EXERCISES

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probability that a repair involves the high repair cost Rc Give the long-run average cost per time unit (c) Verify that the average cost is minimal for the unique solution z to the equation z exp[ (L z)] = Rp /(Rc Rp ) when L > Rp /(Rc Rp ) 223 A group of N identical machines is maintained by a single repairman The machines operate independently of each other and each machine has a constant failure rate Repair is done only if the number of failed machines has reached a given critical level R with 1 R N Then all failed machines are repaired simultaneously Any repair takes a negligible time and a repaired machine is again as good as new The cost of the simultaneous repair of R machines is K + cR, where K, c > 0 Also there is an idle-time cost of > 0 per time unit for each failed machine (a) De ne a regenerative process and identify its regeneration epochs (b) Determine the long-run average cost per time unit 224 The following control rule is used for a slow-moving expensive product No more than one unit of the product is kept in stock Each time the stock drops to zero a replenishment order for one unit is placed The replenishment lead time is a positive constant L Customers asking for the product arrive according to a renewal process in which the interarrival times are Erlang (r, ) distributed Each customer asks for one unit of the product Each demand occurring while the system is out of stock is lost (a) De ne a regenerative process and identify its regeneration epochs (b) Determine the long-run fraction of demand that is lost (c) Determine the long-run fraction of time the system is out of stock (Hint: use part (b) of Exercise 25) 225 Jobs arrive at a station according to a renewal process The station can handle only one job at a time, but has no buffer to store other jobs An arriving job that nds the station busy is lost The handling time of a job has a given probability density h(x) Use renewal-reward theory to verify for this loss system that the long-run fraction of jobs that are rejected is given by 0 M(x)h(x) dx divided by 1 + 0 M(x)h(x) dx, where M(x) is the renewal function in the renewal process describing the arrival of jobs What is the long-run fraction of time that the station is busy Simplify the formulas for the cases of deterministic and Poisson arrivals 226 Use the renewal-reward theorem to prove relation (233) when customers arrive according to a renewal process and the stochastic processes {L(t)} and {Un } regenerate themselves each time an arriving customer nds the system empty, where the cycle lengths have nite expectations For ease assume the case of an in nite-capacity queue Use the following relations: (i) the long-run average reward earned per time unit = (the expected reward earned in one cycle)/(expected length of one cycle), (ii) the long-run average amount paid per customer = (the expected amount earned in one cycle)/(expected number of arrivals in one cycle), (iii) the long-run average arrival rate = (expected number of arrivals in one cycle)/(expected length of one cycle) 227 Let {X(t), t 0} be a continuous-time regenerative stochastic process whose state space is a subset of the non-negative reals The cycle length is assumed to have a nite expectation Denote by P (y) the long-run fraction of time that the process {X(t)} takes on a value larger than y Use the renewal-reward theorem to prove that

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