Queueing Systems

, Volume 28, Issue 4, pp 377-401

First online:

Markov network processes with product form stationary distributions

  • X. ChaoAffiliated withDepartment of Industrial and Manufacturing Engineering, New Jersey Institute of Technology
  • , M. MiyazawaAffiliated withDepartment of Information Sciences, Science University of Tokyo
  • , R.F. SerfozoAffiliated withSchool of Industrial and System Engineering, Georgia Institute of Technology
  • , H. TakadaAffiliated withDepartment of Information Sciences, Science University of Tokyo

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This study concerns the equilibrium behavior of a general class of Markov network processes that includes a variety of queueing networks and networks with interacting components or populations. The focus is on determining when these processes have product form stationary distributions. The approach is to relate the marginal distributions of the process to the stationary distributions of “node transition functions” that represent the nodes in isolation operating under certain fictitious environments. The main result gives necessary and sufficient conditions on the node transition functions for the network process to have a product form stationary distribution. This result yields a procedure for checking for a product form distribution and obtaining such a distribution when it exits. An important subclass of networks are those in which the node transition rates have Poisson arrival components. In this setting, we show that the network process has a product form distribution and is “biased locally balanced” if and only if the network is “quasi-reversible” and certain traffic equations are satisfied. Another subclass of networks are those with reversible routing. We weaken the known sufficient condition for such networks to be product form. We also discuss modeling issues related to queueing networks including time reversals and reversals of the roles of arrivals and departures. The study ends by describing how the results extend to networks with multi-class transitions.

Markov network process product form stationary distribution queueing network negative customer migration process quasi-reversibility biased local balance reversible routing multi-class transitions