A finite-source M/G/1 retrial queue with outgoing calls
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A finite-source M/G/1 retrial queue with outgoing calls Velika I. Dragieva1 · Tuan Phung-Duc2 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper we deal with a single-server, finite-source retrial queue where the server not only accepts incoming calls but after some exponentially distributed idle time makes outgoing calls. The service times of incoming and outgoing calls follow two distinct arbitrary distributions. The outgoing calls are directed not to the customers in the system but outside it, which implies that the model can be considered as a model with vacations or with customers of two types. Along with the standard retrial queue where all customers are allowed to join the orbit we consider also the corresponding queue with restriction on the orbit size. We derive formulas for computing the stationary system state distribution and investigate the influence of the system input parameters on the main macro characteristics of the system performance. Keywords Finite queues · Retrials · Outgoing calls · Restriction on the orbit size
1 Introduction In the queueing models with finite source (also called closed queueing models) it is assumed that the server/servers serve a finite number of customers as it is in most of the real situations. Each of these customers produces its flow of demands which means that the generalized input flow depends on the number of customers able to produce demands. These models have been used to analyze the performance of telephone, computer, communication and other systems (see Balazsfalvi and Sztrik 2008; Biro et al. 2013; Cooper 1981; Jain 1991; Jaiswal 1969). The characteristic feature of queueing systems with retrials concerns the behavior of those unsuccessful demands whose service cannot start at the moment of their arrival. In the models with retrials it is assumed that these customers are not lost or allowed to queue. Instead, they repeat their attempts for service until finding the server idle. Between retrials the customers are said to be in the orbit, or to be sources of repeated (secondary) calls, secondary subscribers. Retrial queues arise in diverse real situations including our daily activity, telephone switching
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Tuan Phung-Duc [email protected] Velika I. Dragieva [email protected]
1
University of Forestry, 1756 Sofia, Bulgaria
2
University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan
123
Annals of Operations Research
systems, telecommunication and computer networks, call centers, cellular and local area networks, etc. (see Aguir et al. 2004; Artalejo and Gómez-Corral 2008; Choi et al. 1992; Deslauriers et al. 2007; Tran-Gia and Mandjes 1997; Van Do et al. 2014). A systematic account of the fundamental methods and the latest results, as well as an classified bibliography on this topic can be found, for example in Artalejo and Gómez-Corral (2008), Gómez-Corral and Phung-Duc (2016), Falin and Templeton (1997), Kim and Kim (2016) and references therein. Single-server retrial queue with a finite numb
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