Abstract
A queueing model consisting of two multi-server service systems is considered. Primary customers arrive at a multi-server queueing system-1 having an infinite buffer. The input flow is described by a MAP (Markovian Arrival Process). The service time of a primary customer has a PH (Phase-type) distribution. Besides the primary customers, a MAP of interruptions arrives to the system. An interruption removes one of the primary customers from the service if the state (phase) of its PH service process does not belong to some given set of so called protected phases when an interruption is successful. The interrupted customer leaves the system permanently with some probability. With complementary probability, the interrupted primary customer moves for service to system-2. This system consists of K independent identical servers and has no buffer. If all K servers are busy at the moment of a primary customer interruption, this customer will be lost. Otherwise, this primary customer starts the service in an arbitrary idle server of system-2. It is assumed that the service time of a primary customer by a server of system-2 has a PH distribution. Upon completion of the service at system-2, the customer becomes priority customer. If, at the service completion moment, there are free servers at system-1, the priority customer immediately starts getting the service at system-1. It is assumed that the service time of a priority customer by a server of system-1 has a PH distribution and this service can not be interrupted. If, at the moment when the primary customer finished the service at system-2, there are no idle servers at system-1, this customer is placed into the finite buffer for priority customers of capacity K. The customers will be picked up for the service according to the FIFO discipline. When a server of system-1 becomes free, it takes for service a priority customer from this buffer, if any. Type-1 customers are picked-up from the infinite buffer only if the buffer for priority customers is empty at the service completion moment at system-1.
Behavior of this system is described by a multi-dimensional Markov chain. Algorithms for checking ergodicity condition and computing the stationary distribution are presented. Formulas for computing important performance measures of the system are derived.
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The Authors thank the referees for the valuable comments and constructive criticism which resulted in improvement in the presentation of the paper.
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Varghese Jacob’s research is supported by the University Grants Commission, Govt. of India, under Faculty Development Programme (Grant No. F.FIP/11th Plan/KLMG003TF03).
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Dudin, A., Jacob, V. & Krishnamoorthy, A. A multi-server queueing system with service interruption, partial protection and repetition of service. Ann Oper Res 233, 101–121 (2015). https://doi.org/10.1007/s10479-013-1318-3
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DOI: https://doi.org/10.1007/s10479-013-1318-3