Tandem Retrial Queueing System with Correlated Arrival Flow and Operation of the Second Station Described by a Markov Chain

Tandem queues are good mathematical models for description of information transmission in various communication systems and networks. These queues play also an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, their investigation is interesting for theory and applications. In this paper, we consider tandem queue suitable for modeling the systems and networks where information flows are correlated and bursty what is typical for many modern telecommunication networks. Possible correlation of customers inter-arrival times and batch arrivals are taken into account via of consideration of the Batch Markovian Arrival Process (BMAP) as input stream to the system. The system consists of two stations. The service time at the station 1 is assumed to be generally distributed. There is no buffer at this station, and customers who meet the busy server repeat attempts to enter the system in random time intervals. The service process at the station 2 is assumed to be described by the continuous time Markov chain with a finite state space. This assumption holds good, e.g., if the station 2 has a finite buffer, consists of a finite number of identical or heterogeneous servers where the service time distribution is assumed to be of PH (PHase) type. Markov chain embedded at service completion epochs at the station 1 and the process of system states at arbitrary time are under study. Ergodicity condition and algorithms for computing the steady state probabilities are presented.

[1]  Valentina Klimenok,et al.  Investigation of the BMAP/G/1→·/PH/1/M tandem queue with retrials and losses , 2010 .

[2]  Armin Heindl,et al.  Decomposition of general queueing networks with MMPP inputs and customer losses , 2003, Perform. Evaluation.

[3]  A. Gómez-Corral,et al.  On a tandem G-network with blocking , 2002, Advances in Applied Probability.

[4]  J. Kemeny,et al.  Denumerable Markov chains , 1969 .

[5]  Paola Inverardi,et al.  A review on queueing network models with finite capacity queues for software architectures performance prediction , 2003, Perform. Evaluation.

[6]  Chelliah Sriskandarajah,et al.  A Survey of Machine Scheduling Problems with Blocking and No-Wait in Process , 1996, Oper. Res..

[7]  Alexander N. Dudin,et al.  The BMAP/G/1/N -> ./PH/1/M tandem queue with losses , 2005, Perform. Evaluation.

[8]  Srinivas R. Chakravarthy.  The Batch Markovian Arrival Process: A Review and Future Work , 2001 .

[9]  David M. Lucantoni,et al.  New results for the single server queue with a batch Markovian arrival process , 1991 .

[10]  Alexander N. Dudin,et al.  A Two-Phase BMAP|G|1|N → PH|1|M – 1 System with Blocking , 2004 .

[11]  Harry G. Perros,et al.  A Bibliography of Papers on Queueing Networks with Finite Capacity Queues , 1989, Perform. Evaluation.

[12]  V. I. Klimenok,et al.  Tandem service system with batch Markov flow and repeated calls , 2010 .

[13]  M. Neuts.  A Versatile Markovian Point Process , 1979 .

[14]  Antonio Gómez-Corral,et al.  A matrix-geometric approximation for tandem queues with blocking and repeated attempts , 2002, Oper. Res. Lett..

[15]  Valentina Klimenok,et al.  A Tandem Queue with Group Occupation of Servers at the Second Station , 2012 .

[16]  Valentina Klimenok,et al.  Queueing system BMAP/G/1 with repeated calls , 1999 .

[17]  Che Soong Kim,et al.  A tandem retrial queueing system with two Markovian flows and reservation of channels , 2010, Comput. Oper. Res..

[18]  Christoph Lindemann,et al.  Modeling IP traffic using the batch Markovian arrival process , 2003, Perform. Evaluation.

[19]  Alexander N. Dudin,et al.  A retrial BMAP/SM/1 system with linear repeated requests , 1999, Queueing Syst. Theory Appl..

[20]  Che Soong Kim,et al.  The BMAP/G/1 -> ./PH/1/M tandem queue with feedback and losses , 2007, Perform. Evaluation.

[21]  Armin Heindl.  Decomposition of general tandem queueing networks with MMPP input , 2001, Perform. Evaluation.

[22]  Che Soong Kim,et al.  Priority tandem queueing model with admission control , 2011, Comput. Ind. Eng..

[23]  A. N. Dudin,et al.  A dual tandem queueing system with a finite intermediate buffer and cross traffic , 2010, QTNA.

[24]  Liming Liu,et al.  A tandem network with MAP inputs , 2008, Oper. Res. Lett..

[25]  Alexander N. Dudin,et al.  Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory , 2006, Queueing Syst. Theory Appl..

[26]  Erhan Çinlar,et al.  Introduction to stochastic processes , 1974 .

[27]  Antonio Gómez-Corral,et al.  A Tandem Queue with Blocking and Markovian Arrival Process , 2002, Queueing Syst. Theory Appl..