Analysis of a discrete-time GI–G–1 queueing model subjected to bursty interruptions

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Abstract

In this contribution, we investigate a discrete-time single-server queue subjected to server interruptions generated by a 2-state Markov process. The model under consideration assumes customers with multiple-slot service times, which leads to the introduction of two different service strategies depending on whether service of an interrupted customer continues or restarts after an interruption. For both alternatives, we establish expressions for the steady-state probability generating functions of the buffer contents, the unfinished work and the customer delay in terms of the effective customer service times. From these results, closed-form expressions for various performance measures, such as the moments of these quantities, can be established. After dealing with some stability issues, we illustrate the impact of both service strategies on the buffer performance with some numerical examples.

Scope and purpose

Discrete-time queueing theory is distinguished from its more developed continuous-time counterpart by the synchronous nature of its service. Time is divided into fixed-length slots and service of a customer typically takes an integer number of slots. Discrete-time queueing models are particularly appropriate to describe the various queueing related phenomena in digital computer and communication systems including mobile and B-ISDN networks based on asynchronous transfer mode (ATM) technology, due to the packetized nature of these transport protocols.

This work focuses on a discrete-time queueing system with server interruptions. Interruptions are an abstraction of temporary server unavailability. The latter can be caused by temporary occupancy of the server by other customers — regulated by a scheduling discipline such as round robin or priority scheduling — or temporary failure of the server (e.g. due to transmission errors, maintenance or repair). Typically, interruptions occurring in these systems have a bursty nature, i.e., they occur in bursts rather than being randomly spread over time. The presented model allows to study the effects of the burstiness of the server interruptions on various performance measures such as mean queue contents and mean customer delay. Two different operation strategies are hereby considered: after an interruption, service may either continue where it left off or it may completely restart.

Introduction

Queueing models with server interruptions [1] have been investigated by several authors, in continuous as well as in discrete time. Server interruptions are an abstraction of temporary server unavailability caused by sharing a common server with other queues — e.g. for polling systems [2] or multi-class queueing systems — or by external causes, e.g. for maintenance [3] or due to failures or interference [4]. etc.

The current contribution investigates a discrete-time single-server queueing model with correlated server interruptions, in the sense that the interruption process is a two-state Markov process. The length of the customer service-times is assumed to be generally distributed. Such a model can be used to assess the performance, for instance, of an IP-based network, where variable-length packets are exchanged between network nodes. The core contribution of this paper is that it takes the possible occurrence of interruptions explicitly into account. We will investigate the cases where after an interruption, service of a customer either continues or restarts. In case the interruptions are caused by servicing higher priority customers in a preemptive priority queue, these two modes correspond to preemptive resume and the preemptive non-resume (without resampling) service disciplines.

Using a generating-functions approach, we present an analysis that allows us to derive expressions for the moments of quantities such as the buffer contents, the unfinished work and the customer delay. In particular, the effect of correlation and of the different service strategies on system performance is investigated. The model extends the results presented in [5], [6] as well as those presented in [7], [8], in that it considers, respectively, correlated server interruptions (compared with no interruptions in [5] and uncorrelated random interruptions in [6]) and generally distributed service times (compared with single slot service times in [7] and fixed-length service times in [8]).

The outline is as follows. In the next section, the system under investigation is described in more detail. Expressions for the probability generating function of the effective service time are derived in Section 3. The results of the latter are used in 4 Buffer contents, 5 Unfinished work and customer delay, where we derive expressions for the probability generating function of the buffer occupancy and of the unfinished work and message delay, respectively. After some remarks on stability in Section 6 and a numerical example in Section 7, we conclude in Section 8.

Section snippets

Model

We consider a discrete-time system, i.e., time is divided into constant-length intervals called slots. During each of these slots, customers that arrive in the system are stored in a buffer with infinite capacity, and are served on a first-in-first-out basis. Service of a customer takes, in general, a number of slots and is synchronized with respect to slot boundaries. This implies that the service of a customer cannot start before the beginning of the slot following its arrival slot. The

Effective service times

The effective service time of a customer is defined as the time period — expressed as an integer number of slots — that the system effectively spends on serving this customer. This includes the time during which the server is blocked when the customer is in service and, for RAI, the time lost by restarting the customer's service from the beginning after an interruption. Due to the independence of the service times of the consecutive customers and the nature of the interruption process described

Buffer contents

In the first step, we derive an expression for the probability generating function of the buffer occupancy at customer departure times. Let un denote the buffer occupancy, i.e., the number of customers in the buffer, at the beginning of the slot following the departure slot of the nth customer. For positive un, service of the (n+1)th customer can start immediately. This implies that — as the previous slot was an A slot since there was a departure — it will take sA slots to the next departure,

Unfinished work and customer delay

The unfinished work at a given time instant is defined as the number of slots it would take to empty the buffer under the assumption that there are no new arrivals. Note that this includes the B-slots during which the server is interrupted, and (in case of RAI) slots lost due to uncompleted interrupted service trials. Consider now a random slot k, and define wk as the unfinished work at the beginning of this slot. During this slot, the unfinished work is diminished by 1 if the buffer is not

Stability

In the remainder of the paper we assume non-heavy-tailed customer arrival and service time distributions, i.e., at least the first and second moments of these random variables are finite.

Consider now the effective service times. It is clear from , , for CAI and from , , for RAI, that as long as α>0 and β<1, the conditional effective service time distributions (conditioned on the state of the server during the slot preceding the start of the customer service) exist and that the corresponding

Numerical example

Let the number of customers arriving in a slot be Poisson distributed with mean λ and assume that the service times for the consecutive customers are shifted geometrically distributed with parameter φ, i.e.,E(z)=eλ(z−1),T(z)=(1−φ)z1−φz.In Fig. 1, the stability conditions for this system for CAI and RAI are plotted versus the interruption parameters (σ,K) as defined by (2) for different φ-values and for λ=0.05. Given φ and λ, the mean buffer occupancy is finite for (σ,K) to the right of the

Conclusions

We investigated a discrete-time single-server queueing system with Markovian interruptions and generally distributed service times, using a probability generating functions approach. We obtained expressions for the probability generating function of the buffer occupancy, unfinished work and customer delay for two service modes, called CAI and RAI. These expressions can be used to obtain closed-form expressions for the moments of the corresponding random variables. Wherever stability

Dieter Fiems received an engineering degree at KAHO-St-Lieven and a post-graduate degree in computer science at Ghent University. He is now a Ph.D. student at the Department of Telecommunications and Information Processing at Ghent university, as a member of the SMACS research group. His main research interests include discrete-time queueing models and stochastic modeling of IP and ATM networks.

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Dieter Fiems received an engineering degree at KAHO-St-Lieven and a post-graduate degree in computer science at Ghent University. He is now a Ph.D. student at the Department of Telecommunications and Information Processing at Ghent university, as a member of the SMACS research group. His main research interests include discrete-time queueing models and stochastic modeling of IP and ATM networks.

Bart Steyaert received the degrees of Licentiate in Physics and Licentiate in Computer Science from Ghent University, Belgium. Since, January 1990, he has been working as a researcher at the SMACS Research Group. His main research interests include discrete-time queueing models, traffic control, and stochastic modeling of ATM and IP networks.

Herwig Bruneel received the M.S. degree in Electrical Engineering, the degree of Licentiate in Computer Science, and the Ph.D. degree in Computer Science from Ghent University, Belgium. He leads the SMACS Research Group within this department. His main research interests include stochastic modeling of digital communication systems, discrete-time queueing theory, and the study of ARQ protocols. He has published more than 150 papers on these subjects and is the coauthor of the book. “Discrete-Time Models for Communication Systems Including ATM”, (besides B.G. Kim).

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