System occupancy in a multiclass batch-service queueing system with limited variable service capacity
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System occupancy in a multiclass batch-service queueing system with limited variable service capacity Jens Baetens1
· Bart Steyaert1 · Dieter Claeys2,3 · Herwig Bruneel1
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In multi-class telecommunications or manufacturing systems, customers belonging to the same class can often be processed together. This results in a service capacity that depends on the classes of the customers in the queue. In this paper, we analyse a discrete-time batchservice queue with two customer classes. The single batch server can group all same-class customers at the head of the queue up to a constant class-dependent maximum service capacity. We focus on the analysis of the system occupancy at service initiation opportunities, and also compute both a light- and heavy traffic approximation in order to reduce the numerical complexity introduced by the maximum service capacities. Additionally, we propose a method for interpolating between these approximations in order to study the behaviour in the intermediate region. We also deduce the system occupancy and its approximations at random slot boundaries. In the numerical experiments, we examine the conditions under which these proposed approximations are accurate. Keywords Batch service · Two-class · Variable service capacity · Generally distributed service times · Correlated customer types
1 Introduction Batch-service queueing systems are often found in manufacturing (Niranjan et al. 2017), transportation systems (Bountali and Economou 2017), and telecommunication systems (Bellalta and Oliver 2009) where packets are grouped together based on similarities in the production process or destination. Due to their wide range of applications, this type of queueing system has been studied extensively, for instance, by Chaudhry and Templeton (1983), Arumuganathan and Jeyakumar (2005), Banerjee and Gupta (2012), Banerjee et al. (2015),
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Jens Baetens [email protected]
1
Department of Telecommunications and Information Processing, SMACS Research Group, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium
2
Department of Industrial Systems Engineering, Ghent University, Technologiepark 903, 9052 Zwijnaarde, Belgium
3
Industrial Systems Engineering (ISyE), Flanders Make, Lommel, Belgium
123
Annals of Operations Research
Banerjee et al. (2014), Chang and Takine (2005), Claeys et al. (2013a), Claeys et al. (2013b), Claeys et al. (2012), Goswami et al. (2006), Janssen and van Leeuwaarden (2005), Pradhan and Gupta (2017). While the capacity of the batch server is assumed to be constant in these contributions, this service capacity often depends on the environment and the content of the queue. An example of a model with variable service capacity has been studied by Chaudhry and Chang (2004) where they analysed the system content in the Geo/G Y /1/N + B model . Server vacations were also incorporated in the previous model by Chang and Choi (2005). Yi et al. (2007) further extended this model by using th
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