This paper deals with the BMAP/G/1 queue service system with server vacation. In this model a batch of customers arrives at the infinite buffer queue according to a batch Markovian arrival process (BMAP). The service times are independent and identically distributed. The server occasionally takes vacations, during which no customer is served. After finishing the vacation, the server continues to serve the queue. If the server finds the queue empty upon return from vacation then it immediately takes the next vacation. The vacation periods are independent and identically distributed. For this vacation model the authors assume that the following properties are valid. The independence property: the arrival process, the customer service times and the length of the vacation periods are independent. The non-preemptive service property: the service is non-preemptive, hence the service of the actual customer is finished before the server goes on vacation. To complete the description of the vacation model, the authors specify the discipline determining the end of service periods. Specifically, they study the BMAP/G/1 vacation model in the following two cases that are defined in the literature: under binomial-gated discipline or under binomial-exhaustive discipline. For these two models the authors prove that a specific form-functional equation can be established for the vector probability generating function (GF) of the stationary number of customers at the start of vacations. They also give a closed-form solution of this vector GF by applying a recursive method. Leonardo Pasini

Recensione dell' articolo: ( Saffer, Zsolt; Telek, Mikl´os - "Closed form results for BMAP/G/1 vacation model with binomial type discipline" - Publ.Math.Debrecen 76 (2010), no.3-4, 359–378.)

PASINI, Leonardo
2011-01-01

Abstract

This paper deals with the BMAP/G/1 queue service system with server vacation. In this model a batch of customers arrives at the infinite buffer queue according to a batch Markovian arrival process (BMAP). The service times are independent and identically distributed. The server occasionally takes vacations, during which no customer is served. After finishing the vacation, the server continues to serve the queue. If the server finds the queue empty upon return from vacation then it immediately takes the next vacation. The vacation periods are independent and identically distributed. For this vacation model the authors assume that the following properties are valid. The independence property: the arrival process, the customer service times and the length of the vacation periods are independent. The non-preemptive service property: the service is non-preemptive, hence the service of the actual customer is finished before the server goes on vacation. To complete the description of the vacation model, the authors specify the discipline determining the end of service periods. Specifically, they study the BMAP/G/1 vacation model in the following two cases that are defined in the literature: under binomial-gated discipline or under binomial-exhaustive discipline. For these two models the authors prove that a specific form-functional equation can be established for the vector probability generating function (GF) of the stationary number of customers at the start of vacations. They also give a closed-form solution of this vector GF by applying a recursive method. Leonardo Pasini
2011
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11581/332185
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