Tandem queueing system with infinite and finite intermediate buffers and generalized phase-type service time distribution

Keywords: Queueing Tandem queueing system Marked Markovian arrival process Phase-type distribution Laplace–Stieltjes transform a b s t r a c t A tandem queueing system with infinite and finite intermediate buffers, heterogeneous customers and generalized phase-type service time distribution at the second stage is investigated. The first stage of the tandem has a finite number of servers without buffer. The second stage consists of an infinite and a finite buffers and a finite number of servers. The arrival flow of customers is described by a Marked Markovian arrival process. Type 1 customers arrive to the first stage while type 2 customers arrive to the second stage directly. The service time at the first stage has an exponential distribution. The service times of type 1 and type 2 customers at the second stage have a phase-type distribution with different parameters. During a waiting period in the intermediate buffer, type 1 customers can be impatient and leave the system. The ergodicity condition and the steady-state distribution of the system states are analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transform of the sojourn time distribution of type 2 customers is derived. Numerical examples are presented. Tandem queueing systems are very important part of the queueing theory that takes into account the possibility that a customer may need service from several sequentially arranged serv-ers. The overwhelming majority of the results in the theory of tandem queues is obtained for systems with stationary Poisson arrival process and exponential service time distribution. Tandem queueing systems with correlated arrival flows were investigated not so extensively. For background information and overview of the present state of the art in the study of tandem queueing systems with correlated arrival flows and phase-type service time distribution see, e. and Gomez-Corral and Martos (2006) are devoted to the MAP=PH=1 ! =G=1 system with blocking. The tandem queue BMAP=G=1=N ! =PH=1=M with losses is considered in Klimenok et al. (2005). The tandem-queue of the BMAP=G=1 ! =PH=1=M type with losses and feedback is investigated in Klimenok et al. (2007). Analysis of the BMAP=G=1 ! =PH=1=M tandem queue with retrials and losses is provided in Kim et al. (2010). In Kim et al. (2013a), a tandem queueing system with more than one ser-ver in each stage, Markovian arrival flow, and phase-type service time distribution at the second stage is considered. In this paper, we analyze a tandem queueing system with two …