Quasireversibility Explained

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

\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

\pi( x)q'( x, x' ) = \pi( x' )q( x' , 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 _{x' \in M _{x}} q( x, x' ) = \sum _{x' \in M _{x}} q'( x, x' )

where M_x is a set such that

\scriptstyle{ x' \in M _{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

Notes and References

1. Muntz. R.R.. 1972. Poisson departure process and queueing networks (IBM Research Report RC 4145). IBM Thomas J. Watson Research Center. Yorktown Heights, N.Y..
2. Kelly . F. P. . Frank Kelly (mathematician). Networks of Queues with Customers of Different Types . Journal of Applied Probability . 12 . 3 . 542–554 . 10.2307/3212869 . 3212869. 1975 . 51917794 .
3. Kelly . F. P. . Frank Kelly (mathematician). Networks of Queues . Advances in Applied Probability . 8 . 2 . 416–432 . 10.2307/1425912 . 1425912. 1976 . 204177645 .
4. Book: Peter G. Harrison. Peter G.. Harrison. Naresh M.. Patel. Performance Modelling of Communication Networks and Computer Architectures. Addison-Wesley. 1992. 288. 0-201-54419-9. registration.
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. . Queueing Systems . 28 . 4 . 377 . 10.1023/A:1019115626557 . 1998 . Markov network processes with product form stationary distributions. 14471818 .
7. Kelly, F.P., Reversibility and Stochastic Networks, 1978 pages 66-67
8. Burke . P. J. . The Output of a Queuing System . 10.1287/opre.4.6.699 . Operations Research . 4 . 6 . 699–704 . 1956 . 55089958 .
9. Burke . P. J. . The Output Process of a Stationary M/M/s Queueing System . 10.1214/aoms/1177698238 . The Annals of Mathematical Statistics . 39 . 4 . 1144–1152 . 1968 . free .
10. O'Connell . N. . Yor . M. . Brownian analogues of Burke's theorem . 10.1016/S0304-4149(01)00119-3 . Stochastic Processes and Their Applications . 96 . 2 . 285–298 . December 2001 . free .
11. Book: Kelly, F.P.. Frank Kelly (mathematician). 1979. Reversibility and Stochastic Networks. Wiley. New York. 2011-12-02. 2012-02-05. https://web.archive.org/web/20120205024930/http://www.statslab.cam.ac.uk/~frank/rsn.html. live.
12. Book: Dao-Thi . T. H. . Mairesse . J. . Zero-Automatic Queues . 10.1007/11549970_6 . Formal Techniques for Computer Systems and Business Processes . Lecture Notes in Computer Science . 3670 . 64 . 2005 . 978-3-540-28701-8 .