Recurrence Relations for Two-Channel Queuing Systems with Erlangian Service Time
- PDF / 136,956 Bytes
- 8 Pages / 594 x 792 pts Page_size
- 27 Downloads / 142 Views
RECURRENCE RELATIONS FOR TWO-CHANNEL QUEUING SYSTEMS WITH ERLANGIAN SERVICE TIME
Yu. V. Zhernovyi
UDC 519.21
Abstract. We propose a method to investigate M / E s / 2 / m and M / E s / 2 / ¥ queueing systems including the case of random dropping of customers. Recurrence relations are obtained for computing the stationary distribution of the number of customers in the system and its steady-state characteristics. The developed algorithms are tested on examples using simulation models constructed with the help of the GPSS World tools. Keywords: two-channel queueing systems, Erlangian service times, random dropping of customers, fictitious phase method, recurrence relations, steady-state characteristics. INTRODUCTION To analyze queueing systems with Erlangian distributions, in particular systems M / E s / n / ¥ , the fictitious phase method developed by A. K. Erlang [1] is applied. For Erlangian distribution of service time of order s, it is supposed that each customer sequentially passes s service phases whose durations are exponentially distributed with parameters m1 , m 2 , K , m s . Taking the phases into account requires appropriate states to be fixed and makes description of queueing system with distributions of phase type more awkward. Immediate solution of the system of equations for steady-state probabilities of states may appear impossible because of large dimension of the matrix of coefficients of the system. The algorithmic approach is most expedient, which assumes obtaining the solution of systems of equations in the form of either recurrence formulas or matrix-recurrence relations and algorithms. The matrix-geometrical approach proposed in [2–4] is based on preliminary analysis of the system of equations being solved; as a result, it is divided into subsystems of smaller order, related to each other by recurrence relations. If the matrices of coefficients of these subsystems are invertible, then due to their recurrent coherence it is possible to obtain recurrence formulas in matrix or scalar form to solve the original system. The matrix recurrence-iterative method developed in [5–7] to solve vector-matrix equations of balance of transitions between states for queueing systems with phase-type distributions has a number of shortcomings: the conditions of convergence of iterations leads to additional constraints imposed on the matrices of transitions between states, and presence of iterations increases the calculation time. The purpose of the present paper is to develop recurrence algorithms to calculate stationary distribution of the number of customers in queueing systems M / E s / 2 / m and M / E s / 2 / ¥ , both standard and with random dropping of customers. Random dropping of customers is used in queueing systems to prevent overloads: each arriving customer can be dropped with certain probability dependent on the queue length at the moment of customer’s arrival even if the buffer is not filled completely yet [8–12]. The proposed method is based on direct recurrence relations that follow immediately f
Data Loading...
