The BMAP/PH/N retrial queue with Markovian flow of breakdowns

We consider a multi-server retrial queue with the Batch Markovian Arrival Process (BMAP). The servers are identical and independent of each other. The service time distribution of a customer by a server is of the phase (PH) type. If a group of primary calls meets idle servers the primary calls occupy the corresponding number of servers. If the number of idle servers is insufficient the rest of calls go to the orbit of unlimited size and repeat their attempts to get service after exponential amount of time independently of each other. Busy servers are subject to breakdowns and repairs. The common flow of breakdowns is the MAP. An event of this flow causes a failure of any busy server with equal probability. When a server fails the repair period starts immediately. This period has PH type distribution and does not depend on the repair time of other broken-down servers and the service time of customers occupying the working servers. A customer whose service was interrupted goes to the orbit with some probability and leaves the system with the supplementary probability. We derive the ergodicity condition and calculate the stationary distribution and the main performance characteristics of the system. Illustrative numerical examples are presented.

[1]  Antonio Gómez-Corral,et al.  A bibliographical guide to the analysis of retrial queues through matrix analytic techniques , 2006, Ann. Oper. Res..

[2]  Hui Li,et al.  Ergodicity of the BMAP/PH/s/s+K retrial queue with PH-retrial times , 2000, Queueing Syst. Theory Appl..

[3]  Che Soong Kim,et al.  Lack of Invariant Property of the Erlang Loss Model in Case of MAP Input , 2005, Queueing Syst. Theory Appl..

[4]  J. Templeton.  Retrial queues , 1999 .

[5]  David W. Lewis,et al.  Matrix theory , 1991 .

[6]  Jesus R. Artalejo,et al.  On the single server retrial queue subject to breakdowns , 1998, Queueing Syst. Theory Appl..

[7]  Srinivas R. Chakravarthy,et al.  A Multi-Server Retrial Queue with BMAP Arrivals and Group Services , 2002, Queueing Syst. Theory Appl..

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

[9]  Oj Onno Boxma,et al.  Teletraffic Analysis and Computer Performance Evaluation , 1988 .

[10]  Alexander Graham,et al.  Kronecker Products and Matrix Calculus: With Applications , 1981 .

[11]  Vidyadhar G. Kulkarni,et al.  Retrial queues revisited , 1998 .

[12]  Jesús R. Artalejo,et al.  A classified bibliography of research on retrial queues: Progress in 1990–1999 , 1999 .

[13]  Alexander N. Dudin,et al.  A Retrial BMAP/PH/N System , 2002, Queueing Syst. Theory Appl..

[14]  Alexander N. Dudin,et al.  A BMAP/PH/n System with Impatient Repeated Calls , 2007, Asia Pac. J. Oper. Res..

[15]  Gunter Bolch,et al.  Modeling Finite-Source Retrial Queueing Systems with Heterogeneous Non-Reliable Servers by MOSEL , 2005 .

[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]  Quan-Lin Li,et al.  A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs , 2006, Ann. Oper. Res..

[19]  Che Soong Kim,et al.  Performance Analysis of a Loss Retrial BMAP/PH/N System , 2004 .

[20]  Kuo-Hsiung Wang,et al.  Cost analysis of the M/M/R machine repair problem with balking, reneging, and server breakdowns , 1999, J. Oper. Res. Soc..

[21]  王國雄,et al.  Cost analysis of the cold-standby M/M/R machine repair problem with multiple modes of failure , 1998 .

[22]  Peter G. Taylor,et al.  Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes , 1995 .

[23]  Jewgeni H. Dshalalow,et al.  Frontiers in Queueing: Models and Applications in Science and Engineering , 1997 .

[24]  Alexander N. Dudin,et al.  Mobile Networks Modeling the access to a wireless network at hot spots , 2005, Eur. Trans. Telecommun..

[25]  Vidyadhar G. Kulkarni,et al.  Retrial queues with server subject to breakdowns and repairs , 1990, Queueing Syst. Theory Appl..

[26]  Quanlin Li,et al.  Reliability Analysis of the Retrial Queue with Server Breakdowns and Repairs , 2001, Queueing Syst. Theory Appl..

[27]  Jesús R. Artalejo,et al.  Accessible bibliography on retrial queues , 1999 .

[28]  G Fayolle.  A simple telephone exchange with delayed feedbacks , 1986 .

[29]  Tao Yang,et al.  A survey on retrial queues , 1987, Queueing Syst. Theory Appl..