A dual tandem queueing system with a finite intermediate buffer and cross traffic

We analyze a dual tandem queue having a single server queueing system with infinite buffer at the first station and a multi-server queueing system with a finite buffer at the second station. Arrival flow is described by the Batch Markovian Arrival Process (BMAP). Service time at the first station is generally distributed while at the second station it is exponentially distributed. In situation when the intermediate buffer between the stations is full at the service completion of a customer at the first station, this customer is lost or blocks the server until service completion in one of the servers at the second station. Besides the customers, which got service at the first station, an additional MAP flow (cross traffic) arrives to the second station directly, not entering the first station. Ergodicity condition for this system is derived. Stationary state distribution of the system at embedded and arbitrary time epochs is computed as well as the main performance measures of the system. Numerical results show possibility of optimization of the system operation by means of appropriate choosing the capacity of an intermediate buffer and the intensity of cross traffic. Necessity of the account of correlation in the arrival processes is illustrated.

[1]  Antonio Gómez-Corral,et al.  Performance of two-stage tandem queues with blocking: The impact of several flows of signals , 2006, Perform. Evaluation.

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

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

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

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

[6]  Antonio Gómez-Corral,et al.  A. B. Clarke's Tandem Queue Revisited—Sojourn Times , 2008 .

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

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

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

[10]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

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

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

[13]  Antonio Gómez-Corral,et al.  Sojourn times in a two‐stage queueing network with blocking , 2004 .

[14]  Valentina Klimenok,et al.  Multi-dimensional quasitoeplitz Markov chains , 1999 .

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

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

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

[18]  Adhir K Basu,et al.  Introduction to Stochastic Process , 2002 .

[19]  B. Conolly.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1991 .

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