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the piece by a new one and letting the piece operate for the present day A preventive replacement takes one day A new piece has working condition i = 1 A piece whose present working condition is i has the next day working condition j with known probability qij where qij = 0 for j < i The following replacement rule is used The current piece is only replaced by a new one when its working condition is greater than the critical value m, where m is a given integer with 1 m < N (a) De ne an appropriate Markov chain and specify its one-step transition probabilities (b) Explain how to calculate the long-run fraction of days the equipment is inoperative and the fraction of replacements occurring in the failure state N 315 Consider a stochastically failing piece of equipment with two identical components that operate independently of each other The lifetime in days of each component has a discrete probability distribution {pj , j = 1, , M} A component in the failure state at the beginning of a day is replaced instantaneously It may be economical to preventively replace the other working component at the same time the failed component has to be replaced The cost of replacing only one component is K1 , while the cost of replacing simultaneously both components equals K2 with 0 < K2 < 2K1 The control rule is as follows Replace a component upon failure or upon reaching the age of R days, whichever occurs rst If a component is replaced and the other component is still working, the other component is preventively eplaced when it has been in use for r or more days The parameters r and R are given integers with 1 r < R (a) De ne an appropriate Markov chain and specify its one-step transition probabilities (b) How can you calculate the long-run average cost per day 316 A transmission channel transmits messages one at a time, and transmission of a message can only start at the beginning of a time slot The time slots have unit length and the transmission time of a message is one time slot However, each transmission can fail with some probability f A failed transmission is tried again at the beginning of the next time slot The numbers of new messages arriving during the time slots are independent random variables with a common discrete distribution {ak , k = 0, 1, } Newly arriving messages are temporarily stored in a buffer of ample capacity It is assumed that the average arrival rate of new messages is smaller than the average number of attempts needed to transmit a message successfully, that is, k kak < 1/f The goal is to nd the long-run average throughput per time unit (a) De ne an appropriate Markov chain with a one-dimensional state space and specify its one-step transition probabilities (b) Can you give a recursive algorithm for the computation of the state probabilities Express the average throughput in terms of the state probabilities 317 Messages arrive at a transmission channel according to a Poisson process with rate The channel can transmit only one message at a time and a new transmission can only start at the beginnings of the time slots t = 1, 2, The transmission time of a message is one time slot The following access-control rule is used The gate is closed for newly arriving messages when the number of messages awaiting transmission has reached the level R and is opened again when the number of messages awaiting transmission has dropped to the level r, where the parameters r and R are given integers with 0 r < R The goal is to study the long-run fraction of lost messages as function of r and R (a) De ne an appropriate Markov chain and specify its one-step transition probabilities (b) Show how to calculate the long-run fraction of lost messages 318 In Example 351 we have determined for the GI /M/1 queue the customer-average probability j denoting the long-run fraction of customers who nd j other customers present upon arrival Denote by the time-average probability pj the long-run fraction of time that j customers are present for j = 0, 1, Use Theorem 333 and Lemma 114.

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