Exact formulas for Markov retrial queues controlled by hysteresis strategies
DOI:
https://doi.org/10.17721/1812-5409.2024/1.4Keywords:
retrial queue, steady-state probabilities, quasi-birth-and-death process, optimal controlAbstract
This paper examines the Markov model for multiserver retrial queues with an input flow rate that depends on the number of calls in orbit and is controlled by hysteresis strategies. The system consists of n identical servers. If an incoming call finds a free server, it occupies it and is served for an exponentially distributed time. If all servers are busy upon arrival, the call joins the orbit and returns for service after a random period of time. The system's service process is described by a three-dimensional continuous-time Markov chain. We first establish the conditions for the existence of a stationary regime. Next, we provide exact vector-matrix formulas for steady-state probabilities. Our investigative technique is based on approximating the input system by the system with a truncated state space and contains an effective computational algorithm. For n=1 and n=2, the representations can be simplified to closed scalar formulas for stationary probabilities using the model parameters. These results are consistent with earlier works. To demonstrate practical significance, we present a multi-objective problem of maximizing total income generated by the system. Considering the economic nature of the problem, we utilized the method of linear convolution of criteria. The obtained representations enable us to determine an optimal strategy that maximizes the objective function. Pages of the article in the issue: 26 - 32 Language of the article: Ukrainian EnglishReferences
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