Fundamentals of Queueing Networks

Volume 46 of the series Stochastic Modelling and Applied Probability pp 69-96

Kelly Networks

  • Hong ChenAffiliated withFaculty of Commerce and Business Administration, University of British Columbia
  • , David D. YaoAffiliated withDepartment of Operations Research and Industrial Engineering, Columbia University

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The central approach used in this chapter, just as in Section 2.6, is to examine a stationary Markov chain together with its time-reversal: another Markov chain defined through reversing the time index of the original chain. Applying this technique to the Jackson network in Section 2.6, we not only recovered the product-form distributions derived earlier in Chapter 2 but also established the Poisson property of the exit processes from the network. In particular, we showed that in equilibrium the Jackson network preserves the Poisson property of the input processes to the network, such that the exit processes are also Poisson. This “Poisson-in-Poisson-out” property is the main motivation here for defining and studying a class of so-called quasi-reversible queues. This class is an extension of the basic M/M(n)/1 queues, which constitute the nodes in a Jackson network, to allow multiclass jobs and more general arrival and service disciplines, while still preserving the Poisson-in-Poisson-out property. It turns out that a general multiclass network, known as a Kelly network, which connects a set of quasi-reversible queues through some very general routing schemes, still enjoys the basic properties of the Jackson network, namely, the product-form equilibrium distribution and the Poisson-in-Poisson-out property.