Quasireversibility

In queueing theory, a discipline within the mathematical theory of probability, quasireversibility (sometimes QR) is a property of some queues. The concept was first identified by Richard R. Muntz and further developed by Frank Kelly. Quasireversibility differs from reversibility in that a stronger condition is imposed on arrival rates and a weaker condition is applied on probability fluxes. For example, an M/M/1 queue with state-dependent arrival rates and state-dependent service times is reversible, but not quasireversible.

A network of queues, such that each individual queue when considered in isolation is quasireversible, always has a product form stationary distribution. Quasireversibility had been conjectured to be a necessary condition for a product form solution in a queueing network, but this was shown not to be the case. Chao et al. exhibited a product form network where quasireversibility was not satisfied.

A queue with stationary distribution ${\displaystyle \pi }$ is quasireversible if its state at time t, x(t) is independent of

for all classes of customer.

Quasireversibility is equivalent to a particular form of partial balance. First, define the reversed rates q'(x,x') by

then considering just customers of a particular class, the arrival and departure processes are the same Poisson process (with parameter ${\displaystyle \alpha }$), so

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