Decay property of stopped Markovian bulk-arriving queues

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λ C is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λ C ]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λ C ]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λ C -transient and some deep properties are revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.

[1]  Mu-Fa Chen,et al.  From Markov Chains to Non-Equilibrium Particle Systems , 1992 .

[2]  Erol Gelenbe,et al.  Product form networks with negative and positive customers , 1991 .

[3]  W. J. Anderson.  Continuous-Time Markov Chains: An Applications-Oriented Approach , 1991 .

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

[5]  Mu-Fa Chen.  Eigenvalues, inequalities and ergodic theory , 2000 .

[6]  E. A. vanDoorn,et al.  Conditions for exponential ergodicity and bounds for the Deacy parameters of a birth-death process , 1982 .

[7]  D. Vere-Jones.  GEOMETRIC ERGODICITY IN DENUMERABLE MARKOV CHAINS , 1962 .

[8]  B. Krishna Kumar,et al.  Density-dependent birth and death process with state-dependent immigration , 1991 .

[9]  W. J. Anderson.  Continuous-Time Markov Chains , 1991 .

[10]  Alexander N. Dudin,et al.  A BMAP/SM/1 queueing system with Markovian arrival input of disasters , 1999 .

[11]  E. Gelenbe.  Product-form queueing networks with negative and positive customers , 1991 .

[12]  Kai Lai Chung,et al.  Markov Chains with Stationary Transition Probabilities , 1961 .

[13]  E. Seneta,et al.  On quasi-stationary distributions in absorbing continuous-time finite Markov chains , 1967, Journal of Applied Probability.

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

[15]  Onno J. Boxma,et al.  Wiener-Hopf analysis of an M/G/1 queue with negative customers and of a related class of random walks , 1996, Queueing Syst. Theory Appl..

[16]  Eric Renshaw,et al.  Markovian bulk-arriving queues with state-dependent control at idle time , 2004, Advances in Applied Probability.

[17]  E. V. Doorn,et al.  Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes , 1991 .

[18]  Peter G. Harrison,et al.  SOJOURN TIMES IN SINGLE-SERVER QUEUES WITH NEGATIVE CUSTOMERS , 1993 .

[19]  Karl Sigman,et al.  A Pollaczek–Khintchine formula for M/G/1 queues with disasters , 1996, Journal of Applied Probability.

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

[21]  P. Pollett.  The determination of quasistationary distributions directly from the transition rates of an absorbing Markov chain , 1995 .

[22]  Hajime Sato,et al.  EIGENVALUE EXPRESSION FOR A BATCH MARKOVIAN ARRIVAL PROCESS , 1997 .

[23]  Quasistationary distributions for continuous time Markov chains when absorption is not certain , 1999 .

[24]  R. Syski.  Passage Times for Markov Chains , 1992 .

[25]  P. K. Pollett,et al.  On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains , 1993, Advances in Applied Probability.

[26]  D. J. Daley,et al.  Quasi-Stationary Behaviour of a Left-Continuous Random Walk , 1969 .

[27]  Masaaki Kijima,et al.  Quasi-limiting distributions of Markov chains that are skip-free to the left in continuous time , 1993, Journal of Applied Probability.

[28]  Uniqueness and Extinction of Weighted Markov Branching Processes , 2005 .

[29]  W. Henderson.  Queueing networks by negative customers and negative queue lengths , 1993 .

[30]  C. D. Litton,et al.  A First Course in Bulk Queues , 1983 .

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

[32]  J. F. C. Kingman,et al.  The Exponential Decay of Markov Transition Probabilities , 1963 .

[33]  Marcel F. Neuts,et al.  Matrix-geometric solutions in stochastic models - an algorithmic approach , 1982 .

[34]  F. P. Kelly.  Probability, Statistics and Analysis: Invariant measures and the q-matrix , 1983 .

[35]  Eric Renshaw,et al.  The M / M /1 queue with mass exodus and mass arrivals when empty , 1997 .

[36]  S. Aoki,et al.  Prediction of galvanic corrosion rates by the boundary element method , 1991 .

[37]  Richard L. Tweedie,et al.  SOME ERGODIC PROPERTIES OF THE FELLER MINIMAL PROCESS , 1974 .

[38]  Wolfgang Stadje.  Some exact expressions for the bulk-arrival queue mx/M/1 , 1989, Queueing Syst. Theory Appl..

[39]  R. Syski,et al.  Fundamentals of Queueing Theory , 1999, Technometrics.

[40]  P. K. Pollett,et al.  Reversibility, invariance and μ-invariance , 1988, Advances in Applied Probability.

[41]  M. L. Chaudhry,et al.  A first course in bulk queues , 1983 .

[42]  Erol Gelenbe,et al.  Queues with negative arrivals , 1991, Journal of Applied Probability.

[43]  David C. Flaspohler.  Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains , 1974 .