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Here we run into subtleties that are well beyond the scope of this book Note that fundamental dif culties may arise when the state space is in nite, but these dif culties are absent in almost all practical applications To avoid the technical problems, we make the following assumption for the given data qij Assumption 412 The rates i =
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qij are positive and bounded in i I
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The boundedness assumption is trivially satis ed when I is nite and holds in most applications with an in nite state space Using very deep mathematics it can be shown that under Assumption 412 the in nitesimal transition rates determine a unique continuous-time Markov chain {X(t)} This continuous-time Markov chain is precisely the Markov jump process constructed according to the above rules (a) and (b), where the leaving rates are given by i = j =i qij and the pij by pij = qij / i The continuous-time Markov chain {X(t)} does indeed have the property P {X(t + t) = j | X(t) = i} = qij t + o( t), 1 i t + o( t), j = i, j = i (411)
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It is noted that Assumption 412 implies that the constructed continuous-time Markov chain {X(t)} automatically satis es Assumption 411 In solving speci c problems it suf ces to specify the in nitesimal transition rates qij We now give two examples In these examples the qij are determined as the result of the interaction of several elementary processes of the Poisson type The qij are found by using the interpretation that qij t represents the probability of making a transition to state j in the next t time units when the current state is i and t is very small Example 411 (continued) Inventory control for an in ammable product The stochastic process {X(t), t 0} with X(t) denoting the stock on hand at time t is a continuous-time Markov chain with state space I = {0, 1, , Q} Its in nitesimal transition rates qij are the result of the interaction of the two independent Poisson processes for the demands and the replenishment opportunities The qij are given by qi,i 1 = for i = 1, , Q, q0Q = and the other qij = 0
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To see this, note that for any state i with i 1, P {X(t + t) = i 1 | X(t) = i} = P {a demand occurs in (t, t + = t + o( t)
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Conditions under which the in nitesimal parameters determine a unique continuous-time Markov chain are discussed in depth in Chung (1967)
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Figure 411 The transition rate diagram for the inventory process
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and P {X(t + t) = Q | X(t) = 0} = P {a replenishment opportunity occurs in (t, t + = t + o( t) for t 0 In the analysis of continuous-time Markov chains, it is very helpful to use a transition rate diagram The nodes of the diagram represent the states and the arrows in the diagram give the possible state transitions An arrow from node i to node j is only drawn when the transition rate qij is positive, in which case the arrow is labelled with the value qij The transition rate diagram not only visualizes the process, but is particularly useful when writing down its equilibrium equations Figure 411 shows the transition rate diagram for the inventory process Example 412 Unloading ships with an unreliable unloader Ships arrive at a container terminal according to a Poisson process with rate The ships bring loads of containers There is a single unloader for unloading the ships The unloader can handle only one ship at a time The ships are unloaded in order of arrival It is assumed that the dock has ample capacity for waiting ships The unloading time of each ship has an exponential distribution with mean 1/ The unloader, however, is subject to breakdowns A breakdown can only occur when the unloader is operating The length of any operating period of the unloader has an exponential distribution with mean 1/ The time to repair a broken unloader is exponentially distributed with mean 1/ Any interrupted unloading of a ship is resumed at the point it was interrupted It is assumed that the unloading times, operating times and repair times are independent of each other and are independent of the arrival process of the ships The average number of waiting ships, the fraction of time the unloader is down, and the average waiting time per ship, these and other quantities can be found by using the continuous-time Markov chain model For any t 0, de ne the random variables X1 (t) = the number of ships present at time t t]} + o( t)
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