On the discrete-time Geo/G/1 queue with randomized vacations and at most J vacations

Abstract

a b s t r a c tThis paper examines a discrete-time Geo/G/1 queue, where the server may take at most J À 1 vacations after the essential vacation. In this system, messages arrive according to Bernoulli process and receive corresponding service immediately if the server is available upon arrival. When the server is busy or on vacation, arriving messages have to wait in the queue. After the messages in the queue are served exhaustively, the server leaves for the essential vacation. At the end of essential vacation, the server activates immediately to serve if there are messages waiting in the queue. Alternatively, the server may take another vacation with probability p or go into idle state with probability (1 À p) until the next message arrives. Such pattern continues until the number of vacations taken reaches J. This queueing system has potential applications in the packet-switched networks. By applying the generating function technique, some important performance measures are derived, which may be useful for network and software system engineers. A cost model, developed to determine the optimum values of p and J at a minimum cost, is also studied.

Study on the Departure Process of Discrete-TimeGeo/G/1Queue with Randomized Vacations

“…Such a modified vacation discipline has potentially applications in practical systems [13], for example, in some stochastic production and inventory control systems such as production to orders. This is a continuation of work by Wang et al(2011) [16] and Luo et al (2013) [17] where they study the queue size distribution for / /1 with randomized vacation and at most vacations by using different methods, respectively. Instead of studying the queue size distribution studied in [16,17], this paper considers another theme-the departure process in that discrete-time model.…”

“…This is a continuation of work by Wang et al(2011) [16] and Luo et al (2013) [17] where they study the queue size distribution for / /1 with randomized vacation and at most vacations by using different methods, respectively. Instead of studying the queue size distribution studied in [16,17], this paper considers another theme-the departure process in that discrete-time model. The investigation of departure process in a queueing system is primarily motivated by the need to analyze queueing network models, in which the departure process of an upstream queue is the arrival process of the downstream queue.…”

“…This pattern does not terminate until the server has taken successive vacations. Wang (2010) [15] and Wang et al (2011) [16] introduce the randomized vacation policy to the discrete-time / /1 queue. Such a modified vacation discipline has potentially applications in practical systems [13], for example, in some stochastic production and inventory control systems such as production to orders.…”

Analysis of a Single Server Retrial Queue with Server Vacation and Two Waiting Buffers Based on ATM Networks

“…Over the past decades, Geo/G/1 queues and Geo/G/1 retrial queues with a variety of vacation policies have been well developed, such as Takagi [2], Zhang and Tian [16], Gao and Wang [17], Wang [18], Wang et al [19], Yue and Zhang [20], Luo et al [21], and Wang [22,23] (not exhaustive list). However, most of these vacation queues deal with the case of single waiting queues; i.e., there is only a waiting queue in the service area in the Geo/G/1 queue or an orbit waiting queue in the Geo/G/1 retrial queue.…”

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

“…A comprehensive review of vacation queues can be found in the surveys of Doshi [5] and Ke et al [12], and the monographs of Takagi [24] and Tian and Zhang [25]. In the recent past, remarkable contributions on discrete-time queueing systems with vacations have been made by many authors (see e.g., Zhang and Tian [32], Tang et al [23], Wang et al [27], Wang [28], Goswami and Mund [10], Laxmi and Jyothsna [20], and Gao and Wang [11]).…”