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Structural perturbation analysis of a single server queue with breakdowns. (English) Zbl 1193.60105
Summary: This paper studies perturbations of the single server queue with breakdowns, where the perturbations consist of introducing dependent breakdowns. Problems that are modeled by queueing models with breakdowns and repairs are often very complicated, and they are resolved only through approximations. Therefore, it is very important to justify these approximations and to estimate the resultant error. For this purpose, we use the strong stability method to approximate the characteristics of the $M/G/1$ queue with dependent breakdowns by those of the $M/G/1$ queue with classical (constant) breakdowns. This latter queue is simpler and more exploitable in practice. Thus, we prove the stability conditions and next obtain stability inequalities with exactly computing of the constants. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results an algorithm allowing one to verify the approximation conditions and to provide the made numerical error. The accuracy of the approach is evaluated by comparison with true distance values.
##### MSC:
 60K25 Queueing theory 68M20 Performance evaluation of computer systems; queueing; scheduling 90B22 Queues and service (optimization) 60J10 Markov chains (discrete-time Markov processes on discrete state spaces)