Journal of Probability and Statistics

Volume 2019, Article ID 6480139, 18 pages

https://doi.org/10.1155/2019/6480139

## Analytically Simple and Computationally Efficient Results for the *GI*^{X}/*Geo*/*c* Queues

^{1}Department of Mathematics and Computer Science, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston Ont, Canada K7K 7B4^{2}Royal Canadian Air Force (RCAF), 86 Moreuil Wood Blvd, Petawawa Ont, Canada K8H 1A5^{3}School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Permanent Campus Argul, Jatni, Khurda-725 050, Odisha, India

Correspondence should be addressed to James J. Kim; ac.cmr@21452s

Received 21 March 2019; Revised 15 July 2019; Accepted 29 July 2019; Published 3 September 2019

Academic Editor: Alessandro De Gregorio

Copyright © 2019 Mohan L. Chaudhry et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple solution to determine the distributions of queue-lengths at different observation epochs for the model *GI*^{X}/*Geo*/*c* is presented. In the past, various discrete-time queueing models, particularly the multiserver bulk-arrival queues, have been solved using complicated methods that lead to incomplete results. The purpose of this paper is to use the roots method to solve the model *GI*^{X}/*Geo*/*c* that leads to a result that is analytically elegant and computationally efficient. This method works well even for the case when the inter-batch-arrival times follow heavy-tailed distributions. The roots of the underlying characteristic equation form the basis for all distributions of queue-lengths at different time epochs.

#### 1. Introduction

The study of discrete-time queues is relatively recent compared to its continuous-time counterpart. Over the last few decades, the field quickly gained value among queueing theorists and researchers due to the digitization of information technology, particularly in the area of signal processing devices, microcomputers, and computer networks. In analyzing the discrete-time queues, many researchers have recognized that the continuous-time models are no longer suitable to accurately measure the performance of systems in which the basic operational units are digital, such as machine cycle time, bits, and packets (see Miyazawa and Takagi [1] for details). For further details on the discrete-time models and telecommunications, one may see Bruneel and Kim [2]. In this connection, see also Takagi [3].

In discrete-time queueing theory, various techniques have been introduced by many researchers to analyze the *GI*/*Geo*/1 queues. Some of their techniques include the imbedded Markov chain, supplementary variable, semi-Markov process, birth and death process, matrix-analytic, combinatorial, and numerical methods. What is pervasive across these techniques is in their final product: The *GI*/*Geo*/1 queue has three distinct queue-length distributions at three different time epochs. Though each queue-length distribution has its own importance, they all have other purposes as well. For instance, the queue-length distribution at a pre-arrival time epoch is used to compute the actual waiting-time distribution, and in the case of queues with finite capacity, it is used to evaluate the blocking probabilities (see Chaudhry and Gupta [4]). The queue-length distribution at a random time epoch is needed to compute the virtual waiting-time distribution, whereas the queue-length distribution at an outside observer’s time epoch is used to obtain the mean waiting-time-in-queue using Little’s law. Moreover, relations between the queue-length distributions at different time epochs involve interesting mathematical derivations. For some of the earlier work on the discrete-time single-server queueing models, see Zukerman and Rubin [5].

In comparing the *GI*/*Geo*/1 queue with its continuous-time counterpart *GI*/*M*/1, the main difference between the two models is in the measurement of time. While a continuous-time model has a time parameter that is a real number, a discrete-time model has a time parameter that is an integer. In the model *GI*/*Geo*/1, the time axis is divided into individual time slots where the duration of one time slot is a single unit of time. In each individual time slot, two events may occur: arrival and departure. When an arrival occurs before a departure, it is called *GI*/*Geo*/1 with early arrival system (EAS), and when a departure occurs before an arrival, it is called *GI*/*Geo*/1 with late arrival system (LAS). Further, in *GI*/*Geo*/1 with LAS, if a server is idle and a customer arrives, then either his service can start immediately or in the next time slot. In the former case, it is known as an immediate access (IA), whereas in the latter case, it is known as a delayed access (DA). When discussing the *GI*/*Geo*/1 type queues, there exist six queue-length distributions: pre-arrival, random, and outside observer’s time epochs (three for EAS and three for LAS-DA).

The queueing model *GI*^{X}/*Geo*/1 considers the model *GI*/*Geo*/1 with a batch arrival. The earliest work on *GI*^{X}/*Geo*/1 appears to be that of Vinck and Bruneel [6] who use the complex contour integration technique. Though they provide an analytical solution to *GI*^{X}/*Geo*/1 with EAS at different time epochs, they do not provide the corresponding numerical results. Furthermore, in their work, the analysis of *GI*^{X}/*Geo*/1 with LAS-DA is missing, yet it is deemed to be an important aspect of *GI*^{X}/*Geo*/1 when considering applications in telecommunications (see Takagi [3]). In response to this, Chaudhry and Gupta [7] provide the first complete solution to *GI*^{X}/*Geo*/1 using the supplementary variable technique. One of the main contribution of their work is that they do not stop after finding the probability generation function (p.g.f.) which was a common way to conclude the analysis of a queueing model at that time, but perform the inversion of the p.g.f. Doing so enables the finding of steady-state queue-length distributions in terms of the roots of the model’s characteristic equation. This technique is referred to as the roots method.

Historically, the roots method was dismissed by some queueing theorists due to perceived difficulties in computing the roots of a model’s characteristic equation. Neuts states (see Stidham [8]) “in discussing matrix-analytic solutions, I had pointed out that when the Rouché roots coincide or are close together, the method of roots could be numerically inaccurate. When I finally got copies of Crommelin’s papers, I was elated to read that, for the same reasons as I, he was concerned about the clustering of roots. In 1932, Crommelin knew; in 1980, many of my colleagues did not….” Readers can refer to Neuts [9] for his other comments on the roots method. In the 1980s, commercial computing software such as MAPLE were not able to find the roots (they do now). To address the issue of root-finding in queueing theory, the QROOT software was developed by M. L. Chaudhry in the early 1990s and demonstrated that such roots can be found (see Chaudhry [10]). The roots method was then successfully adopted to solve a wide variety of queueing problems as noted by Janssen and van Leeuwaarden [11] who write “initially, the potential difficulties of root-finding were considered to be a slur on the unblemished transforms since the determination of the roots can be numerically hazardous and the roots themselves have no probabilistic interpretation. However, Chaudhry et al. [12] have made every effort to dispel the skepticism towards root-finding in queueing theory….”

While the roots method is simply another way of solving queueing problems, there are added advantages to it as well. Gouweleeuw [13] states “it is more efficient to use the roots method to get explicit expressions for probabilities from generating functions.” Furthermore, a recent paper by Maity and Gupta [14] compares the spectral theory approach against the roots method. Maity and Gupta [14] identify several difficulties in getting results using the spectral theory approach, an approach which may be simpler than the matrix-geometric approach as stated in several papers by Chakka (see, e.g., Chakka [15]). As well, Daigle and Lucantoni [16] state “whenever the roots method works, it works blindingly fast.”

However, while the roots method historically only dealt with queues with light-tailed distributions, more recent research by Harris et al. [17] conclude that the roots method cannot solve queues with heavy-tailed inter-arrival times. The heavy-tailed distributions constitute a class of probability distributions that are characterized by slower decay rate than the exponential or geometric distribution. When considering the heavy-tailed distributions as an inter-arrival time distribution, the consensus among some researchers is that the roots method cannot be applied due to the unique probabilistic properties of the heavy-tailed random variables. In sharing this view, Harris et al. [17] state that “the standard root-finding problem gets complicated particularly when the inter-arrival time distribution possesses a complicated non-closed form or non-analytic Laplace-Stieltjes transform (L-ST).” The same difficulty persists in discrete-time queues since the discrete heavy-tailed probability distributions such as Weibull and Log-Normal distributions do not have a closed form of probability generating function (p.g.f.). In addition, the discrete Pareto distribution, for certain values of its parameter, has an infinite mean just like the continuous Pareto distribution.

Nevertheless, the heavy-tailed distributions are useful tools in modeling real-life examples (see Willinger and Paxon [18], Leland et al. [19], Park et al. [20, 21], and Pitkow [22]). In particular, the heavy-tailed distributions (or synonymously referred to as the power, long, or fat-tailed distribution) are particularly useful when modeling the inter-arrival times of network packets and connection sizes under heavy traffic congestion (see Harris et al. [17]). While the use of heavy-tailed distributions has predominantly been in the continuous-time realm, the use of discrete heavy-tailed distributions in probabilistic modeling is evident in a number of synchronous Time Division Multiplexed (TDM) communication systems such as slotted wireless communications channels, asynchronous transfer mode, and optical packet switching (see Hernández et al. [23]). In the field of medical science and biostatistics, Para and Jan [24] have recently discovered that the Burr-type XIII and Lomax distributions are helpful in modeling a discrete data which exhibits heavy tails.

In this paper, we show that the model *GI*^{X}/*Geo*/*c* including the heavy-tailed inter-batch-arrival times can be solved through roots. In doing so, we express all queue-length distributions of *GI*^{X}/*Geo*/*c* in terms of the roots of the model’s characteristic equation.

Our analytical proof on the existence of roots (see Appendix A) and the root-finding algorithm (see Section 11) form the foundation of our contribution. In our numerical analysis, we demonstrate how the roots of the characteristic equation behave within the contour of a unit circle. In doing so, several interesting results appear as we vary the traffic intensity, the batch size distribution, inter-batch-arrival times, and the maximum batch size.

While solving the model *GI*^{X}/*Geo*/*c* through roots directly addresses some statements made by several authors, there are added benefits to it as well. Queueing theory, under the scope of algorithmic efficiency, places heavy emphasis on achieving a simpler and more efficient solution procedure as it saves memory usage and computing-time. In our method, the roots are quickly found and the queue-lengths (which are all in terms of roots) can be computed efficiently. Efficiency equates to higher engineering productivity, whereas in a purely mathematical sense, a new approach to *GI*^{X}/*Geo*/*c* through roots reveals something new in the literature. For instance, the real root from the characteristic equation of *GI*^{X}/*Geo*/*c* can also be used to approximate the asymptotic loss probability in *GI*^{X}/*Geo*/*c/S* with a moderately sized , where is the maximum capacity of the system. Also, the characteristic equation of *GI*^{X}/*Geo*/*c* can be transformed into that of *GI*^{X}/*M*/*c*. Doing so enables us to compute the roots of the characteristic equation of *GI*^{X}/*M*/*c* with an inter-batch-arrival time distribution that has nonclosed and nonexistent L-STs. Lastly, it is identified that that some of the preexisting derivations in a model with light-tailed distributions are no longer applicable if the mean inter-batch-arrival time is infinite.

The roots method significantly simplifies and extends the similar work done by others using different techniques: In solving the non-bulk-arrival queue *GI*/*Geo*/*c*, Chan and Maa [25] use the imbedded Markov chain technique to derive the queue-length distribution for EAS at the pre-arrival epoch only. In this connection, one may also see Gao et al. who use the generating function approach to solve their non-bulk queues *GI*/*Geo*/*c* [26, 27] and *GI*/*D*/*c* [28, 29]. We have avoided that approach by using the difference-equation approach. Besides, none of the above authors consider heavy-tailed inter-arrival times. Similarly, Wittevrongel et al. [30] solve using complex analysis and contour integration while only considering the EAS with no numerical results. Lastly, Chaudhry et al. [31] solve using the supplementary variable technique that consider the light-tailed inter-batch-arrival times only. The results by Chaudhry et al. [31] are in terms of iterative relations between different queue-length distributions which can be analytically laborious and numerically inefficient. Our work also responds to the comments made by several mathematicians such as Kendall, Kleinrock, and Neuts [9]. Kendall [32] made a famous remark that queueing theory wears the Laplacian curtain. Kleinrock [33] states “one of the most difficult parts of this method of spectrum factorization is to solve for the roots.”

#### 2. Model Description

Consider the model *GI*^{X}/*Geo*/*c* in which customers arrive in batches of size with a maximum size . The probability mass function (p.m.f.) of is with mean and p.g.f. . The -th inter-batch-arrival time, say , is a discrete-time period that is measured from the moment just before the *n*-th batch-arrival (say at time to the moment just before the -th batch-arrival (say at time ). Inter-batch-arrival times are independent identically distributed random variables (i.i.d.r.v.’s) distributed as which is divided into time slots. The random variable has a p.m.f. , mean , and p.g.f. . The batch size and the inter-batch-arrival times are independent. There are servers in the model where each has service time that is independent to one another and geometrically distributed as

The mean service time of one server is and the traffic intensity of the model is . In addition, let be the probability of an event that customers are served out of customers within a single time slot. The conditional probability then follows the binomial distributionwith and for or . Lastly, given that *GI*^{X}/*Geo*/*c* is a discrete-time queueing model, it has two different but related aspects (EAS and LAS-DA). Since the model is solved under the steady-state condition, the queue-length distribution of the Late Arrival System with Immediate Access (LAS-IA) is identical to that of the EAS.

#### 3. *GI*^{X}/*Geo*/*c* with EAS at a Pre-Arrival Time Epoch

In *GI*^{X}/*Geo*/*c* with EAS, the -th batch arrival occurs before the departures in the -th inter-batch arrival time. This is illustrated in Figure 1.