Networks of infinite-server queues with multiplicative transitions

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Abstract

This paper considers a network of infinite-server queues with the special feature that, triggered by specific events, the network population vector may undergo a linear transformation (a ‘multiplicative transition’). For this model we characterize the joint probability generating function in terms of a system of partial differential equations; this system enables the evaluation of (transient as well as stationary) moments. We show that several relevant systems fit in the framework developed, such as networks of retrial queues, networks in which jobs can be rerouted when links fail, and storage systems. Numerical examples illustrate how our results can be used to support design problems.

Introduction

The vast majority of queueing network models studied in the literature are of the following form: there is a set of nodes that are fed by streams of external arrivals, and a routing mechanism that determines to which queue served clients are forwarded or whether the client leaves the system altogether. The most common queueing disciplines are of single-server type (entailing that clients may have to wait until they get into service) and of infinite-server type (in which all customers present at a node are served in parallel).

A key feature of the conventional class of models described above is that clients join and leave queues one by one. In many applications, however, triggered by specific events, the full population of individual queues may move around the network (or leave the system altogether). Particularly in the reliability and availability context, there are many relevant examples of such dynamics. We could for instance think of a data communication network with unreliable nodes: at the moment that a node goes down, all traffic residing at the node may be instantly lost. Another example concerns rerouting: triggered, for instance, by a link failure, clients are moved from one set of resources to an alternative set (the ‘backup route’). Due to the fact that they correspond to transitions of the entire population of specific queues, the dynamics of the above two examples do not align with those of conventional queueing models.

Scope, object of study. Motivated by the above examples, the main objective of the present paper is to analyze queueing networks with multiplicative transitions. These multiplicative transitions effectively entail that the network dynamics include transitions by which the network’s population vector, say m, is (pre-)multiplied by a matrix A with integer-valued, nonnegative entries, so that the network population after the transition becomes Am. For instance, choosing A to be a diagonal matrix with [A]ii=0 and [A]kk=1 for all ki would correspond to the event of all clients at node i being lost. Relocation of clients can be taken care of in a similar manner: the full population of queue i moving to queue j corresponds to [A]ji=1, [A]kk=1 for all ki, and all other entries equal to 0.

In this paper the queues considered are of infinite-server type. This type of queue is particularly relevant in contexts where the sojourn time at a node of each client is not (or hardly) affected by other clients. As such, the model has a broad variety of applications, ranging from the number of websurfers simultaneously present at a set of websites, to the number of messenger RNA molecules simultaneously present in a collection of cells. A specific application that features in the present article concerns the optimal design of storage networks. To make the model as widely applicable as possible, we assume that all relevant transition rates (i.e., arrival rates and departure rates) are affected by an external autonomously evolving Markovian environmental process; the resulting model is therefore of a Markov modulated nature. As will become clear, in a reliability context such an environmental process can be used to model the state of the nodes of the network (i.e., ‘up’ or ‘down’).

Contributions. The paper has two main contributions. (i) In the first place we set up a general model of a network of infinite-server queues with multiplicative transitions. For this model we derive a system of partial differential equations that describe its time-dependent behavior (in terms of the probability generating function of the joint queue length distribution), as well as a procedure to evaluate the corresponding moments. The model turns out to have a non-trivial stability condition (under which the system’s stationary behavior is well-defined), which we establish using the expression we found for the time-dependent mean. (ii) In the second place, we point out that various natural, practically relevant models fit in our framework. Most notably, we concentrate on a network of retrial queues, a network with rerouting, and a storage network. Our results can be used to support various design decisions. In the storage system application, for instance, interesting tradeoffs can be numerically assessed: files are typically stored on multiple locations so as to mitigate the risk of loss, but evidently one wants to do so without using an unnecessarily large amount of storage space.

Literature. As mentioned above, in typical queueing network models the number of clients per queue changes by one at a time; see e.g. the standard textbooks [[1], [2]]. Several papers, such as [[3], [4], [5], [6], [7], [8]], consider queues with batch arrivals and batch services and find product-form results, but these typically neither cover our multiplicative update rule nor allow the transition rates to be affected by an environmental process. We also refer to the related papers [[9], [10], [11]].

As mentioned above, a relevant special case of our model corresponds to the context of reliability. In many situations, when a network resource (a node or a link) fails, all clients using it will be lost. Such models are known as queueing models with catastrophes; for a fairly complete account of such models, we refer to the recent literature review in [12, Section 1]. The models studied are typically (but not always) one-dimensional; interesting contributions include [[13], [14]].

Queueing models for which the underlying infrastructure alternates between being ‘up’ and ‘down’ can be seen as examples of stochastic processes on dynamically evolving graphs. Despite the sizeable literature on random graphs, the body of work on dynamic random graphs is considerably smaller, and (evidently) the body of work covering stochastic processes on dynamic random graphs is even smaller. In recent contributions, results on dynamic random graphs have been reported; see e.g. [[15], [16], [17], [18], [19]]. Our paper can be regarded as being among the first to facilitate describing queueing processes on a randomly evolving graph (but it is noticed that the model we propose is substantially more general, as the multiplicative transitions are not restricted to node failures and repairs).

As mentioned, our model covers various practically relevant models as special cases. In each of the corresponding application areas there is a large collection of papers and textbooks available; in Section 4 we include a number of domain-specific references.

Organization. The paper is organized as follows. Section 2 presents the model in its generic form, and after some preliminaries, the results in terms of partial differential equations characterizing the joint probability generating function and ordinary linear differential equations characterizing the moments. In addition, the stability condition is provided, under which stationary moments exist, which can be found by solving systems of linear equations. Section 3 gives an indication of the width of our framework: we show that it covers a network of retrial queues, a network with rerouting, and a storage network. Section 4 demonstrates a couple of design issues that can be resolved by using our machinery. Finally, Section 5 provides a discussion and concluding remarks.

Section snippets

Analysis

This section studies our generic model: a network of infinite-server queues with multiplicative transitions. We first introduce the model, then study its time-dependent behavior, derive its stability condition, and conclude by commenting on numerical issues.

Retrial queues, rerouting, storage systems

In this section we show the power of the framework introduced in the previous section, by pointing out how it facilitates the modeling of all sorts of relevant phenomena. We specifically focus on: (i) systems in which nodes go down but in which lost customers attempt reentry, (ii) systems in which customers are rerouted when one of the links along the route goes down, and (iii) storage systems.

Numerical experiments

To illustrate the potential of our results, in this section we provide two examples: one on a retrial queue, and another one on storage networks.

Discussion and concluding remarks

In this paper we studied a network of Markov-modulated infinite-server queues with the distinguishing feature that it also incorporates events by which the network population vector makes multiplicative transitions (at which it changes from m to Am, for some matrix A). As we argued, the resulting framework covers various relevant models as special cases; for example, it enables the modeling of retrial queues, networks with rerouting, and storage systems.

Our results for the system’s transient

Acknowledgments

The authors wish to thank Peter Taylor (The University of Melbourne) and Ross McVinish (The University of Queensland) for useful remarks. The research for this paper is partly funded by the NWO Gravitation Programme Networks, Grant Number 024.002.003 (Mandjes, Patch), an NWO Top Grant, Grant Number 613.001.352 (Mandjes), the ARC Centre of Excellence for Mathematical and Statistical Frontiers under grant number CE140100049 (Patch), and an Australian Government Research Training Program (RTP)

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