Modelling M/G/1 queueing systems with server vacations using stochastic Petri nets

Abstract

The theory of non-Markovian stochastic Petri nets is employed in this paper to derive an alternative method for studying the steady state behaviour of the M/G/1 vacation queueing system with a limited service discipline. Three types of vacation schemes are considered, and sytems with both a finite population and those with an infinite population (but finite capacity) are considered. Simple numerical examples are also provided to illustrate the functionality of the methods and some useful performance measures for the system are obtained. Key words:Markov regenerative stochastic Petri nets, vacation queueing models, limited service discipline, single, multiple and hybrid vacation schemes. IntroductionQueueing systems in which the server is sometimes inactive (taking vacations or working elsewhere) while customers are waiting for service may find many applications in the performance modelling of computer and communication systems. For example, in many digital systems, the processor is multiplexed among a number of jobs and is hence not available all the time for a single job type. So, if we take any one job type as a reference point, the processor is alternatively busy handling that job type and absent doing work elsewhere [12].Server vacation has been used successfully to model polling systems [13], multiclass priority scheduling systems [18], maintenance models [17] and processor failure [2]. For a survey on the use of vacation models, see Doshi [7].Most of the papers on vacation models assume the source population of the queueing system to be infinite. However, Trivdei et al. [11] have studied several vacation models in which they have taken the population to be finite. They assume an exponentially * Corresponding author: School of Mathematics, Madurai Kamaraj University, Madurai 625021, India, email: kasturey@rediffmail.com † School of Mathematics, Madurai Kamaraj University, Madurai 625021, India 131 132 K Ramanath & P Lakshmi distributed service time. This paper is an extension of their work to the case where the service times are generally distributed.We obtain Markov regenerative stochastic Petri net (MRSPN) representations of the M/G/1/N queueing systems with server vacations under a limited service discipline. For an excellent introduction to the theory of M/G/1 queueing systems, see Gross and Harris [9] and Takagi [14]. The book by Takagi also contains information about queueing systems with server vacations. We consider three types of vacation schemes, namely the multiple vacation scheme, the single vacation scheme and the hybrid vacation scheme. We also show how the representations may be extended to the case where the population is infinite, but the system has a finite capacity.The chief advantages of an MRSPN representation are:1. The representation is aesthetically appealing and simple to understand, even for a layman.2. Several other features of vacation queueing systems may be included in this representation, whose inclusion in other classical models would have led to analytically intract...

MRSPN analysis of Semi-Markovian finite source retrial queues

“…Oliver and Kishor (1991) study an M/M/1//N queue with vacation by means of GSPN. Furthermore, performance analysis of the queueing system M/G/1//N with different vacation schemes is given by Ramanath and Lakshmi (2006) by using the MRSPN tool.…”