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Markov network processes with string transitions. (English) Zbl 0948.60085

The authors consider Markov network processes with string transitions. They assume that it is described by a Markov process \(\{X(t)\), \(t\geq 0\}\) that represents the numbers of units at \(m\) nodes. The state space consists of \(m\)-dimensional vectors \(x= (x_1, x_2,\dots, x_m)\) where \(x_j\) denotes the number of units at node \(j\). A transition of the network from \(x\) to \(y\) has a rate of the form \(\sum_{sa} \lambda_{sa} r_{sa} (x,y)\). \(\lambda_{sa}\) is the rate of selecting a string of vectors \(s\) and another add-on vector \(a\) as the increments in the state. \(r_{sa} (x,y)\) is a system-dependent transition-initiation rate that may represent service rates at the nodes plus other transition information. The authors give necessary and sufficient conditions for a string-net to have an invariant measure of the form \(\Phi(x) \prod_{j=1}^m w_j^{x_j}\). \(\Phi\) is determined by the transition-initiation rates \(r_{sa} (x,y)\), and the parameters \(w_j\) are solutions to certain polynomial equations involving the string-generation rates \(\lambda_{sa}\). They also give sufficient conditions for the existence of a solution to these so-called traffic equations and show that the equations are equivalent to equalities of certain average flows in the network (a partial balance property).

MSC:

60K25 Queueing theory (aspects of probability theory)
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