Transient Behavior of a Queueing Model with Hyper-Exponentially Distributed Processing Times and Finite Buffer Capacity

Abstract

In the paper, a finite-capacity queueing model is considered in which jobs arrive according to a Poisson process and are being served according to hyper-exponential service times. A system of equations for the time-sensitive queue-size distribution is established by applying the paradigm of embedded Markov chain and total probability law. The solution of the corresponding system written for Laplace transforms is obtained via an algebraic approach in a compact form. Numerical illustration results are attached as well.

Transient Behavior of a Two-Phase Queuing System with a Limitation on the Total Queue Size

“…A similar situation arises when studying new-generation 5G and 6G networks [9]. Due to the relevance of this problem, in recent years the number of works devoted to the study of the transient operating mode of QSs and their nonstationary Markov models has increased [10][11][12][13][14][15][16][17][18][19]. One of the first works where such a problem for a two-phase QS with a Poisson input flow, an infinite buffer in the first phase, and a zero buffer in the second phase was considered is the paper of 1967 [20].…”

“…By substituting (27) and expression (11), we can find the probabilities of states of a two-phase QS in the transition mode under given initial conditions. These expressions make it possible to calculate and analyze the performance indicators of the system under consideration at an arbitrary moment of time t in both transient and stationary modes: the time before the system enters stationary mode, the probability of losses, throughput, and the number of requests served in each phase.…”

“…One of the first works where such a problem for a two-phase QS with a Poisson input flow, an infinite buffer in the first phase, and a zero buffer in the second phase was considered is the paper of 1967 [20]. In further works, more complex systems are studied, such as systems with phase-type service time [11,12] and various types of tandem networks [14,15]. It should be noted that in most of these works, the authors do not provide final expressions that allow analyzing the performance characteristics of the QS in the transient mode, but use ready-made numerical methods of existing software packages.…”