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**One-dimensional circuit-switched networks.**
*(English)*
Zbl 0626.60102

In a one-dimensional (linear) circuit-switched network arriving calls request for an interconnection between two nodes of the network. If such a “route” which consists of a sequence of successive circuits is not available the call is lost. (The constraint on the availability originates from the finite capacity condition for the links between two neighboured nodes.)

Calls for different routes arrive in independent Poisson processes. The stochastic process describing the number of occupied routes for each type of route has a well-known “product form” steady-state distribution. A much more complicated problem is to describe the process which records the joint number of occupied circuits on the links of the network.

The author proves that under some restrictions on the arrival intensities the steady state of that process is an inhomogeneous Markov chain which additionally can be identified to be a quasistationary distribution of a suitable homogeneous Markov chain.

Under the additional restriction that each link comprises one circuit the steady-state distribution can be reduced in an analogous way to distributions obtained from alternating renewal processes.

Calls for different routes arrive in independent Poisson processes. The stochastic process describing the number of occupied routes for each type of route has a well-known “product form” steady-state distribution. A much more complicated problem is to describe the process which records the joint number of occupied circuits on the links of the network.

The author proves that under some restrictions on the arrival intensities the steady state of that process is an inhomogeneous Markov chain which additionally can be identified to be a quasistationary distribution of a suitable homogeneous Markov chain.

Under the additional restriction that each link comprises one circuit the steady-state distribution can be reduced in an analogous way to distributions obtained from alternating renewal processes.

Reviewer: H.Daduna

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60K30 | Applications of queueing theory (congestion, allocation, storage, traffic, etc.) |

90B15 | Stochastic network models in operations research |