RENEWAL-REWARD PROCESSES

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RENEWAL THEORY

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As a generalization of the Poisson process, renewal theory concerns the study of stochastic processes counting the number of events that take place as a function of time Here the interoccurrence times between successive events are independent and identically distributed random variables For instance, the events could be the arrival of customers to a waiting line or the successive replacements of light bulbs Although renewal theory originated from the analysis of replacement problems for components such as light bulbs, the theory has many applications to quite a wide range of practical probability problems In inventory, queueing and reliability problems, the analysis is often based on an appropriate identi cation of embedded renewal processes for the speci c problem considered For example, in a queueing process the embedded events could be the arrival of customers who nd the system empty, or in an inventory process the embedded events could be the replenishment of stock when the inventory position drops to the reorder point or below it Formally, let X1 , X2 , be a sequence of non-negative, independent random variables having a common probability distribution function F (x) = P {Xk x}, x 0

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for k = 1, 2, Letting 1 = E(Xk ), it is assumed that 0 < 1 < The random variable Xn denotes the interoccurrence time between the (n 1)th and nth event in some speci c probability problem De ne

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Then Sn is the epoch at which the nth event occurs For each t 0, let N (t) = the largest integer n 0 for which Sn t Then the random variable N (t) represents the number of events up to time t De nition 211 The counting process {N (t), t 0} is called the renewal process generated by the interoccurrence times X1 , X2 , It is said that a renewal occurs at time t if Sn = t for some n For each t 0, the number of renewals up to time t is nite with probability 1 This is an immediate consequence of the strong law of large numbers stating that Sn /n E(X1 ) with probability 1 as n and thus Sn t only for nitely many n The Poisson process is a special case of a renewal process Here we give some other examples of a renewal process

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RENEWAL THEORY

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Example 211 A replacement problem Suppose we have an in nite supply of electric bulbs, where the burning times of the bulbs are independent and identically distributed random variables If the bulb in use fails, it is immediately replaced by a new bulb Let Xi be the burning time of the ith bulb, i = 1, 2, Then N (t) is the total number of bulbs to be replaced up to time t Example 212 An inventory problem Consider a periodic-review inventory system for which the demands for a single product in the successive weeks t = 1, 2, are independent random variables having a common continuous distribution Let Xi be the demand in the ith week, i = 1, 2, Then 1 + N (u) is the number of weeks until depletion of the current stock u 211 The Renewal Function An important role in renewal theory is played by the renewal function M(t) which is de ned by M(t) = E[N (t)], t 0 (211)

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For n = 1, 2, , de ne the probability distribution function Fn (t) = P {Sn t}, Note that F1 (t) = F (t) A basic relation is N (t) n if and only if Sn t This relation implies that P {N (t) n} = Fn (t), Lemma 211 For any t 0,

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