Question

Melanie is the manager of the Clean Machine car wash and has gathered the following information. Customers arrive at a rate of eight per hour according to a Poisson distribution. The car washer can service an average of ten cars per hour with service times described by an ex- ponential distribution. Melanie is concerned about the number of customers waiting in line. She has asked you to calculate the following system characteristics:

(a) Average system utilization 8/10=0.8= 80%

(b) Average number of customers in the system 8/(10-8)=8/2= 4

(c) Average number of customers waiting in line 8^2/10(10-8)= 64/20= 3.2 customers

2. Melanie realizes that how long the customer must wait is also very important. She is also concerned about customers balking when the waiting line is too long. Using the arrival and service rates in Problem 1, she wants you to calculate the following system characteristics:

(a) The average time a customer spends in the system 1/(10-8)= ½ hours

(b) The average time a customer spends waiting in line0.8/(10-8)= 0.8/2 = 0.4 hours

(c) The probability of having more than three customers in the system 0.8^3 =0.512

(d) The probability of having more than four customers in the system 0.8^4 =0.4096

If Melanie adds an additional server at Clean Machine car wash, the service rate changes to an average of 16 cars per hour. The customer arrival rate is 10 cars per hour. Melanie has asked you to calculate the following system characteristics:
(a) Average system utilization
(b) Average number of customers in the system

(c) Average number of customers waiting in line

Melanie is curious to see the difference in waiting times for customers caused by the additional server added in Problem 3. Calculate the following system characteristics for her :

(a) The average time a customer spends in the system

(b) The average time a customer spends waiting in line

(c) The probability of having more than three customers in the system

(d) The probability of having more than four customers in the system

After Melanie added another car washer at Clean Machine (service rate is an average of 16 customers per hour), business improved. Melanie now estimates that the arrival rate is 12 customers per hour. Given this new information, she wants you to calculate the fol- lowing system characteristics:

(a) Average system utilization

(b) Average number of customers in the system

(c) Average number of customers waiting in line

As usual, Melanie then requested you to calculate sys- tem characteristics concerning customer time spent in the system.

(a) Calculate the average time a customer spends in the system.

(b) Calculate the average time a customer spends waiting in line.

(c) Calculate the probability of having more than four customers in the system.

Answer 1