Two-way Communication Orbit Queues with Server Vacation
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Two-way Communication Orbit Queues with Server Vacation Sweta Dey1 · T. G. Deepak1 © Springer Nature India Private Limited 2020
Abstract We consider a single server retrial model with two streams of calls namely, incoming and outgoing calls. Each stream consists of multiple classes of calls. As part of the internal work load, presence of outgoing calls are always assumed in the system. Arrival of incoming calls obey the Poisson law. Upon seeing a busy server at its arrival epoch, an incoming call will be directed to an orbit according to the class it belongs to and tries to get an idle sever in gap of exponential amount of time with class dependent mean. Similar kind of attempt is being made by the outgoing calls also to reach an idle server. Once the sever becomes idle, if neither an incoming nor an outgoing call turns up for an exponential amount of time, the server goes for vacation and the vacation time is assumed to be exponential. Within each stream, service times of multiple classes of calls are assumed to be independent exponential with class dependent means. Matrix Analytic Method and regenerative approach are used to derive the explicit form of the steady state probabilities. Many performance measures are computed to analyse the system performance. Keywords Retrial queue · Multiclass customers · Regenerative approach · Matrix analytic methods · Vacation model Mathematics Subject Classification 60J27 · 60K25 · 90B22
Introduction Loss models are those where a customer, upon seeing the server busy at its arrival, takes a decision to leave the system. These customers are called blocked customers. There are other cases where a customer sees the server busy at its arrival gets service at a later time by waiting for its turn in an infinite buffer in the system. However, there are some real life situations in which the blocked customers are not patient enough to wait and may decide to leave the system initially, but may try after some random time to get a free server from outside the
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T. G. Deepak [email protected] Sweta Dey [email protected]
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Department of Mathematics, Indian Institute of Space Science and Technology, Thiruvananthapuram 695547, India 0123456789().: V,-vol
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Int. J. Appl. Comput. Math
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system. In such cases, we may assume a blocked customer waits in a virtual waiting space outside the system before retrying to get the server back. These situations are mainly modeled as retrial queues. Retrial queues are broadly used in modelling many practical problems such as those related to call centres, computer networks, cellular networks, random access protocol in LAN etc. A detailed review of retrial queueing literature can be found in Artalejo [2,3], Falin [16], Falin and Templeton [17] , Kim and Kim [18] , Yang and Templeton [29] and in the references there on. Most of the literature related to retrial set up consider the situation where server remains idle after a service completion till a new customer turns up or a successful retrial happens. In the
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