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A 2-class maintenance model with dynamic server behavior. (English) Zbl 1434.90045

Summary: We analyze a 2-class maintenance system within a single-server polling model framework. There are \(C+f\) machines in the system, where \(C\) is the cap on the number of machines that can be turned on simultaneously (and hence, be at risk of failure), and the excess \(f\) machines comprise a maintenance float which can be used to replace machines that are taken down for repair. The server’s behavior is dynamic, capable of switching queues upon a machine failure or service completion depending on both queue lengths. This generalized server behavior permits the analysis of several classic service policies, including preemptive resume priority, non-preemptive priority, and exhaustive. More complicated polices can also be considered, such as threshold-based ones and a version of the Bernoulli service rule. The system is modeled as a level-dependent quasi-birth-and-death process and matrix analytic methods are used to find the steady-state joint queue length distribution, as well as the distribution for the sojourn time of a broken machine. An upper bound on the expected number of working machines as a function of \(C\) is derived, and Little’s Law is used to find the relationship between the expected number of working machines and the expected sojourn time of a failed machine when \(f=0\) or \(f \ge 1\). Several numerical examples are presented, including how one might optimize an objective function depending on the mean number of working machines, with penalty costs attributed to increasing \(C\) or \(f\).

MSC:

90B25 Reliability, availability, maintenance, inspection in operations research
60K25 Queueing theory (aspects of probability theory)
60J28 Applications of continuous-time Markov processes on discrete state spaces

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