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4. The processing time of a service system is gamma distributed with parameters n and (the pdf of Gamma(n, A) is given in Problem 3) when there are n customers in the system. The rate parameter A is a random variable and given that there are n customers in the system, 1/A follows an exponential distribution with rate n. Furthermore, the number of customers in the system follows a Poisson distribution with rate 5. What is the mean processing time? Hint: Denote by Y the processing time, and N the number of customers in the system. Then Y N n, A = x ~ Gamma(n, x), AN = n~ Exp(n), and N~ 11 Poisson(5). You may start with computing E(Y N = n).

4. The processing time of a service system is gamma distributed with parameters n and (the pdf of Gamma(n, A) is given in Problem 3) when there are n customers in the system. The rate parameter A is a random variable and given that there are n customers in the system, 1/A follows an exponential distribution with rate n. Furthermore, the number of customers in the system follows a Poisson distribution with rate 5. What is the mean processing time? Hint: Denote by Y the processing time, and N the number of customers in the system. Then Y N n, A = x ~ Gamma(n, x), AN = n~ Exp(n), and N~ 11 Poisson(5). You may start with computing E(Y N = n).

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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4. The processing time of a service system is gamma distributed with parameters n and (the pdf of Gamma(n, A) is given in Problem 3) when there are n customers in the system. The rate parameter A is a random variable and given that there are n customers in the system, 1/A follows an exponential distribution with rate n. Furthermore, the number of customers in the system follows a Poisson distribution with rate 5. What is the mean processing time? Hint: Denote by Y the processing time, and N the number of customers in the system. Then Y N n, A = x ~ Gamma(n, x), AN = n~ Exp(n), and N~ 11 Poisson(5). You may start with computing E(Y N = n).
Transcribed Image Text:4. The processing time of a service system is gamma distributed with parameters n and (the pdf of Gamma(n, A) is given in Problem 3) when there are n customers in the system. The rate parameter A is a random variable and given that there are n customers in the system, 1/A follows an exponential distribution with rate n. Furthermore, the number of customers in the system follows a Poisson distribution with rate 5. What is the mean processing time? Hint: Denote by Y the processing time, and N the number of customers in the system. Then Y N n, A = x ~ Gamma(n, x), AN = n~ Exp(n), and N~ 11 Poisson(5). You may start with computing E(Y N = n).
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