A New and Pragmatic Approach to the GI X /Geo/c/N Queues Using Roots
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A New and Pragmatic Approach to the GIX/Geo/c/N Queues Using Roots J. J. Kim 1
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& M. L. Chaudhry & V. Goswami & A. D. Banik
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Received: 12 March 2019 / Revised: 25 April 2020 Accepted: 29 October 2020 # Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract
A simple and complete solution to determine the distributions of queue lengths at different observation epochs for the model GIX/Geo/c/N is presented. In the past, various discrete-time queueing models, particularly the multi-server bulk-arrival queues with finite-buffer have been solved using complicated methods that lead to results in a non-explicit form. The purpose of this paper is to present a simple derivation for the model GIX/Geo/c/N that leads to a complete solution in an explicit form. The same method can also be used to solve the GIX/Geo/c/N queues with heavy-tailed inter-batch-arrival time distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue lengths at different time epochs. All queue-length distributions are in the form of sums of geometric terms. Keywords Queueing . multi-server . discrete-time . bulk-arrivals . finite-buffer . heavy-tailed . and roots
1 Introduction A queue forms whenever and wherever demand exceeds supply. It is for this reason that a study of queues naturally emerged as a practical field of study known as the queueing theory. Among many different type of queues, the multi-server bulk-arrival queues are particularly useful in
* J. J. Kim [email protected] M. L. Chaudhry [email protected] V. Goswami [email protected] A. D. Banik [email protected] Extended author information available on the last page of the article
Methodology and Computing in Applied Probability
modeling cases where a group of customers joins a queue in a system that consists of one or more simultaneous servers. Queues of this type are easily seen in our daily lives: A barbershop, supermarket, hospital admissions, seaports, and many more. In particular, in the field of information transmission systems, signal processing, and engineering applications, extensive studies have been carried out on the discrete-time multi-server bulk-arrival queues (see Miyazawa and Takagi (Miyazawa and Takagi 1994)). The GIX/Geo/c queues are often the model of choice when measuring the performance of various digital technologies. Such queues are better suited than their continuous time counterparts when evaluating performance measures such as packet loss and delay in digital communication networks because of the clock-driven operation of those systems. One may see Goswami and Samanta (Goswami and Samanta 2007) for further examples. The model GIX/Geo/c consists of two systems: early arrival system (EAS) and late arrival system with delayed access (LAS-DA). For each system the queue-length distribution can be observed at three different time epochs: pre-arrival, random, and outside observer. Though each time epoch has its own importance, they all have other purposes as well. For instance,
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