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which a cost or reward structure is imposed Many stochastic processes have the property of regenerating themselves at certain points in time so that the behaviour of the process after the regeneration epoch is a probabilistic replica of the behaviour starting at time zero and is independent of the behaviour before the regeneration epoch A formal de nition of a regenerative process is as follows De nition 221 A stochastic process {X(t), t T } with time-index set T is said to be regenerative if there exists a (random) epoch S1 such that: (a) {X(t + S1 ), t T } is independent of {X(t), 0 t < S1 }, (b) {X(t + S1 ), t T } has the same distribution as {X(t), t T } It is assumed that the index set T is either the interval T = [0, ) or the countable set T = {0, 1, } In the former case we have a continuous-time regenerative process and in the other case a discrete-time regenerative process The state space of the process {X(t)} is assumed to be a subset of some Euclidean space The existence of the regeneration epoch S1 implies the existence of further regeneration epochs S2 , S3 , having the same property as S1 Intuitively speaking, a regenerative process can be split into independent and identically distributed renewal cycles A cycle is de ned as the time interval between two consecutive regeneration epochs Examples of regenerative processes are: (i) The continuous-time process {X(t), t 0} with X(t) denoting the number of customers present at time t in a single-server queue in which the customers arrive according to a renewal process and the service times are independent and identically distributed random variables It is assumed that at epoch 0 a customer arrives at an empty system The regeneration epochs S1 , S2 , are the epochs at which an arriving customer nds the system empty (ii) The discrete-time process {In , n = 0, 1, } with In denoting the inventory level at the beginning of the nth week in the (s, S) inventory model dealt with in Example 213 Assume that the 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