Analysis of M/M/S queues with 1-limited service
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#2002 Operational Research Society Ltd. All rights reserved. 0160-5682/02 $15.00 www.palgrave-journals.com/jors
Analysis of M/M/S queues with 1-limited service A Tyagi1, J Harms2 and A Kamal3* 1 Nortel Networks, Ottawa, ON, Canada; 2University of Alberta, Edmonton, AB, Canada; and 3Iowa State University, Ames, IA, USA
In this paper, a multiple server queue, in which each server takes a vacation after serving one customer is studied. The arrival process is Poisson, service times are exponentially distributed and the duration of a vacation follows a phase distribution of order 2. Servers returning from vacation immediately take another vacation if no customers are waiting. A matrix geometric method is used to find the steady state joint probability of number of customers in the system and busy servers, and the mean and the second moment of number of customers and mean waiting time for this model. This queuing model can be used for the analysis of different kinds of communication networks, such as multi-slotted networks, multiple token rings, multiple server polling systems and mobile communication systems. Journal of the Operational Research Society (2002) 53, 752–768. doi:10.1057/palgrave.jors.2601364 Keywords: communications; queuing; matrix geometric; method queues with vacations; multiple service queues
Introduction Queuing systems that allow servers to be on vacation arise in many computer and communication systems. Server vacations may be due to lack of work, server failure or another task being assigned to the server. In these systems the server is not always available to serve its primary customers. Some of the applications which can be modelled using these systems are computer maintenance and testing,1 CPU scheduling, TDMA networks,2 and priority queues.3 Another important application of vacation systems are polling systems or cyclic queues.4,5 Cyclic queues arise in multi-processor systems,6 multiple token ring and multislotted networks.5,7,8 Single server queues with vacations have been used to study single server cyclic queues,4 similarly multiple server queues with vacation can be used to study multiple server cyclic queues. There are several variations of vacation models that can be defined. There may be a single server or multiple servers. The server may take a vacation at a random time (nonexhaustive), after serving at most k customers (k-limited) or after all the customers in the queue are served (exhaustive). Also, depending on the application, when the server finishes a vacation and there is no customer to be served, the server may take another vacation (multiple vacation model VM) or it may wait, ready to serve, until a customer arrives (single vacation model VS). In this paper a 1-limited M/M/S/VM vacation queuing model is studied. The customer interarrival and service times are assumed to be exponential; while the vacation *Correspondence: A Kamal, Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011, USA. E-mail: [email protected]
times obey a phase distrib
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