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The non-negative numbers qij are called the in nitesimal transition rates of the continuous-time Markov chain {X(t)} Note that the qij uniquely determine the sojourn-time rates i and the one-step transition probabilities pij by i = j =i qij and pij = qij / i The qij themselves are not probabilities but transition rates However, for t very small, qij t can be interpreted as the probability of moving from state i to state j within the next t time units when the current state is state i In applications one usually proceeds in the reverse direction The in nitesimal transition rates qij are determined in a direct way They are typically the result of the interaction of two or more elementary processes of the Poisson type Contrary to the discrete-time case in which the one-step transition probabilities determine unambiguously a discrete-time Markov chain, it is not generally true that the in nitesimal transition rates determine a unique continuous-time Markov chain

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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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)

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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