Research Article | Open Access
Analytically Simple and Computationally Efficient Results for the GIX/Geo/c Queues
A simple solution to determine the distributions of queue-lengths at different observation epochs for the model GIX/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 GIX/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.
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  for details). For further details on the discrete-time models and telecommunications, one may see Bruneel and Kim . In this connection, see also Takagi .
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 ). 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 .
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 GIX/Geo/1 considers the model GI/Geo/1 with a batch arrival. The earliest work on GIX/Geo/1 appears to be that of Vinck and Bruneel  who use the complex contour integration technique. Though they provide an analytical solution to GIX/Geo/1 with EAS at different time epochs, they do not provide the corresponding numerical results. Furthermore, in their work, the analysis of GIX/Geo/1 with LAS-DA is missing, yet it is deemed to be an important aspect of GIX/Geo/1 when considering applications in telecommunications (see Takagi ). In response to this, Chaudhry and Gupta  provide the first complete solution to GIX/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 ) “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  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 ). The roots method was then successfully adopted to solve a wide variety of queueing problems as noted by Janssen and van Leeuwaarden  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.  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  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  compares the spectral theory approach against the roots method. Maity and Gupta  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 ). As well, Daigle and Lucantoni  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.  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.  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 , Leland et al. , Park et al. [20, 21], and Pitkow ). 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. ). 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. ). In the field of medical science and biostatistics, Para and Jan  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 GIX/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 GIX/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 GIX/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 GIX/Geo/c through roots reveals something new in the literature. For instance, the real root from the characteristic equation of GIX/Geo/c can also be used to approximate the asymptotic loss probability in GIX/Geo/c/S with a moderately sized , where is the maximum capacity of the system. Also, the characteristic equation of GIX/Geo/c can be transformed into that of GIX/M/c. Doing so enables us to compute the roots of the characteristic equation of GIX/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  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.  solve using complex analysis and contour integration while only considering the EAS with no numerical results. Lastly, Chaudhry et al.  solve using the supplementary variable technique that consider the light-tailed inter-batch-arrival times only. The results by Chaudhry et al.  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 . Kendall  made a famous remark that queueing theory wears the Laplacian curtain. Kleinrock  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 GIX/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 GIX/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. GIX/Geo/c with EAS at a Pre-Arrival Time Epoch
In GIX/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.
Let be the number of customers in the system, including those, if any, receiving service just before the -th batch arrival (i.e., at the -th pre-arrival time epoch). The stochastic process forms a homogenous Markov chain:where is the number of customer departures during and is the size of the -th batch arrival. In considering the steady-state aspect of GIX/Geo/c with EAS, becomes as with a p.m.f. and becomes as . Let be the n-step transition probabilities of . We define and then using the imbedded Markov chain analysis, we have the following transition probabilities:where for . As becomes with p.m.f. which is defined asfor The ’s given above can be derived by replacing the single-arrival notion with a batch-arrival condition in the one-step transition probabilities of GI/Geo/c which are available in Chan and Maa . Furthermore, the same results are also available in Chaudhry et al. . Lastly, in expressing in terms of roots, the Chapman-Kolmogorov equation,is used extensively. Since (6) is a set of first-order linear difference equations with being the unknown functions to be determined, we can assume a solution of a general formwhere is a nonzero constant. Note that is the pre-arrival queue-length distribution for the overloaded servers of GIX/Geo/c with EAS since indicates that all servers are busy throughout . Substituting the above into (6) yields the characteristic equation of GIX/Geo/cwhere . Since (8) has roots inside the unit circle (see Appendix A for proof), let these roots be . Hence, for becomes -fold and is expressed aswhere (yet to be evaluated) for are the nonzero constants. In completely finding , we replace in (6) with (9) such that
The normalizing conditionis used in conjunction with (10) such that when we let in (10) to have equations that are required to solve for and . On the contrary, when we let in (10), and with the normalizing condition we have equations that are required to solve for and (the ’s when are different from those when ). Solving these equations leads to the solution
4. GIX/Geo/c with EAS at a Random Time Epoch
In the steady-state GIX/Geo/c with EAS, a pre-arrival time epoch falls just before a batch-arrival, whereas a random time epoch falls at any instant between two consecutive batch arrivals. Let be the discrete-time period measured from to . In addition, are i.i.d.r.v.’s distributed as with p.m.f. . Visual illustration of and for GIX/Geo/c with EAS are provided in Figure 2, where is the duration of time measured from the -th pre-arrival time epoch to the -th random time epoch, whereas is the duration of time measured from the -th pre-arrival time epoch to the -th pre-arrival time epoch.
Based on discrete renewal theory and the length-biased sampling phenomenon (see Chaudhry and Templeton  for definition and Kim  for similar application in continuous-time queues), if consists of time slots, then its p.m.f. can be expressed in terms of such that
As , the above expression becomes
In addition, let be the steady-state p.m.f. of the number of customers in the system at a random time epoch for GIX/Geo/c with EAS. Using the above relation, we can determine with the following Chapman-Kolmogorov equation:where are already known and ’s are the transition probabilities from Section 3, except that is replaced with . In addition, holds. The GASTA property (Geometric Arrivals See Time Averages; see Takagi ) holds if is substituted into the relation between and . Doing so leads to ; hence, due to this property, in GeoX/Geo/c.
5. GIX/Geo/c with EAS at an Outside Observer’s Time Epoch
By definition, an outside observer’s time epoch in GIX/Geo/c with EAS falls anywhere just after a batch-arrival and immediately before an event of departures. Let be the steady-state queue-length distribution of GIX/Geo/c with EAS at an outside observer’s time epoch.
In Figure 3, the -th inter-batch-arrival time period is a single time slot ; hence, the -th random time epoch and the -th pre-arrival time epoch coincide at (this can be checked by letting in Figure 2).
In Figure 3, if there are customers in the system at an outside observer’s time epoch (with probability ) and there are customers in the system at a random time epoch (with probability ), then there must be an event of customer departures (with probability ) between an outside observer’s time epoch and a random time epoch. Based on this notion, we form the relation
Since (16) is an set of first-order linear difference equations with being the unknown functions to be determined, we assume that for is a geometric sum that consists of the same roots of (8) but with different constant coefficients (hence they are unknown). Let these unknown constant terms be , then we havewhich is the p.m.f. of the steady-state number of customers in the system for the overloaded servers of GIX/Geo/c with EAS at an outside observer’s time epoch. By replacing in (16) with the above geometric sum, we havewith the normalizing condition
When we let in (18), and with (19) it leads to equations that are required to solve for and . On the contrary, when we let in (18), and with (19) it leads to equations that are required to solve for and (the ’s when are different from those when ). Solving these equations leads to an expression of for as
6. GIX/Geo/c with LAS-DA at a Pre-Arrival Time Epoch
In GIX/Geo/c with LAS-DA, the -th batch arrival occurs after the departures in the -th inter-batch arrival time. This is illustrated in Figure 4.
Let be the number of customers in the system, including the ones, if any, in service just before the -th batch-arrival. The random variable becomes and has a p.m.f. as we assume the steady-state condition . Before proceeding to find , it is worth noting that can be found independently of (as done by Chaudhry and Kim  in solving the model GIX/Geo/1). However, we leverage from Section 3 to determine for the purpose of demonstrating continuity in our solution procedure. Let be the p.m.f. of there being customers in the system GIX/Geo/c with LAS-DA at time , and let be the p.m.f. of there being customers in the system GIX/Geo/c with EAS at time . Figure 5 illustrates that in EAS, the -th batch arrives at the beginning of the time slot , whereas in LAS-DA, the same -th batch would arrive at the end of the time slot .
If up to customers depart during , then we can develop a relation between and as
As a remark, indicates that if no customers depart during , then with probability , whereas any customer departures during result in with probability . By letting and , so that we have the steady-state queue-length distributions, we have
Since (22) is a set of first-order linear difference equations with being the unknown functions to be determined, we can make a similar assumption as the one made in Sections 3 and 5, and letbe the p.m.f. of the steady-state number of customers in the system for the overloaded servers of GIX/Geo/c with LAS-DA at a pre-arrival time epoch. The ’s in (23) are the unknown nonzero constants. As a remark, (23) differs from (9) since we have in (23) but in (9). This difference is due to the property of GIX/Geo/c with LAS-DA: assume that there are initially no customers in the system and then a batch of customers arrives after some inter-batch-arrival time. If the next inter-batch-arrival time is one time slot (with probability ), then no customers access any of the servers, whereas if the inter-batch-arrival time is at least two time slots (with probability ), then customers start accessing the idle servers at the beginning of the second time slot. This property results in the definition of the overloaded servers of GIX/Geo/c with LAS-DA to come into effect when . To solve for and s, (22) is alternatively expressed as
We also consider the normalizing conditionsuch that when we let in (24), and with (25) it leads to equations that are required to solve for and . On the contrary, when we let in (24), and with (25) it leads to equations that are required to solve for and (the ’s when are different from those when ). Solving these equations leads to as
7. GIX/Geo/c with LAS-DA at Random and Outside-Observer’s Time Epochs
Let and be the p.m.f.’s of the number of customers in system in GIX/Geo/c with LAS-DA at random and outside observer’s time epochs, respectively. In LAS-DA, an outside observer’s time epoch falls in a time interval that begins just after a departure and immediately before a batch-arrival. Based on this notion, is the same as . In addition, we havewhere is available in (20).
8. p.m.f. of Waiting-Time-in-Queue and Performance Measures
Let be the amount of time an arbitrary customer of an incoming batch spends in queue upon batch-arrival and until entering service. As mentioned by Chaudhry and Templeton , if the position of an arbitrary customer of an incoming batch is , then its p.m.f. is . Since the system capacity of the GIX/Geo/c queues is infinite, the waiting time distribution remains the same in both LAS-DA and EAS (see Hunter  and Chaudhry and Gupta ). Let the p.m.f. of be . As stated by Chaudhry et al. , iswhere .
9. Performance Measures
The model GIX/Geo/c has various performance measures. Since the distributions are known, various moments can be calculated. In particular, denote as the mean number of customers in the system and as the mean number of customers in queue, both at an outside observer’s time epoch. and can be found in terms of such that
The mean waiting-time in the system and the mean waiting-time-in-queue can also be found using Little’s law:
In addition, can be found separately by using the result in Section 8, which is . Furthermore, it can be shown that the average number of idle servers can be found using the expression
The left-hand side of (32) determines the average number of idle servers using the queue-length-distribution at an outside observer’s time epoch. The right-hand side of (32) determines the same number, except that it is independent of (hence independent of roots).
10. Approximation of Loss Probability in GIX/Geo/c/S Using the Loss Formula
In finite-buffer queues, the loss probability is defined as the probability of a customer being rejected from the system. In the case of finite-buffer queues with bulk-arrivals, an incoming batch of customers may be partially or totally rejected, depending on how close the number of customers in the system is to the finite-buffer immediately prior to a batch-arrival. Gouweleeuw  and Gouweleeuw and Tijms  approximate the loss probability of a wide range of queueing models with general input such as BMAP/G/1/S and GI/G/1/S. In particular, while discussing the discrete-time model GIX/G/c/S, Gouweleeuw  states that “… an enormous amount of methods to calculate or approximate the loss probability of a customer in such a queueing system is present in the literature…. An overview is presented of those methods which explore the relation between the finite-buffer queue and its infinite-buffer counterpart.” From this statement, it is understandable that Gouweleeuw’s  main effort is in presenting an efficient and accurate approximation of loss probabilities using the steady-state queue-length distribution of the infinite-buffer counterpart.
The asymptotic loss probability is the loss probability as becomes a large integer. In discussing the asymptotic loss probability of GIX/M/c/S, Kim and Choi  express the asymptotic loss probabilities of GIX/M/c/S in terms of the roots of the model’s characteristic equation. They prove that there is exactly one positive real root (say ) that has the largest modulus among the roots. Kim and Choi  go further to claim that with its corresponding constant coefficient can accurately approximate the asymptotic loss probability, even for a moderately sized . By this claim, we can also accurately approximate the loss probability of GIX/Geo/c/S (both total and partial rejections). Depending on whether the model involves EAS or LAS-DA, we can easily identify from (8) as well as and from Sections 3 and 6, respectively, and approximate the asymptotic loss probabilities of GIX/Geo/c/S.
11. GIX/Geo/c Involving Heavy-Tailed Inter-Batch-Arrival Times
The heavy-tailed distributions distinguish themselves from the light-tailed distributions by having a significantly slower rate of decay. In GIX/Geo/c, a slow decaying arrival-time distribution renders a very lengthy (or an infinite) mean inter-batch-arrival time that equates to a very small (or a zero) arrival rate . The arrival rate is directly proportional to , and as , the roots of the characteristic equation of GIX/Geo/c tend towards the origin (see, e.g., Table 1) which was a concern to some researchers.
The heavy-tailed distributions are believed to impose another challenge on the roots method due to the nonclosed form of p.g.f.’s. While the light-tailed distributions have closed form of (see Table 2), this is not the case for Weibull, Standard Log-Normal (SLN), and Pareto distributions. Moreover, directly solving (8) with an that follows the Pareto distribution with will consume a lengthy computing time (or not compute at all) due to the infinite series of a p.m.f. that decays at an extremely slow rate.