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# hw7sln - IEOR 161 Operations Research II University of...

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IEOR 161 Operations Research II University of California, Berkeley Spring 2008 Homework 7 Suggested Solution Chapter 5. 70. (a) Let N 1 (t) be the number of departures by time t, and S be the service time for the first customer. From example 5.25, we know that the departing process of the infinite server queue having Poisson arrivals and general service distribution G is a non-homogeneous Poisson process with intensity function λ ( t ) = λG ( t ). Pr { first customer to arrive is also the first to depart } = Pr { N 1 ( S ) = 0 } = Pr { N 1 ( t ) = 0 | S = t } dG ( t ) = e - t y =0 λ ( y ) dy dG ( t ) = e - t y =0 λG ( y ) dy dG ( t ) (b) Given N(t), the number of arrivals by t, the arrival times are iid and are uniform distributed U(0,t). The service time, T, for each customer is thus independent and follows the same distribution. The total remaining time is just the sum of all the service times and is independent of N(t). (c) Let R be the remaining service after time t, R=T-(t-s) if T > t-s, R=0 if T < t-s.

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