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).
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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