Transient Behavior of the MAP/M/1/N Queuing System

Abstract

This paper investigates the characteristics of the MAP/M/1/N queuing system in the transient mode. The matrix method for solving the Kolmogorov equations is proposed. This method makes it possible, in general, to obtain the main characteristics of the considered queuing system in a non-stationary mode: the probability of losses, the time of the transient mode, the throughput, and the number of customers in the system at time t. The developed method is illustrated by numerical calculations of the characteristics of the MAP/M/1/3 system in the transient mode.

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

“…The transition time is the time during which the QS goes into stationary mode. In accordance with [22], the time of the transition mode is determined by the smallest absolute value of the real part of the pole of the state probability images:…”

“…Here ∀α j ∈ Γ: (Γ = α j , α j α min =⇒ α j = α min ) and k > 0, k ∈ R. The value of k is selected based on the formulation of a specific problem. It was shown in [22] that the transition mode can be considered completed when k = (3 ÷ 5).…”

“…An analysis of the stability of non-stationary Markov processes with continuous time, describing the functioning of the main classes of QSs with non-stationary input flows, including those varying according to a sinusoidal law, was carried out in [10,13,18,19]. In [21][22][23], the main performance characteristics of single-line and multi-line QSs with Poisson and correlated flows in transient mode are analyzed. This work studies the non-stationary performance characteristics of a two-phase QS with a limitation on the total buffer size of the two phases.…”