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.

Definition

A queue with stationary distribution $\pi$ is quasireversible if its state at time ''t'', ''x(t)'' is independent of

* the arrival times for each class of customer subsequent to time ''t'',
* the departure times for each class of customer prior to time ''t''

for all classes of customer.

Partial balance formulation

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

:$\pi(\mathbf x)q'(\mathbf x,\mathbf) = \pi(\mathbf)q(\mathbf,\mathbf x)$

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

:$\alpha = \sum_ q(\mathbf x,\mathbf) = \sum_ q'(\mathbf x,\mathbf)$

where ''M_x'' is a set such that $\scriptstyle$ means the state x' represents a single arrival of the particular class of customer to state x.

Examples

* Burke's theorem shows that an M/M/m queueing system is quasireversible.
* Kelly showed that each station of a BCMP network is quasireversible when viewed in isolation.
* G-queues in G-networks are quasireversible.

See also

:*Time reversibility

References

{{Queueing theory

Queueing theory