This paper deals with an with the analysis of a two-phase queueing system with one server in each phase and with a Markov arrival process. Customers in each phase are served according to the FIFO discipline. A customer is served in each server a random time with an arbitrary distribution function B1(x) in the first phase and B2(x) in the second one. There are buffers of finite capacity in each phases. An arriving customer who finds the first buffer full is lost. Similarly, if the second server and its buffer are completely occupied, a customer who goes from the first stage to the second one is lost. After the service completion of a customer in the first phase, independently of the fact that he has passed to the second phase or has lost, the service of the customer in the second server is interrupted and this customer begins his service again and will be served a random time with the same distribution function B2(x). For the analysis of such a queueing system we have introduced a Markov chain generated by the moments of completion of the service in the first phase. We have obtained the stationary distribution of the Markov chain. This allowed us, using notions of renewal theory, to derive the stationary distributions of the queue length in each phase for an arbitrary instant.

### Analysis of a two-phase queueing system with a Markov arrival process and losses

#### Abstract

This paper deals with an with the analysis of a two-phase queueing system with one server in each phase and with a Markov arrival process. Customers in each phase are served according to the FIFO discipline. A customer is served in each server a random time with an arbitrary distribution function B1(x) in the first phase and B2(x) in the second one. There are buffers of finite capacity in each phases. An arriving customer who finds the first buffer full is lost. Similarly, if the second server and its buffer are completely occupied, a customer who goes from the first stage to the second one is lost. After the service completion of a customer in the first phase, independently of the fact that he has passed to the second phase or has lost, the service of the customer in the second server is interrupted and this customer begins his service again and will be served a random time with the same distribution function B2(x). For the analysis of such a queueing system we have introduced a Markov chain generated by the moments of completion of the service in the first phase. We have obtained the stationary distribution of the Markov chain. This allowed us, using notions of renewal theory, to derive the stationary distributions of the queue length in each phase for an arbitrary instant.
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2005
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/1068067`
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