Quasireversibility

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]

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.

• 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]

• Time reversibility