488 An Introduction to Stochastic Modeling Exercises 9.5.1 Consider the three-server network pictured here: P 21 = 0.2 P 12 = 0.6 Server #2 2n = 3, n 1 Departs with probability 0.8 Arrivals Poisson, Rate = 2 Server #1 1n = 6, n 1 P 13 = 0.4 Server #3 3n = 2, n 1 In the long run, what fraction of time is server #2 idle while, simulta- neously, server #3 is busy? Assume that all service times are exponentially distributed. 9.5.2 Refer to the network of Exercise 9.5.1. Suppose that server #2 and server #3 share a common customer waiting area. If it is desired that the total number of customers being served and waiting to be served not exceed the waiting area capacity more than 5% of the time in the long run, how large should this area be? Problem 9.5.1 Suppose three service stations are arranged in tandem so that the departures from one form the arrivals for the next. The arrivals to the first station are a Poisson process of rate = 10 per hour. Each station has a single server, and the three service rates are µ 1 = 12 per hour, µ 2 = 20 per hour, and µ 3 = 15 per hour. In-process storage is being planned for station 3. What capacity C 3 must be provided if in the long run, the probability of exceeding C 3 is to be less than or equal to 1%? That is, what is the smallest number C 3 = c for which lim t Pr{X 3 (t) > c} 0.01? 9.6 General Open Networks The preceding section covered certain memoryless queueing networks in which a cus- tomer could visit any particular server at most once. With this assumption, the depar- tures from any service station formed a Poisson process that was independent of the number of customers at that station in steady state. As a consequence, the numbers