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^[1] and further developed by Frank Kelly.^[2][3] 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.^[4]

A network of queues, such that each individual queue when considered in isolation is quasireversible, always has a product form stationary distribution.^[5] 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.^[6]

Definition

A queue with stationary distribution ${\displaystyle \pi }$ 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.^[7]

Partial balance formulation

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

${\displaystyle \pi (\mathbf {x} )q'(\mathbf {x} ,\mathbf {x'} )=\pi (\mathbf {x'} )q(\mathbf {x'} ,\mathbf {x} )}$

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

${\displaystyle \alpha =\sum _{\mathbf {x'} \in M_{\mathbf {x} }}q(\mathbf {x} ,\mathbf {x'} )=\sum _{\mathbf {x'} \in M_{\mathbf {x} }}q'(\mathbf {x} ,\mathbf {x'} )}$

where M_x is a set such that ${\displaystyle \scriptstyle {\mathbf {x'} \in M_{\mathbf {x} }}}$ 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.^[8][9][10]
• Kelly showed that each station of a BCMP network is quasireversible when viewed in isolation.^[11]
• G-queues in G-networks are quasireversible.^[12]

See also

• Time reversibility