Question

A store has a car wash facility for cleaning service. The arrival rate of cars is 15 per hour. The average service time is 3 minutes. Assume that the cars arrive in a poisson process and the service time distribution is exponential. There is only one facility providing service, and the parking space is only enough for two cars. If there is no car, the arrival car will enter this store. If there is one car in the parking space, 80% of arrival cars will enter this store. If there are two cars in the store, the arrival cars will just leave.

(a) Draw the rate diagram to show the birth-death process. (5 marks)

(b) Calculate the utilization rate of the wash facility. (5 marks)

(c) If the store manager wants to spend $50 per day to rent parking space for one more car, and the profit of washing one car is $5, is it worth renting? The store opens for 12 hours per day. After renting, when there are two cars in the store, 60% of arrival cars will enter it, and when there are three cars in the store, the arrival cars will just leave. (10 marks)

Answer 1

ANSWER :

Given data,

Average rate of cars ( ) = 15/hr

Average service time = 3minutes

Average service rate ( ) = 60/3/hr

= 12/hr

Now,

This problem of quening model P_n(t) denotes the probability that there are n person in syatem at ttime t.

   (independent of time)

If there is no car the arrival car will enter this store.

If there is one car, 80% of arrival car enter this store.

If there are two cars, arrival cars, arrival car will leave the store.

no of server = 1

It is a   model with discourage arrival rate.

a).

transition diagram :

b). Now, we know at any state'

inflow = outflow

   for n = 0

15 P₀(t) = 20 P₁(t)

as t

P₁ = 15/20 P₀

  

   

_{Again for n = 1}

_as

  

  

  

We have ,

   

  

  

  

.

  

  

  

  

   

Average number of person insystem,

  

Average no of persons in queue'

  

  

effective =

   

  

  

  

   

  

c). Now, another parking slot our model will be =

  

  

  

  

  

n=2

  

  

  

Now'

  

  

working hour for store = 12hr

Active time for server = 12* (p₁+p₂+p₃)

= 12* (1-P₀)

= 12*21/25

= 252/25/hr

therefore no of cars served each day =

  

therefore profit = $5

= $1008

Rent of parking space = $50

therefore, Net profit= $1008-$50

= $958

Profit of condition state on part (b) =$12* (1-p₀ )*20*5

= $12* 6/11*100

= $654.4

As profit in (c) > profit in (b)

therefore, It is worth renting.