# 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 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

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

where *M _{x}* is a set such that 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

## References

- ↑ 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. - ↑ 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. - ↑ Kelly, F. P. (1976). "Networks of Queues".
*Advances in Applied Probability*.**8**(2): 416–432. doi:10.2307/1425912. JSTOR 1425912. - ↑ Harrison, Peter G.; Patel, Naresh M. (1992).
*Performance Modelling of Communication Networks and Computer Architectures*. Addison-Wesley. p. 288. ISBN 0-201-54419-9. - ↑ 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 - ↑ 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. - ↑ Kelly, F.P., Reversibility and Stochastic Networks, 1978 pages 66-67
- ↑ Burke, P. J. (1956). "The Output of a Queuing System".
*Operations Research*.**4**(6): 699–704. doi:10.1287/opre.4.6.699. - ↑ 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. - ↑ 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. - ↑ Kelly, F.P. (1979).
*Reversibility and Stochastic Networks*. New York: Wiley. - ↑ 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.