Methods for Analysis of Multi-Channel Queueing System with Instantaneous and Delayed Feedbacks
- PDF / 230,542 Bytes
- 13 Pages / 594 x 792 pts Page_size
- 98 Downloads / 153 Views
METHODS FOR ANALYSIS OF MULTI-CHANNEL QUEUEING SYSTEM WITH INSTANTANEOUS AND DELAYED FEEDBACKS V. S. Koroliuk,1 A. Z. Melikov,2 L. A. Ponomarenko,3 A. M. Rustamov4
UDC 519.872
Abstract. The authors propose a mathematical model for a multi-channel queueing system with feedback in which one part of calls instantaneously enters the system for repeated service and the other part either retries in some random time or finally leaves the system. The behavior of the serviced calls is randomized. Both exact and asymptotic methods are developed to calculate the characteristics of the proposed model. The results of numerical experiments are presented. Keywords: multi-channel queueing system, feedback, quality of service. INTRODUCTION Classical models of queueing systems (QS) do not take into account feedback. This substantially reduces the adequacy of classical QS models used for the mathematical analysis of real call handling processes where a part of calls already passed the stage of service feedbacks for various reasons, for example, depending on the quality of service, holding time, etc. For example, in multi-agent systems, even a call that have obtained satisfactory service may arrive again for repeated service by these agents [1–3]. Two types of models of QS with feedback are distinguished: models without orbit and models with orbit. In models without orbit, some calls instantaneously return after obtaining primary service, while in models with orbits, calls “think over” a decision before the return. According to these properties, models without orbit are called models with instantaneous feedback and models with orbit are called models with delayed feedback. Models of QS with feedback are poorly studied. After the classical studies by Takacs [4, 5], where the generating functions method is used to analyze two-dimensional Markov models of one-channel QS with unbounded queue and infinite orbit volume, models of QS with feedback have not drawn attention of contributors for a long time. Models without orbit are analyzed in [6–8]. For example, models with infinite number of channels are investigated in [6, 7], where the method of generating functions is used to calculate the distribution (stationary and nonstationary) of primary, retrial, and total flows. The model of a one-channel QS with a bounded queue is analyzed in [8], where the matrix-geometrical Neuts method is applied to calculate the stationary distribution of the corresponding four-dimensional Markov chain (MC) [9]. (Hereinafter, by stationary distribution of a model, in particular, MC, we understand the set of stationary probabilities of states of the described model.) The studies [4–8] assume that the times of handling of primary calls and retrials are identical, and the probabilities of leaving the system and retrial are constant (i.e., do not depend on the state of the system). These assumptions limit the fields of application of the proposed models. In all the above-mentioned studies, models of QS without orbit and with orbit are studied separately, which a
Data Loading...
