Quasireversibility


In queueing theory, a discipline within the mathematical theory of probability, quasireversibility 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.

Definition

A queue with stationary distribution is quasireversible if its state at time t, x is independent of
for all classes of customer.

Partial balance formulation

Quasireversibility is equivalent to a particular form of partial balance. First, define the reversed rates q by
then considering just customers of a particular class, the arrival and departure processes are the same Poisson process, so
where Mx is a set such that means the state x' represents a single arrival of the particular class of customer to state x'.

Examples