Instability of LAS multiclass queueing networks

Abstract

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the least attained service (LAS) service protocol. It is a reentrant line with two servers and six customer classes. The customer interarrival times in our system are bounded below and have finite exponential moments, while the corresponding service times are deterministic. As a special case, we obtain a deterministic, strictly subcritical unstable LAS network.

Introduction

An important question in the theory of multiclass queueing networks is whether a given network is stable. The intuitive meaning of network stability is that the system performs well under reasonable workload: the queue lengths do not grow linearly with time and do not oscillate “wildly”, there is no mutual blocking and forced idleness of the servers when work is present in the system. Thus, stability of a network is a basic indicator of its proper design. It appears that there is no general criterion for this behavior. In particular the usual necessary condition that the network be strictly subcritical (i.e., the traffic intensity ${\rho }_j$ is strictly less than $1$ at each station $j$) is not sufficient. On the positive side, the condition ${\rho }_j<1$ for all $j$ is sufficient for generalized Jackson networks [31] and multiclass networks with some disciplines, including first-in-first-out (FIFO) in networks of Kelly type [7], head-of-the-line proportional processor sharing (HLPPS) [8], first-buffer-first-served and last-buffer-first-served [16], [21], as well as earliest-deadline-first (EDF) [10], [27]. Dai [16], generalizing and systematizing earlier work of Rybko and Stolyar [33], provided a general framework for proving such stability results by showing stability of the corresponding fluid model, a deterministic analog of the network under consideration. Dai and Meyn [20] extended the scope of this method to assuring convergence of the corresponding queue length moments to their steady-state values.

On the other side of the spectrum we have unstable, strictly subcritical multiclass networks. Following Bramson [11], p. 53, we say that a queueing system is unstable if, for some initial state, the number of jobs in the network tends to infinity with positive probability as the time parameter $t\to \infty$. First examples of such systems were given in a deterministic setting. Kumar and Seidman [28] showed instability of a clearing policy in two nonacyclic networks. Lu and Kumar [29] provided an example (suggested by Seidman) of an unstable reentrant line with a preemptive static buffer priority (SBP) discipline. A random counterpart of this network was later analyzed in [11], Section 3.1. To our knowledge, the first example of an unstable stochastic network was a preemptive SBP system due to Rybko and Stolyar [33], with the same topology as one of the clearing systems considered by Kumar and Seidman [28]. Subsequently, it was found that even strictly subcritical FIFO queueing networks might be unstable. Examples of such systems were given by Seidman [35] and Bramson [5], [6] in a deterministic and random case, respectively. Further examples of unstable queueing networks may be found in [3], [9], [21], [22] or [18], [19]. Meyn [30] and Dai [17] provided instability criteria of multiclass networks based on instability of their fluid models. Unfortunately, the assumptions made in the latter papers appear to be rather strong and they are not satisfied for some systems of interest.

So far, most of the work in this area has been concentrated on the investigation of head-of-the-line (HL) disciplines, in which only the first job in each class may receive service, and hence the tasks are served in the FIFO order within each class. (One notable exception is EDF, which is, in general, not HL, even in the nonpreemptive case.) In particular, little attention has been devoted to stability of multiclass queueing networks with size-based scheduling policies in which the order of service is established on the basis of the customer service times (either initial, remaining or attained). Service policies of this type have been investigated thoroughly in the single-server setting, especially in the M/G/1 case, in which their performance is relatively well understood; see the last three chapters in [23] and the references given there. Interest in such protocols stems from the fact that a proper size-based scheduling policy can substantially improve the performance of a queueing system. For instance, a classic result of Schrage [34] assures that the shortest remaining processing time (SRPT) policy, giving preemptive priority to the job which can be completed first, minimizes the queue length in a single server system at each point in time. In order to implement SRPT, the scheduler must know the job sizes. This knowledge is usually available, e.g., in Web servers with static contents [24], but it is not known in many other applications, like Internet routers, switches or Web servers with dynamic pages. To overcome this difficulty, service protocols based on the amount of service attained by the jobs, rather than their sizes, were developed. The most popular of them is least attained service (LAS) policy, also known as foreground–background (FB) [26] or shortest elapsed time (SET) first [15] scheduling. It was initially proposed and studied in the late 60s, in the context of time-sharing computers. LAS is a preemptive scheduling policy that gives service at each station to the job that has received the least service. In the event of ties, processor sharing of the jobs that have received the least service is used. A considerable number of studies has been devoted to the investigation of the LAS performance. The interested reader is referred to [32] for a survey and references. For samples of more recent research in this area see [2] or [25].

It is natural to ask how well size-based disciplines perform in multiclass, multiserver networks. To our knowledge, there are few mathematically rigorous results in this direction. Verloop, Borst and Núñez Queija [36] investigated this issue in the context of resource sharing networks. A fundamental difference between the latter systems and multiclass queueing networks is that jobs in a resource sharing network need access to all the resources on their routes simultaneously, while customers of a multiclass queueing network visit different servers along their routes in succession. Verloop, Borst and Núñez Queija [36] found that linear, strictly subcritical resource sharing networks with Poisson arrivals and generally distributed document sizes, working under the SRPT, shortest expected remaining processing time (SERPT) and the LAS scheduling, may be unstable. Brown [12] investigated a packet level model (as opposed to flow level models used in [36]) and obtained stability conditions for aged-based policies, including LAS, in data networks. As far as multiclass queueing networks are concerned, Banks and Dai [4] conducted a simulation study demonstrating that a three station reentrant line with nine customer classes can be unstable under the SERPT protocol. Moreover, their simulations suggested that a variant of the Rybko–Stolyar (Kumar–Seidman) network may be unstable under the shortest mean remaining processing time first discipline, an analog of SERPT in which the priority of a customer class is established on the basis of the sum of the mean remaining processing times along its path rather than the mean processing time for this class. Chen and Yao [13], Section 8.6, presented a simulation indicating that a variant of the Rybko–Stolyar (Kumar–Seidman) network may not be stable under the preemptive SERPT discipline, called also shortest (expected) service (processing) time first and abbreviated as SPT. They also stated, without providing any details, that both the simulation and the analysis of Sections 8.1–8.3 in [13] indicated that the above-mentioned network might not be stable under SRPT. Recently, Chojecki and Kruk [14] provided an example of a strictly subcritical multiclass queueing network unstable under the SRPT, SERPT and the shortest job first (SJF) service protocols, where the latter one is a non-preemptive variant of SRPT.

In this paper, we provide an example of a strictly subcritical multiclass queueing network which is unstable under the LAS service protocol. The underlying network topology is a reentrant line with two servers and six customer classes which may be regarded, in a suitable sense, as a variant of the Lu–Kumar network [29]. Let us note that in [14], a similar reentrant line, with two servers and eight customer classes, was used as a counterexample. The requirements for the customer interarrival times are milder than those in [14], allowing, in particular, for deterministic periodic arrivals or spread-out (even continuous) interarrival distributions. As in [14], the customer service times in our system are deterministic.

In spite of similarities between the queueing networks considered here and in [14], our analysis is necessarily different from the one in the latter paper. One of the key ideas underlying the proofs in [14] was to choose lattice interarrival times and deterministic service times in such a way that, with probability one, no customer of the resulting SRPT system was ever preempted. Under the LAS service protocol, such a choice is impossible, because every new arrival to a class results in a preemption. Moreover, in the SRPT case, FIFO is used as a tie-breaking rule, making the system’s evolution easier to track than the one of the corresponding LAS network, utilizing processor sharing arbitration. The key to overcoming these difficulties is to focus on the amounts of service attained by the customers present at the network’s servers and to compare them with the amounts of service necessary for these customers to leave the servers, taking new arrivals into account. Such comparisons allow for an estimation of the job transition times between the stations of the network, which are crucial for our instability proof.

Together with the findings of [14], [36], our results indicate that size-based service policies may not use the available resources efficiently in a multiserver network setting and in fact cause instability effects. This is in sharp contrast to their good performance in single server queues. Accordingly, using LAS (or other size-based protocol) only within each customer class, with class priorities arbitrated by another discipline, known to be stable in multiclass networks (e.g., HLPPS), might be advisable. See [1] for a similar idea in the context of resource sharing networks and the SRPT service protocol.

This paper is organized as follows. In Section 2, we provide our example of an unstable LAS network. The proof of its instability is given in Section 3. Section 4 concludes.

The following notation will be used throughout the paper. Let $\mathbb{N}=\{1,2,\dots \}$ and let $\mathbb{R}$ denote the set of real numbers. For $a\in \mathbb{R}$, we write $\lfloor a\rfloor$ for the largest integer less than or equal to $a$. For a vector $a=(a_1,\dots ,a_n)\in {\mathbb{R}}^n$, let $\left|a\right|\triangleq {\sum }_{i=1}^n\left|a_i\right|$. For a left-continuous function $f:[0,\infty )\to \mathbb{R}$ and $t>0$, we write $f(t-)$ for ${lim}_{s\uparrow t}f\left(s\right)$.