a) average time a car spends in the service system.

(b) average number of cars in line.

(c) average time a car waits before it is washed.

(d) probability that no cars are in the system.

Answer 1

Arrival rate=15

Service rate=60/4=15

1. Average time a car spends in the service system= Arrival rate/( Service rate- Arrival rate)

Average time a car spends in the service system=15/(15-15)=15/0=Indefinite or ∞

b)

Average number of cars in line=arrival rate^2/( Service rate*( Service rate- Arrival rate))

Average number of cars in line= 15^2/(15*(15-15))=Indefinite or ∞

c)

Average time a car waits before it is washed= Average number of cars in line/ Arrival rate

Average time a car waits before it is washed=∞/15=∞ or Indefinite.

d)

Probability that no cars are in the system=1-(Arrival rate/ Service rate)

Probability that no cars are in the system=1-(15/15) =0 or Zero