A BMAP/G/1 Retrial Queue with a Server Subject to Breakdowns and Repairs

In this paper, we consider a BMAP/G/1 retrial queue with a server subject to breakdowns and repairs, where the life time of the server is exponential and the repair time is general. We use the supplementary variable method, which combines with the matrix-analytic method and the censoring technique, to study the system. We apply the RG-factorization of a level-dependent continuous-time Markov chain of M/G/1 type to provide the stationary performance measures of the system, for example, the stationary availability, failure frequency and queue length. Furthermore, we use the RG-factorization of a level-dependent Markov renewal process of M/G/1 type to express the Laplace transform of the distribution of a first passage time such as the reliability function and the busy period.

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

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

[3]  A. W. Kemp,et al.  Applied Probability and Queues , 1989 .

[4]  Marcel F. Neuts,et al.  Structured Stochastic Matrices of M/G/1 Type and Their Applications , 1989 .

[5]  Attahiru Sule Alfa,et al.  Matrix analytical methods for M/PH/1 retrial queues , 1995 .

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

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

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

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

[10]  S. Wittevrongel,et al.  Queueing Systems , 2019, Introduction to Stochastic Processes and Simulation.

[11]  Bara Kim,et al.  MAP1, MAP2/M/c retrial queue with guard channels and its application to cellular networks , 1999 .

[12]  Hui Li,et al.  TheM/G/1 retrial queue with the server subject to starting failures , 1994, Queueing Syst. Theory Appl..

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

[14]  Srinivas R. Chakravarthy,et al.  Analysis of a retrial queuing model with MAP arrivals and two types of customers , 2003 .

[15]  B. Avi-Itzhak,et al.  A Many-Server Queue with Service Interruptions , 1968, Oper. Res..

[16]  Marcel F. Neuts,et al.  Some steady-state distributions for the MAP /SM /1 queue , 1994 .

[17]  A. N. Dudin,et al.  BMAP|SM⥻1 model with Markov modulated retrials , 1999 .

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

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

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

[21]  Marcel F. Neuts,et al.  A Markovian Queue with N Servers Subject to Breakdowns and Repairs , 1979 .

[22]  Quan-Lin Li,et al.  A CONSTRUCTIVE METHOD FOR FINDING fl-INVARIANT MEASURES FOR TRANSITION MATRICES OF M=G=1 TYPE , 2002 .

[23]  Attahiru Sule Alfa,et al.  The map/ph/1 retrial queue , 1998 .

[24]  W. Whitt,et al.  The transient BMAP/G/l queue , 1994 .

[25]  V. G. Kulkarni,et al.  Stability condition for a single-server retrial queue , 1993, Advances in Applied Probability.

[26]  Jin-Fu Chang,et al.  Departure Processes of BMAP/G/1 Queues , 2001, Queueing Syst. Theory Appl..

[27]  J. Dshalalow.  Advances in Queueing Theory, Methods, and Open Problems , 1995 .

[28]  Gennadi Falin,et al.  A survey of retrial queues , 1990, Queueing Syst. Theory Appl..

[29]  Xiuli Chao,et al.  Reliability analysis of M/G/1 queueing systems with server breakdowns and vacations , 1997, Journal of Applied Probability.

[30]  Quan-Lin Li,et al.  β-Invariant Measures for Transition Matrices of GI/M/1 Type , 2003 .

[31]  B. V. Gnedenko,et al.  Introduction to the Second Edition , 1989 .

[32]  David M. Lucantoni,et al.  The BMAP/G/1 QUEUE: A Tutorial , 1993, Performance/SIGMETRICS Tutorials.

[33]  Winfried K. Grassmann,et al.  Equilibrium distribution of block-structured Markov chains with repeating rows , 1990, Journal of Applied Probability.

[34]  Tom Burr,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 2001, Technometrics.

[35]  V. Ramaswami.  THE N/G/1 QUEUE AND ITS DETAILED ANALYSIS , 1980 .

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

[37]  Jongwoo Jeon,et al.  Analysis of an N/G/1 finite queue with the supplementary variable method , 1999 .

[38]  Boris Gnedenko,et al.  Introduction to queueing theory , 1968 .

[39]  Quan-Lin Li,et al.  Reliability approximation of a Markov queueing system with server breakdown and repair , 1997 .

[40]  Amar Aissani,et al.  A retrial queue with redundancy and unreliable server , 1994, Queueing Syst. Theory Appl..

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

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

[43]  Marcel F. Neuts,et al.  Numerical investigation of a multiserver retrial model , 1990, Queueing Syst. Theory Appl..

[44]  Jongwoo Jeon,et al.  A new approach to an N/G/1 queue , 2000, Queueing Syst. Theory Appl..

[45]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[46]  Jesús R. Artalejo,et al.  New results in retrial queueing systems with breakdown of the servers , 1994 .

[47]  Igor N. Kovalenko,et al.  Introduction to Queuing Theory , 1989 .

[48]  Attahiru Sule Alfa,et al.  Matrix analytic methods for a multi-server retrial queue with buffer , 1999 .

[49]  Peter G. Taylor,et al.  Matrix-analytic methods: theory and applications , 2002 .

[50]  Alexander N. Dudin,et al.  The BMAP/SM/1 retrial queue with controllable operation modes , 2001, Eur. J. Oper. Res..

[51]  Quan-Lin Li,et al.  THE RG-FACTORIZATION IN BLOCK-STRUCTURED MARKOV RENEWAL PROCESSES , 2004 .

[52]  Upendra Dave,et al.  Applied Probability and Queues , 1987 .

[53]  Peter G. Taylor,et al.  Advances in Algorithmic Methods for Stochastic Models , 2000 .

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

[55]  Xue-Ming Yuan,et al.  First passage times for Markov renewal processes and applications , 2000 .