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**Limit results for switching Markov queues with finitely many sources.**
*(English.
Russian original)*
Zbl 0815.60085

Cybern. Syst. Anal. 30, No. 1, 59-63 (1994); translation from Kibern. Sist. Anal. 1994, No. 1, 79-84 (1994).

Summary: Queueing methods are used on an ever increasing scale for the study of the behavior of complex systems. In many cases, the jobs to be processed arrive from a finite source (e.g., a finite number of failing machines assigned to an operator, the case of information and computer systems, reliability problems). We present limit results for switching Markov systems with fast service based on the general property of the sojourn time of a Markov process in a subset of states. We show that the limit distributions of the first failure time of the system are exponential, and the stream of failures weakly converges to a Poisson process.

### MSC:

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60K25 | Queueing theory (aspects of probability theory) |

### Keywords:

reliability problems; sojourn time of a Markov process; limit distributions of the first failure time; Poisson process### References:

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