This paper fits into a greater scientific effort which aims at finding explicit analytic solutions for the joint probability distribution (or probability generating function) of the numbers of customers in a system of two coupled discrete-time queues. Various instances of such systems have been studied before, both differing in the cause of the coupling between the two queues or in the scientific perspective taken in the study.

With no claim on completeness, we mention a number of possible causes for the presence of coupling between queues. A first cause may be that the arrival streams into the queues are mutually interdependent or state-dependent, i.e., dependent on the system contents, i.e., the numbers of customers present in the queues. Mutual dependence between arrivals occurs, for instance, in the context of communications networks, where the nodes of the network contain switching systems which have to forward digital packets from many different origins to many different destinations. In such switches, each destination has (at least, conceptually) its own dedicated buffer to temporarily store arriving packets, and, since packets destined to one destination do not enter the output buffer associated to another destination, the arrivals within such output buffers are mutually correlated. Buffered slotted switches were studied, e.g., in [1,2,3,4,5,6]. State-dependence of arrivals occurs, for instance, in join-the-shortest-queue systems, where arriving customers adapt their behavior at the entrance of the system to the system contents upon arrival; see, e.g., [7,8,9]. More conceptual studies of queues with interdependent arrivals include [5,6,10,11].

Another (major) cause of coupling may be that the queues of the system have to share the same service facilities. This situation occurs, for instance, in polling systems, where one server periodically visits multiple customer queues to serve a number of customers and then go to the next queue; various variants of polling systems have been studied quite intensively in the past (see, e.g., [12,13,14,15,16,17,18,19]). Sharing of servers also occurs in so-called alternating service systems, where one server is allocated for alternating random durations of time to either of the two queues, regardless of the states of these queues, (see, e.g., [11,20,21,22,23,24,25,26,27,28]), or in priority queues, where one common service facility gives preferential service to one class of customers over the other class(es) of customers; a large body of research results is available on various types of priority queueing systems (see, e.g., [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]). Similar ideas are also implemented in so-called (generalized) processor sharing (GPS) systems, whereby the service facility is randomly allocated to multiple queues according to preset weights; as opposed to randomly alternating service systems, however, GPS-systems usually allow the server to deliver service to customers of a queue to which it is not allocated when the queue to which it is allocated is empty, thus making the system work-conserving. In fact, GPS-systems can also be viewed as systems with alternating priorities; some papers dealing with GPS-systems are [25,47,48,49,50,51]. We should also mention serve-the-longest-queue systems, where, upon a service completion, a server can autonomously decide to give preference to the queue that contains the largest number of customers; see, e.g, [52,53]. Recently, some authors have examined the combined join-the-shortest-queue and serve-the-longest-queue scenario; see [54,55].

A third possible cause of coupling in two-queue systems can be that (part of) the output stream of one queue constitutes (part of) the input stream into the other queue, such as in the context of tandem queues (see, e.g., [45,56,57]), or, more generally, in a network environment.

As far as the scientific perspective taken by various authors in the literature is concerned, we see a main difference between considering the involved (two-queue) queueing system as the basic concept of the study, where the determination of the joint (or, total) system-content distribution of both queues, the overflow probabilities, the customer delays, etc. is the main objective, versus a more fundamental, mathematically-oriented point of view, whereby the underlying random walks that model the system contents of both queues are the basic concepts of the study (see, e.g., [58,59,60,61,62]). Usually, the aim of such papers is to shed more light on the structural properties of these random walks required to admit elegant solutions. Moreover, the involved random walks are often of nearest-neighbor type, which is rather restrictive in a queueing context, and the structure of their transition probabilities may be quite arbitrary and not necessarily reflect the behavior of a queueing system.

The present paper does not take the mathematical study of the random walk that models the two-queue system explicitly as a major point of interest, but rather concentrates on the explicit determination of the joint pgf of the two system contents in the two queues of the system. Specifically, we consider a conceptually very simple system of two coupled parallel discrete-time queues. The queues are named queue 1 and queue 2, both have their own dedicated server and infinite storage capacity. Customers arriving to queue 1 and to queue 2 are referred to as type-1 and type-2 customers, respectively. The service times of the customers are deterministically equal to one time slot, regardless of the customer type. New customer arrivals of both types occur independently from slot to slot, but are possibly type-interdependent within a slot. This is the only source of coupling in this model. Earlier studies of various instances of this type of two-queue system include [1,2,3,4,5,6,10,11,63].

It is well-known that, in general, determining the steady-state joint pgf $U(z_1,z_2)$ of the system contents in a system of two coupled queues is a formidable task, because it requires the solution of a possibly complicated, nonlinear functional equation for $U(z_1,z_2)$, which contains the unknown boundary functions $U(z_1,0)$ and/or $U(0,z_2)$. In many previous papers, the analysis consists of first determining these boundary functions through various complex-analysis techniques, upon which $U(z_1,z_2)$ can be computed from the functional equation by subsequent substitution of the expressions found for $U(z_1,0)$ and/or $U(0,z_2)$. In this paper, we use a different, purely algebraic, technique, which can be best described as a two-step process: first, we make an educated guess of the solution for $U(z_1,z_2)$, and, next, we prove that the proposed $U(z_1,z_2)$ indeed satisfies the functional equation. Of course, in this approach, making an educated guess of the solution is crucial. In fact, this step is essentially a process of trial and error, based on the intuition gained from the preliminary study of a large number of simple special cases in earlier papers.

For the specific coupled-queues system considered in this paper, explicit results have been obtained thus far only in a number of isolated cases, for very specific choices of the arrival pgf $A(z_1,z_2)$ (see, e.g., [5]). Furthermore, these special cases are of a rather simple nature: either the arrivals of both types should be mutually independent, or the two queues should receive identical numbers of arrivals in each slot, or one of both queues should receive no more than one single arrival per slot, implying that in this queue no accumulation of customers occurs. Some initial indications to extend the class of “solvable” arrival pgfs $A(z_1,z_2)$ were also given in [5], but the extensions are limited.

In this paper, we introduce a multi-parameter, generic class of arrival pgfs $A(z_1,z_2)$, for which we succeed to explicitly determine the joint pgf $U(z_1,z_2)$, using the algebraic approach described above. By making specific choices for the many parameters of the model, we also define three interesting subclasses of arrival pgfs which lead to even more explicit solutions. We find that for arrival pgfs of the classes considered in this paper, the bivariate joint system-content pgf $U(z_1,z_2)$ has a denominator which is a product, say $r_1\left(z_1\right)r_2\left(z_2\right)$ of two univariate functions. This property allows a straightforward inversion of the pgf $U(z_1,z_2)$ by means of an inversion technique we developed in a previous paper [11], resulting in a pmf $u(m,n)$ which can be expressed as a finite linear combination of bivariate geometric terms. We observe that our generic model encompasses most of the previously known results as special cases. In fact, it was by studying these special cases that we developed the intuition needed to be able to identify the class of arrival pgfs introduced in this paper.

The rest of this paper is organized as follows. In Section 2 we introduce the detailed mathematical model of the system under study and establish a functional equation for the joint pgf $U(z_1,z_2)$. The solution of this functional equation is, in fact, the main purpose of the paper. Section 3 defines the generic class of arrival pgfs $A(z_1,z_2)$ that will be studied in this paper. In Section 4, we present and prove the main result of the paper in the form of Theorem 1 which gives an explicit expression for the joint system-content pgf $U(z_1,z_2)$ associated to the joint arrival pgf $A(z_1,z_2)$ defined in Section 3. Section 5 defines three interesting subclasses, named A, B and C, of the generic class of arrival pgfs $A(z_1,z_2)$ defined in Section 3, and establishes even more explicit formulas for the associated system-content pgfs $U(z_1,z_2)$ in these cases, in the form of three corrolaries, also named A, B and C, of Theorem 1. In Section 6, we consider several instances of subclasses A, B and C, whereby specific choices are made for the various parameters and functions appearing in the formulations of Corrolaries A, B and C. Section 7 discusses a fundamental method to invert the system-content pgf $U(z_1,z_2)$, i.e., to determine the pmf $u(m,n)$ from the pgf $U(z_1,z_2)$, and illustrates this techniques by means of specific examples within subclasses A, B and C. Finally, we state some concluding remarks in Section 8.

We define the random variables $a_{1,k}$ and $a_{2,k}$ as the numbers of type-1 and type-2 arrivals, respectively, during slot k. Their joint probability mass function (pmf) $a(i,j)$ and probability generating function (pgf)$A(z_1,z_2)$ are indicated as which are independent of k. The (marginal) pgfs of $a_{1,k}$ and $a_{2,k}$ are given by respectively. The mean number of arrivals of type i per slot is denoted as ${\lambda }_i\triangleq A_i^{\prime }\left(1\right)$. A graphical representation of the system under study is shown in Figure 1.

Let $u_{1,k}$ and $u_{2,k}$ indicate the system contents, i.e., the total numbers of customers present, in queue 1 and queue 2, respectively, including the customer(s) in service, if any, at the beginning of slot k. We indicate their joint pgf as Furthermore, let $q_{1,k}$ and $q_{2,k}$ indicate the queue contents, i.e., the numbers of waiting customers, in queue 1 and queue 2, respectively, excluding the customer(s) in service, if any, at the beginning of slot k. We indicate their joint pgf as It is not difficult to see that the following relationships exist between the system contents and the queue contents: where we have introduced the notation ${\left(x\right)}^{+}$ to indicate the quantity $max(0,x)$.

The main purpose of the paper is to analyze the steady-state behavior of the queueing system under study, i.e., we are interested in determining the steady-state joint pgfs of the two system contents and/or queue contents, provided that such a steady state exists. Specifically, we wish to study the following limit functions: if they exist. A steady state exists if and only if both queues are stable, i.e., receive, on average, less customers per slot than they can serve, i.e., if and only if the following stability conditions are fulfilled: As mentioned in e.g., [5,6], the evolution of the system contents is described by the following system equations: Using standard z-tranform techniques, the equations (8) can be translated into one corresponding transform equation between the joint pgfs $U_k(z_1,z_2)$ and $U_{k+1}(z_1,z_2)$, by using definition (3). Assuming the system reaches a steady state, i.e., assuming the stability conditions (7) are met, letting the time parameter k go to infinity, and using the definitions (3) and (6), the latter transform equation translates into the following functional equation for the steady-state system-content pgf $U(z_1,z_2)$: where the unknown function $L(z_1,z_2)$ is defined as and the kernel $K(z_1,z_2)$ is given by

Although, in general, the functional Equation (9) is hard to solve for $U(z_1,z_2)$, it is fairly easy to derive explicit expressions for the marginal pgfs $U(z_1,1)$ and $U(1,z_2)$ of the individual system contents in queues 1 and 2, by choosing either $z_2=1$ or $z_1=1$ in (9), because such choices greatly simplify the L-function. As a result, we then obtain Invoking the normalization condition $U(1,1)=1$ yields $U(0,1)=1-{\lambda }_1$ and $U(1,0)=1-{\lambda }_2$. The expressions in (12) are very well-known in the context of discrete-time queueing theory; see, for instance [64]; they will be very useful further in this paper.

We now turn our attention to the queue contents. Using (5) in (8), we readily get Transforming these relationships to pgfs, we obtain, on account of the definitions (3) and (4), In view of (6), this implies that Equation (13) makes clear that $Q(z_1,z_2)$ is known as soon as $U(z_1,z_2)$ is known, and vice versa. In the remainder of this paper, we mainly concentrate on the determination of $U(z_1,z_2)$.

Let $f_1\left(z_1\right)$, $f_2\left(z_2\right)$, $g_1\left(z_1\right)$, $g_2\left(z_2\right)$, $h_1\left(z_1\right)$, $h_2\left(z_2\right)$ denote one-dimensional probability generating functions. Furthermore, let $n_{11}$, $n_{12}$, $n_{21}$, $n_{22}$ denote a set of normalized probabilities, i.e., and $d_1$ and $d_2$ two nonnegative real parameters. We then use all the above quantities to define a whole class of bivariate functions $A(z_1,z_2)$ as follows: We now show that the above function is a genuine joint pgf, i.e., it can be developed as a two-dimensional power series in $z_1$ and $z_2$ with nonnegative coefficients which add up to 1.

Let us denote the numerator and the denominator of (14) as $n(z_1,z_2)$ and $d(z_1,z_2)$, respectively, i.e., with

It is clear that both $n(z_1,z_2)$ and $d(z_1,z_2)$ are normalized, i.e., $n(1,1)=1$ and $d(1,1)=1$, which implies that $A(z_1,z_2)$ is also normalized, i.e., $A(1,1)=1$. The numerator $n(z_1,z_2)$ is a probabilistic mixture of four valid pgfs, and, therefore, is a valid pgf too. The function $1/d(z_1,z_2)$ is also a genuine pgf since it can be developed as a two-dimensional power series in $z_1$ and $z_2$ with nonnegative coefficients as follows: where the probabilies ${\pi }_1$ and ${\pi }_2$ have been defined as Equation (15) thus shows that $A(z_1,z_2)$ can be expressed as the product of two valid joint pgfs, and, therefore, is a valid joint pgf too.

In this paper, we will examine a parallel-queues system with joint arrival pgf $A(z_1,z_2)$ defined in (14). The corresponding marginal arrival pgfs $A_1\left(z_1\right)$ and $A_2\left(z_2\right)$ are The mean arrival rates ${\lambda }_1$ and ${\lambda }_2$ are We assume that ${\lambda }_1<1,{\lambda }_2<1$.

According to (12), the marginal system-content pgfs are given by where

We are now ready to formulate the main result of this paper.

In the stable parallel-queues system with joint arrival pgf

where $f_1\left(z_1\right)$, $f_2\left(z_2\right)$, $g_1\left(z_1\right)$ and $g_2\left(z_2\right)$ are arbitrary one-dimensional pgfs, and

the steady-state joint system-content pgf $U(z_1,z_2)$ is given by

provided that the functions $h_1\left(z_1\right)$ and $h_2\left(z_2\right)$ are genuine one-dimensional pgfs, given by

where $\{{\alpha }_1,{\beta }_1,{\gamma }_1\}$ and $\{{\alpha }_2,{\beta }_2,{\gamma }_2\}$ are two sets of “normalized constants”, i.e.,

satisfying the restrictions

Theorem 1 provides an explicit solution for the steady-state joint system-contents pgf $U(z_1,z_2)$, for any joint arrival pgf $A(z_1,z_2)$ that satisfies the shape specified in equations (21), (24), (25) and (26), whereby $h_1\left(z_1\right)$ and $h_2\left(z_2\right)$ are valid pgfs. We now define some interesting subclasses of arrival pgfs that correspond to specific choices of the (many) parameters that the model contains.