HW6Soln - IEOR 161 Introduction to Stochastic Processes...

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IEOR 161 - Introduction to Stochastic Processes Spring 2010 HW6 Solutions ** Note that the numbering is from Ross 9th Edition 5.48 Given T , the time until the next arrival, the number of busy servers found by the next arrival, N , is a binomial random variable with parameters n and p = e - μT . (a). E[ N ] = Z 0 E[ N | T = t ] λe - λt dt = Z 0 ne - μt λe - λt dt = λ + μ . (b). P { N = 0 } = Z 0 P { N = 0 | T = t } λe - λt dt = Z 0 ( 1 - e - μt ) n λe - λt dt = - Z 0 ( 1 - e - μt ) n de - λt = - ( 1 - e - μt ) n e - λt | 0 + Z 0 e - λt n ( 1 - e - μt ) n - 1 μe - μt dt = Z 0 ( 1 - e - μt ) n - 1 e - ( λ + μ ) t dt = λ + μ - Z 0 ( 1 - e - μt ) n - 1 de - ( λ + μ ) t = λ + μ - (1 - e - μt ) n - 1 e - ( λ + μ ) t | 0 + Z 0 e - ( λ + μ ) t ( n - 1) e - μt ( - μ ) = ( )(( n - 1) μ ) λ + μ Z 0 e - ( λ +2 μ ) t ( n - 1) e - μt ( - μ ) = · · · = n Y j =1 ( n - j + 1) μ λ + ( n - j + 1) μ Alternatively, let T j denotes the inter-departure time of customers in the system, j = 1 , ..., n . It’s easy to see that T j is exponential with rate ( n - j + 1) μ , then we have, P ( T > T 1 + ... + T n ) = n Y j =1 P ( T > T j ) = n Y j =1 ( n - j + 1) μ λ + ( n - j + 1) μ 1
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(c). By the same method used in part (b) P { N = i } = λ λ + ( n - i ) μ i Y j =1 ( n - j + 1) μ
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