State-dependent Coupling of Quasireversible Nodes

The seminal paper of Jackson began a chain of research on queueing networks with product-form stationary distributions which continues strongly to this day. Hard on the heels of the early results on queueing networks followed a series of papers which discussed the relationship between product-form stationary distributions and the quasireversibility of network nodes. More recently, the definition of quasireversibility and the coupling mechanism between nodes have been extended so that they apply to some of the later product-form queueing networks incorporating negative customers, signals, and batch movements.In parallel with this research, it has been shown that some special queueing networks can have arrival and service parameters which depend upon the network state, rather than just the node state, and still retain a generalised product-form stationary distribution.In this paper we begin by offering an alternative proof of a product-form result of Chao and Miyazawa and then build on this proof by postulating a state-dependent coupling mechanism for a quasireversible network. Our main theorem is that the resultant network has a generalised product form stationary distribution. We conclude the paper with some examples.

[1]  P. Pollett.  Connecting reversible Markov processes , 1986, Advances in Applied Probability.

[2]  Richard J. Boucherie,et al.  Product forms for queueing networks with state-dependent multiple job transitions , 1991, Advances in Applied Probability.

[3]  Peter G. Taylor,et al.  Product form in networks of queues with batch arrivals and batch services , 1990, Queueing Syst. Theory Appl..

[4]  P. K. Pollett,et al.  Preserving partial balance in continuous-time Markov chains , 1987, Advances in Applied Probability.

[5]  Michael Pinedo,et al.  Queueing networks - customers, signals and product form solutions , 1999, Wiley-Interscience series in systems and optimization.

[6]  Ronald W. Wolff,et al.  SYMMETRIC QUEUES WITH BATCH DEPARTURES AND THEIR NETWORKS , 1996 .

[7]  Peter G. Taylor,et al.  Quasi-reversibility and networks of queues with nonstandard batch movements , 2000 .

[8]  William A. Massey,et al.  Stochastic ordering for Markov processes on partially ordered spaces with applications to queueing networks , 1991 .

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

[10]  Richard F. Serfozo,et al.  Markovian network processes: Congestion-dependent routing and processing , 1989, Queueing Syst. Theory Appl..

[11]  Peter G. Taylor,et al.  STATE-DEPENDENT SIGNALLING IN QUEUEING NETWORKS , 1994 .

[12]  D. R. Axelrad.  2 – Markov Processes and Markov Random Fields , 1984 .

[13]  Peter G. Taylor,et al.  Triggered batch movement in queueing networks , 1995, Queueing Syst. Theory Appl..

[14]  Michael Pinedo,et al.  On Generalized Networks of Queues with Positive and Negative Arrivals , 1993, Probability in the Engineering and Informational Sciences.

[15]  J. R. Jackson.  Networks of Waiting Lines , 1957 .

[16]  Peter G. Taylor,et al.  CONNECTING INTERNALLY BALANCED QUASI , 1992 .

[17]  Peter Whittle,et al.  Systems in stochastic equilibrium , 1986 .

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

[19]  Xiuli Chao,et al.  Queueing networks , 1999 .

[20]  Erol Gelenbe,et al.  Stability of Product Form G-Networks , 1992 .

[21]  E. Gelenbe.  G-networks by triggered customer movement , 1993 .

[22]  Xiuli Chao,et al.  Networks of Queues with Customers, Signals and Arbitrary Service Time Distributions , 1995, Oper. Res..

[23]  J. Ben Atkinson,et al.  An Introduction to Queueing Networks , 1988 .

[24]  Xiuli Chao,et al.  A result on networks of queues with customer coalescence and state-dependent signaling , 1998 .

[25]  Peter G. Taylor,et al.  Geometric equilibrium distributions for queues with interactive batch departures , 1994, Ann. Oper. Res..

[26]  J. Kingman.  Markov population processes , 1969, Journal of Applied Probability.

[27]  Masakiyo Miyazawa,et al.  Traffic Flows and Product Form Solutions in Stochastic Transfer Networks , 2001, Queueing Syst. Theory Appl..

[28]  Xiuli Chao,et al.  A Note on Queueing Networks with Signals and Random Triggering Times , 1994, Probability in the Engineering and Informational Sciences.

[29]  J. Walrand,et al.  Interconnections of Markov chains and quasi-reversible queuing networks , 1980 .

[30]  Masakiyo Miyazawa,et al.  Markov network processes with product form stationary distributions , 1998, Queueing Syst. Theory Appl..

[31]  Erol Gelenbe.  G-Networks with Signals and Batch Removal , 1993 .

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

[33]  Frank Kelly,et al.  Networks of queues with customers of different types , 1975, Journal of Applied Probability.

[34]  Masakiyo Miyazawa,et al.  A GEOMETRIC PRODUCT-FORM DISTRIBUTION FOR A QUEUEING NETWORK WITH NON-STANDARD BATCH , 1997 .

[35]  Michael Pinedo,et al.  Networks of queues with batch services and customer coalescence , 1996 .

[36]  Masakiyo Miyazawa,et al.  Queueing networks with instantaneous movements: a unified approach by quasi-reversibility , 2000, Advances in Applied Probability.

[37]  Frank Kelly,et al.  Reversibility and Stochastic Networks , 1979 .

[38]  C. E. M. Pearce,et al.  Closed queueing networks with batch services , 1990, Queueing Syst. Theory Appl..

[39]  Nico M. van Dijk,et al.  A note on jackson networks with state-dependent rejection probabilities , 1994, Oper. Res. Lett..

[40]  Nico M. van Dijk,et al.  On Jackson's product form with 'jump-over' blocking , 1988 .

[41]  Richard F. Serfozo,et al.  Markov network processes with string transitions , 1998 .

[42]  Guy Pujolle,et al.  Introduction to queueing networks , 1987 .