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A powerful tool in the analysis of numerous applied probability models is the renewal-reward model This model is also very useful for theoretical purposes In
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s 3 and 4, ergodic theorems for Markov chains will be proved by using the renewal-reward theorem The renewal-reward model is a simple and intuitively appealing model that deals with a so-called regenerative process on 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 inventory level equals S at epoch 0 The regeneration epochs are the beginnings of the weeks in which the inventory level is ordered up to the level S Let us de ne the random variables Cn = Sn Sn 1 , n = 1, 2, , where S0 = 0 by convention The random variables C1 , C2 , are independent and identically distributed In fact the sequence {C1 , C2 , } underlies a renewal process in which the events are the occurrences of the regeneration epochs Hence we can interpret Cn as Cn = the length of the nth renewal cycle, n = 1, 2,
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Note that the cycle length Cn assumes values from the index set T In the following it is assumed that 0 < E(C1 ) < In many practical situations a reward structure is imposed on the regenerative process {X(t), t T } The reward structure usually consists of reward rates that are earned continuously over time and lump rewards that are only earned at certain state transitions Let Rn = the total reward earned in the nth renewal cycle, n = 1, 2,
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It is assumed that R1 , R2 , are independent and identically distributed random variables In applications Rn typically depends on Cn In case Rn can take on both positive and negative values, it is assumed that E(|R1 |) < Let R(t) = the cumulative reward earned up to time t The process {R(t), t 0} is called a renewal-reward process We are now ready to prove a theorem of utmost importance Theorem 221 (renewal-reward theorem)
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E(R1 ) R(t) = t E(C1 )
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