# 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 Mx 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.

## References

1. ^ Muntz, R.R. (1972). Poisson departure process and queueing networks (IBM Research Report RC 4145) (Technical report). Yorktown Heights, N.Y.: IBM Thomas J. Watson Research Center.
2. ^ Kelly, F. P. (1975). "Networks of Queues with Customers of Different Types". Journal of Applied Probability. 12 (3): 542–554. doi:10.2307/3212869. JSTOR 3212869.
3. ^ Kelly, F. P. (1976). "Networks of Queues". Advances in Applied Probability. 8 (2): 416–432. doi:10.2307/1425912. JSTOR 1425912.
4. ^ Harrison, Peter G.; Patel, Naresh M. (1992). Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. p. 288. ISBN 0-201-54419-9.
5. ^ Kelly, F.P. (1982). Networks of quasireversible nodes. In Applied Probability and Computer Science: The Interface (Ralph L. Disney and Teunis J. Ott, editors.) 1 3-29. Birkhäuser, Boston
6. ^ Chao, X.; Miyazawa, M.; Serfozo, R. F.; Takada, H. (1998). "Markov network processes with product form stationary distributions". Queueing Systems. 28 (4): 377. doi:10.1023/A:1019115626557.
7. ^ Kelly, F.P., Reversibility and Stochastic Networks, 1978 pages 66-67
8. ^ Burke, P. J. (1956). "The Output of a Queuing System". Operations Research. 4 (6): 699–704. doi:10.1287/opre.4.6.699.
9. ^ Burke, P. J. (1968). "The Output Process of a Stationary M/M/s Queueing System". The Annals of Mathematical Statistics. 39 (4): 1144. doi:10.1214/aoms/1177698238.
10. ^ O'Connell, N.; Yor, M. (December 2001). "Brownian analogues of Burke's theorem". Stochastic Processes and their Applications. 96 (2): 285–298. doi:10.1016/S0304-4149(01)00119-3.
11. ^ Kelly, F.P. (1979). Reversibility and Stochastic Networks. New York: Wiley.
12. ^ Dao-Thi, T. H.; Mairesse, J. (2005). "Zero-Automatic Queues". Formal Techniques for Computer Systems and Business Processes. Lecture Notes in Computer Science. 3670. p. 64. doi:10.1007/11549970_6. ISBN 978-3-540-28701-8.