Multiple class G-networks with iterated deletions

Abstract

We present a new type of multiclass generalized networks of queues with a steady-state product form solution. At its arrival into a queue, a negative customer (or a signal) starts an iteration. At each step of the iteration, a customer is deleted according to a probability which may depend on the type of customer. The iteration stops when the deletion fails. We study networks with multiple classes of positive customers, one class of signals and three service disciplines: FIFO, LIFO/PR and PS.

Introduction

Since Gelenbe’s seminal paper [12], Generalized networks have received considerable attention (see for instance [2], [15], [16], [18], [20] and references therein). Generalized networks (G-networks for short) consist of queues, ordinary customers and signals (which have been also denoted as negative customers). Signals interact with the customers at their arrival into a queue. But signals do not receive service, they are never queued and they disappear immediately. Here, we prefer to denote them as signals rather than as negative customers to emphasis that they are instantaneous.

The effect of a signal is limited to the interaction with the customers already queued at its arrival. But, at the completion of its service, a customer may join another queue as a signal. Thus, G-networks exhibit much more general synchronized transitions than Jackson’s networks.

Several effects of signals have been studied and shown to preserve a product form solution. The first interaction, studied by Gelenbe [12] was the destruction of a customer. Then, in [14] the product form was generalized to networks where a signal triggers a customer movement to a third queue. Gelenbe [13] and Henderson et al. [19] have independently studied batch destruction triggered by signals. If the batch size is infinite, a signal flushes out, with probability 1, the queue it enters. This effect, denoted as a catastrophe, has been studied by Chao in [4]. Several extensions have been proposed by Henderson et al. [21] to allow state-dependent batch movements and triggers. However, most of these results only apply to single-class G-networks.

Multiclass G-networks with single deletion have been studied by Fourneau et al. [6], [7]. As usual with G-networks, only exponential service time distribution were allowed with class dependent service rate (except for FIFO queues). A generalization to Coxian services time distribution was proposed by Chao in [3] with a slightly different effect for a signal. Finally, a multiclass G-network with queue flushing and processor sharing discipline was considered by Fourneau et al. [9].

In this paper, we generalize this last result. We consider G-networks with multiple classes of customers, one type of signals, three service disciplines: FIFO, LIFO/PR and PS, and more complex effects of signals. When a signal enters a queue, it starts to iterate on the deletion of customers in this queue. At each step, it tries to delete one customer. The customer is chosen according to the service discipline. The deletion succeeds following a Bernoulli process whose rate may depend on the class of the selected customer. If the deletion fails, the iteration stops.

Clearly, if the probability of success is 1, then an arriving signal flushes out a queue. As this probability may be class-dependent, we may obtain a much more complex behavior than queue flushing. Note however that, as the selection of the customer to be deleted must be consistent with the service discipline, deletion of arbitrary batches of customers do not occur with probability 1 even if the probability of success is 1. Due to the complexity of the deletion mechanism, we only consider exponential service time distribution. Indeed, the effect studied by Chao [5] to allow general service time becomes very complex when it is iterated.

G-networks were originally designed to model neural networks [11]. Signals and customers, respectively, represent inhibitory and excitatory signals in models of neural networks, while queue lengths represent the neurons input potentials. Recently, new applications of these networks have been proposed in the field of reliability and performability (see [8], [17] for instance). In these models, the deletion capabilities of signals is used to model breakdowns which cause the loss of some customers (or even all the customers) in a queue. As we add more complexity in the deletion mechanisms, we expect that more realistic models of systems with breakdown and failures will be achieved using multiclass G-networks with iterated deletions.

The paper is organized as follows. In the next section we present the model and prove the product form solution in Section 3. For the sake of readability, the detailed proofs of some technical lemmas are postponed into Appendix A. Examples are presented in Section 4. Section 5 is devoted to stability, (i.e. the existence of solution for the flow equation). Like in usual G-networks, these equations are non-linear and the same kind of technique is used to establish existence of a solution. We also study multiclass networks of queues with flushing which exhibit a more interesting behavior in term of stationarity constraints.