A single-server G-queue in discrete-time with geometrical arrival and service process

Abstract

Negative arrivals are used as a control mechanism in many telecommunication and computer networks. This paper analyses a discrete-time single-server queue with geometrical arrivals of both positive and negative customers. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, the two sub-cases in which a customer currently being served can and cannot be killed by a negative customer. Thus, we carry out a complete study of these systems, including the ergodicity condition as well as exact formulae for the associated stationary distribution. The effect of several parameters on the systems is shown numerically.

Introduction

During the last decade there has been an increasing interest in queueing systems and networks with negative arrivals and their applications. Negative customers systems were introduced by Gelenbe [9] with a view to modelling neural networks where a node represents a neuron. Since 1989, the name G-queue has been adopted for queues with negative customers in acknowledgement of Gelenbe. The first researches about G-networks and single node queues with negative customers can be found in [8], [9], [10], [19]. The main G-network [10] was extended to the case of multiple classes of positive and negative customers in [7], [14], [23]. The papers [11], [12], [18] study G-networks with positive customers and signals. Gelenbe and Labed [22] generalize the results presented in [7] including multiple classes of triggers. Artalejo [1] and Gelenbe [15] survey the main results and related ideas with the G-networks field. The G-networks systems have a rich and wide range of real applications in areas such as computers, communications and manufacturing [5], [6], [13], [16], [17], [20], [21], [24], [25].

In its simplest version, a negative customer removes a positive customer in the queue (or being served) according to a specified killing discipline. Chakka and Harrison [4] consider two variants of the RCE killing discipline (removal of the customer from the end of the queue), where the most recent positive arrivals are removed first [27], [28]. The first variant, RCE-inimmune servicing, removes the most recent positive arrival in spite of whether it is in service or waiting; a negative arrival has no effect only when it finds an empty queue and all servers idle. The second variant does not allow a customer currently in service to be removed: a negative customer that arrives when there are no positive customers waiting to start service has no effect. We say that customers in service are immune from killing and that the service itself is RCE-immune servicing.

The first variant is that of the traditional negative customer, suited to the modelling of killing signals in speculative parallelism, for example. It can also be used to model cell losses caused by the arrival of a corrupted cell or one encountering a full buffer, when the preceding cells of a packet would be discarded. The second variant is a modification suitable for the modelling of load balancing where work is transferred from overloaded queues but never work that is actually in progress.

There exists a further killing discipline: RCH (removal of the customer from the head of the queue). Here, customers are removed from the head of the queue, i.e., the earliest arrivals go first. This discipline is suitable for modelling server breakdowns.

On the other hand, the analysis of discrete-time queueing models has received considerable attention in the scientific literature over the past years in view of its applicability in the study of many computer and communication systems in which time is slotted, see for instance [3], [29], [30], [31] and the references therein. Discrete-time queueing models are particularly appropriate to describe the various queueing related phenomena in digital computer and communication systems including mobile and BISDN networks based on asynchronous transfer mode (ATM) technology, due to the packetized nature of these transport protocols.

Many continuous-time queueing models with negative arrivals have been discussed during the last years. However, the only work about negative arrivals in discrete-time can be found in [2], where the authors regard a flow of positive customers which may be transformed into a flow of negative customers when the server is busy. The present paper studies a discrete-time single-server queue with both positive and negative arrival streams; that is, in addition to the positive customers, a second flow of negative customers following a geometrical process is also considered. Thus, the contribution of this work is to analyse a discrete-time queue with two types of customers, positive and negative, in accordance with the RCH, RCE-inimmune servicing and RCE-immune servicing policies. Nevertheless, in our case RCH is equivalent to RCE-inimmune servicing because of the memoryless property of the geometrical distribution.

The rest of the paper is organized as follows. In the next section, we describe the mathematical model. Sections 3 The RCE-inimmune servicing killing policy, 4 The RCE-immune servicing killing policy investigate a $Geo/Geo/1$ queue with negative customers in conformity with the RCE-inimmune servicing and immune servicing disciplines. The study concludes in Section 5 with a discussion of some numerical examples illustrating the influence of the parameters on the system performance.