Product-forms and functional rates

A recent result of Fourneau et al. derives conditions under which the joint state of a set of Markov chains with functional rates has a product-form solution for its equilibrium state probabilities, when they exist. The present note shows how the Reversed Compound Agent Theorem (RCAT) obtains the same result in a few lines. In fact a more general result also follows.

[1]  K. Banasiewicz.  Economic and organizational effects of different legal and organizational forms of enterprises , 2006 .

[2]  Jean-Michel Fourneau,et al.  Product form for stochastic automata networks , 2007, Valuetools 2007.

[3]  Erol Gelenbe,et al.  G-networks with multiple classes of signals and positive customers , 1998, Eur. J. Oper. Res..

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

[5]  Erol Gelenbe,et al.  Learning in the Recurrent Random Neural Network , 1992, Neural Computation.

[6]  Richard J. Boucherie.  A Characterization of Independence for Competing Markov Chains with Applications to Stochastic Petri Nets , 1994, IEEE Trans. Software Eng..

[7]  Jane Hillston,et al.  A compositional approach to performance modelling , 1996 .

[8]  Jean-Michel Fourneau.  Product Form Steady-State Distribution for Stochastic Automata Networks with Domino Synchronizations , 2008, EPEW.

[9]  Peter G. Harrison,et al.  Reversed processes, product forms and a non-product form☆ , 2004 .

[10]  Erol Gelenbe,et al.  Random Neural Networks with Negative and Positive Signals and Product Form Solution , 1989, Neural Computation.

[11]  Peter G. Harrison,et al.  Separable equilibrium state probabilities via time reversal in Markovian process algebra , 2005, Theor. Comput. Sci..

[12]  James R. Jackson,et al.  Jobshop-Like Queueing Systems , 2004, Manag. Sci..

[13]  K. Mani Chandy,et al.  Open, Closed, and Mixed Networks of Queues with Different Classes of Customers , 1975, JACM.

[14]  Peter G. Harrison,et al.  Turning back time in Markovian process algebra , 2003, Theor. Comput. Sci..

[15]  Erol Gelenbe,et al.  G-Networks with Multiple Classes of Negative and Positive Customers , 1996, Theor. Comput. Sci..

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

[17]  Peter G. Harrison.  Compositional reversed Markov processes, with applications to G-networks , 2004, Perform. Evaluation.

[18]  Erol Gelenbe,et al.  G-Networks with resets , 2002 .