## 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)

### Keywords:

Markov queueing; network
Full Text:

### References:

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