A BMAPPH1 queue with feedback operating in a random environment

Feedback queues play an important role in real-life service systems, where customers may require repeated services. In this paper, we consider a feedback queue with batch Markovian arrivals and phase type services. We further assume that both the arrival process and service times are influenced by an external finite state Markovian environment. The stationary state distributions of the queue and the sojourn time are calculated and numerical examples are presented.

[1]  Ralph L. Disney.  A note on sojourn times in M/G/1 queues with instantaneous, bernoulli feedback , 1981 .

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

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

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

[5]  Gerard Hooghiemstra,et al.  The M/G/1 processor sharing queue as the almost sure limit of feedback queues , 1990 .

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

[7]  Jeffrey J. Hunter,et al.  Filtering of Markov renewal queues, III: semi-Markov processes embedded in feedback queues , 1984, Advances in Applied Probability.

[8]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[9]  Ralph L. Disney,et al.  Stationary queue-length and waiting-time distributions in single-server feedback queues , 1984 .

[10]  J. J. Hunter,et al.  Sojourn time problems in feedback queues , 1989, Queueing Syst. Theory Appl..

[11]  V. Ramaswami,et al.  Advances in Probability Theory and Stochastic Processes , 2001 .

[12]  V. Ramaswami.  A stable recursion for the steady state vector in markov chains of m/g/1 type , 1988 .

[13]  Jeffrey J. Hunter,et al.  Filtering of Markov renewal queues, II: Birth-death queues , 1983, Advances in Applied Probability.

[14]  Jeffrey J. Hunter,et al.  Filtering of Markov renewal queues, I: Feedback queues , 1983, Advances in Applied Probability.

[15]  Hans van den Berg,et al.  TheM/G/1 queue with processor sharing and its relation to a feedback queue , 1991, Queueing Syst. Theory Appl..

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

[17]  Jeffrey J. Hunter,et al.  Filtering of Markov renewal queues, IV: Flow processes in feedback queues , 1985, Advances in Applied Probability.

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

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

[20]  L. Takács.  A single-server queue with feedback , 1963 .