The M/M/c queue with mass exodus and mass arrivals when empty

In this paper we consider an MIMIc queue modified to allow both mass arrivals when the system is empty and the workload to be removed. Properties of queues which terminate when the server becomes idle are firstly developed. Recurrence properties, equilibrium distribution, and equilibrium queue-size structure are studied for the case of resurrection and no mass exodus. All of these results are then generalized to allow for the removal of the entire workload. In particular, we obtain the Laplace transformation of the transition probability for the absorptive MIMIc queue.

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

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

[3]  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..

[4]  Junping Li,et al.  Markov branching processes with immigration-migration and resurrection , 2011 .

[5]  Eric Renshaw,et al.  THE MIMI1 QUEUE WITH MASS EXODUS , 1997 .

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

[7]  A. N. Dudin,et al.  BMAP/SM/1 queue with Markovian input of disasters and non-instantaneous recovery , 2001, Perform. Evaluation.

[8]  S. Asmussen,et al.  Applied Probability and Queues , 1989 .

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

[10]  Carl M. Harris,et al.  Fundamentals of queueing theory (2nd ed.). , 1985 .

[11]  Alexander I. Zeifman,et al.  Perturbation Bounds for M t /M t /N Queue with Catastrophes , 2012 .

[12]  Anyue Chen,et al.  Markov branching processes with instantaneous immigration , 1990 .

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

[14]  Junping Li,et al.  Markovian bulk-arrival and bulk-service queues with state-dependent control , 2010, Queueing Syst. Theory Appl..

[15]  Virginia Giorno,et al.  A note on birth–death processes with catastrophes , 2008 .

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

[17]  J.R. Artalejo,et al.  G-networks: A versatile approach for work removal in queueing networks , 2000, Eur. J. Oper. Res..

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