In a factory, there are m operating machines and s

Chapter , Problem 14

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In a factory, there are m operating machines and s machines used as spares and ready to operate. The factory has k repairpersons, and each repairperson repairs one machine at a time. Suppose that, (i) each machine works, independent of other machines, for a time period which is exponentially distributed with mean 1/, then it breaks down; (ii) the time that it takes to repair an out-of-order machine is exponential with mean 1/, independent of repair times for other machines; (iii) at times when all repairpersons are busy repairing machines, the newly broken down machines will wait for repair; and (iv) when a machine breaks down, one of the spares will be used unless there is no spare machine available. Let X(t) be the number of machines operating or ready to operate at time t. Show that $ X(t): t 0 % is a birth and death process and find the birth and death rates.