Mean Characteristics of Markov Queueing Systems

Markov models of the queueing theory described by birth-death processes (BDP) have been investigated and applied for a long time (see, e.g., [1, 2]). At the same time, creation of new methods to study BDP and enlargement of the range of the solved problems invoked appearance of a new series of works [3–11]. More realistic models of queueing systems described by nonstationary Markov chains are actively studied in B.V. Gnedenko’s works, early [12, 13] and the following ones [14–18]. As a rule, it is impossible to design the limiting mode and find explicit formulas for state probabilities of these models; therefore, the main interest is focused on questions of characteristic approximation for these systems [19–22]. The interest in nonstationary Markov models of queueing systems has increased in the last years after the creation of new investigation methods (e.g., [23–27]). Two main characteristics, limiting mean and double mean, are introduced and studied for the processes with periodic rates in [27]. In this paper, we study a classic queueing system (QS) Mn(t)/Mn(t)/S with rates of customer arrival and service close to periodic, obtain estimates of the means, indicate the technique for their approximate computation, and consider several examples. The work is an expanded and supplemented variant of the preliminary report [28].