Multiclass Exponential Clocks. Consider a jump process {X(t) : t ≥ 0} with countable state space S that evolves as follows. For each pair of states i, j, there is a countable set of sources Y(i, j)...

Multiclass Exponential Clocks. Consider a jump process {X(t) : t ≥ 0} with countable state space S that evolves as follows. For each pair of states i, j, there is a countable set of sources Y(i, j) that may trigger a transition from i to j, provided that such a transition is possible. Specifically, whenever the process X(t) is in state i, the time for source y to “potentially” trigger a transition to j is exponentially distributed with rate qy(i, j), independent of everything else. Then the time for a transition from i to j is the minimum of these independent exponential times, and so the potential transition time from i to j is exponentially distributed with rateThus, as in Example 9, we know that X(t) is a CTMC with transition rates qij , provided they are regular with respect to pij = qij/qi, where qi = j qij . In this setting, the sources that trigger the transitions are of interest, especially if there are costs or rewards associated with the sources. By properties of exponential variables, the probability that source y is the one that triggers the transition is qy(i, j)/qij . Let Yn denote the source that triggers the transition at time Tn+1. Consider the processExample 9