Customers arrive at a rate of 8 per hour according to a Poisson distribution. Th
ID: 446937 • Letter: C
Question
Customers arrive at a rate of 8 per hour according to a Poisson distribution. The car washer (one server) can service an average of 10 cars per hour with service times described by an exponential distribution. Melanie is concerned with the number of customers waiting in line. She has asked you to calculate the following system characteristics:
(a) Average system utilization =.8
(b) Average number of customers in the system =4
(c) Average number of customers waiting in line =3.2
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 information in Problem 1, calculate the following (a to i) and answer the questions (j and k):
(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 3 customers in the system
(d) The probability of having 3 customers waiting in line
(e) The probability of having less than 3 customers in the system
(f) The probability of having more than 2 customers in the system
(g) The probability of having at least 4 customers in the system
(h) The probability of having at least 3 customers waiting in line
(i) The probability of having no customer waiting in line
(j) A customer arrives at 11:00 AM, at what time is the customer going to leave the Clean Machine car wash?
(k) A customer arrives at 2:00 PM, what is the expected time of arrival for the next customer?
Explanation / Answer
The arrival rate A= 8
The service rate S= 10
The average time a unit spends in the system 1/(S-A) = 1(10-8) =1/2 HOUR = 30 minutes
Average time a units spends in waiting line = A/S*(S-A) = 8/10(10-8) = 8/20 =0.4 HOUR = 24 minutes
The probability of having three customers in the system = (1-A/S)(A/S)3 = (1-8/10)*(8/10)3 = 2/10* 512/1000 = 0.1024 HOURS = 6.14 SECON DS
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.