Product-form solution

In probability theory, a product-form solution is a particularly efficient form of solution for determining some metric of a system with distinct sub-components, where the metric for the collection of components can be written as a product of the metric across the different components. Using capital Pi notation a product-form solution has algebraic form

${\displaystyle {\text{P}}(x_{1},x_{2},x_{3},\ldots ,x_{n})=B\prod _{i=1}^{n}{\text{P}}(x_{i})}$

where B is some constant. Solutions of this form are of interest as they are computationally inexpensive to evaluate for large values of n. Such solutions in queueing networks are important for finding performance metrics in models of multiprogrammed and time-shared computer systems.

Equilibrium distributions

The first product-form solutions were found for equilibrium distributions of Markov chains. Trivially, models composed of two or more independent sub-components exhibit a product-form solution by the definition of independence. Initially the term was used in queueing networks where the sub-components would be individual queues. For example, Jackson's theorem gives the joint equilibrium distribution of an open queueing network as the product of the equilibrium distributions of the individual queues.^[1] After numerous extensions, chiefly the BCMP network it was thought local balance was a requirement for a product-form solution.^[2][3]

Gelenbe's G-network model was the first to show that this is not the case. Motivated by the need to model biological neurons which have a point-process like spiking behaviour, he introduced the precursor of G-Networks, calling it the random neural network.^[4] By introducing "negative customers" which can destroy or eliminate other customers, he generalised the family of product form networks.^[5] Then this was further extended in several steps, first by Gelenbe's "triggers" which are customers which have the power of moving other customers from some queue to another.^[6] Another new form of customer that also led to product form was Gelenbe's "batch removal".^[7] This was further extended by Erol Gelenbe and Jean-Michel Fourneau with customer types called "resets" which can model the repair of failures: when a queue hits the empty state, representing (for instance) a failure, the queue length can jump back or be "reset" to its steady-state distribution by an arriving reset customer, representing a repair. All these previous types of customers in G-Networks can exist in the same network, including with multiple classes, and they all together still result in the product form solution, taking us far beyond the reversible networks that had been considered before.^[8]

Product-form solutions are sometimes described as "stations are independent in equilibrium".^[9] Product form solutions also exist in networks of bulk queues.^[10]

J.M. Harrison and R.J. Williams note that "virtually all of the models that have been successfully analyzed in classical queueing network theory are models having a so-called product-form stationary distribution"^[9] More recently, product-form solutions have been published for Markov process algebras (e.g. RCAT in PEPA^[11][12]) and stochastic petri nets.^[13][14] Martin Feinberg's deficiency zero theorem gives a sufficient condition for chemical reaction networks to exhibit a product-form stationary distribution.^[15]

The work by Gelenbe also shows that product form G-Networks can be used to model spiking random neural networks, and furthermore that such networks can be used to approximate bounded and continuous real-valued functions.^[16][17]

Sojourn time distributions

The term product form has also been used to refer to the sojourn time distribution in a cyclic queueing system, where the time spent by jobs at M nodes is given as the product of time spent at each node.^[18] In 1957 Reich showed the result for two M/M/1 queues in tandem,^[19] later extending this to n M/M/1 queues in tandem^[20] and it has been shown to apply to overtake–free paths in Jackson networks.^[21] Walrand and Varaiya suggest that non-overtaking (where customers cannot overtake other customers by taking a different route through the network) may be a necessary condition for the result to hold.^[21] Mitrani offers exact solutions to some simple networks with overtaking, showing that none of these exhibit product-form sojourn time distributions.^[22]

For closed networks, Chow showed a result to hold for two service nodes,^[23] which was later generalised to a cycle of queues^[24] and to overtake–free paths in Gordon–Newell networks.^[25][26]

Extensions

• Approximate product-form solutions are computed assuming independent marginal distributions, which can give a good approximation to the stationary distribution under some conditions.^[27][28]
• Semi-product-form solutions are solutions where a distribution can be written as a product where terms have a limited functional dependency on the global state space, which can be approximated.^[29]
• Quasi-product-form solutions are either
  • solutions which are not the product of marginal densities, but the marginal densities describe the distribution in a product-type manner^[30] or
  • approximate form for transient probability distributions which allows transient moments to be approximated.^[31]

References

1. Jackson, James R. (1963). "Jobshop-like queueing systems". Management Science. 10 (1): 131–142. doi:10.1287/mnsc.10.1.131.
2. Boucherie, Richard J.; van Dijk, N. M. (1994). "Local balance in queueing networks with positive and negative customers". Annals of Operations Research. 48 (5): 463–492. doi:10.1007/BF02033315. hdl:1871/12327. S2CID 15599820.
3. Chandy, K. Mani; Howard, J. H. Jr; Towsley, D. F. (1977). "Product form and local balance in queueing networks". Journal of the ACM. 24 (2): 250–263. doi:10.1145/322003.322009. S2CID 6218474.
4. Gelenbe, Erol (1989). "Random Neural Networks with Negative and Positive Signals and Product Form Solution". Neural Computation. 1 (4): 502–510. doi:10.1162/neco.1989.1.4.502. S2CID 207737442.
5. Gelenbe, Erol (1991). "Product-form queueing networks with negative and positive customers". Journal of Applied Probability. 28 (3): 656–663. doi:10.2307/3214499. JSTOR 3214499.
6. Gelenbe, Erol (1993). "G-networks with triggered customer movement". Journal of Applied Probability. 30 (3): 742–748. doi:10.2307/3214781. JSTOR 3214781.
7. Gelenbe, Erol (1993). "G-Networks with triggered customer movement". Probability in the Engineering and Informational Sciences. 7 (3): 335–342. doi:10.1017/S0269964800002953.
8. Gelenbe, Erol; Fourneau, Jean-Michel (2002). "G-Networks with resets". Performance Evaluation. 49 (1): 179–191. doi:10.1016/S0166-5316(02)00127-X.
9. Harrison, J. M.; Williams, R. J. (1992). "Brownian models of feedforward queueing networks: quasireversibility and product-form solutions". Annals of Applied Probability. 2 (2): 263–293. CiteSeerX 10.1.1.56.1572. doi:10.1214/aoap/1177005704.
10. Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services". Queueing Systems. 6: 71–87. doi:10.1007/BF02411466. S2CID 30949152.
11. Hillston, J.; Thomas, N. (1999). "Product form solution for a class of PEPA models" (PDF). Performance Evaluation. 35 (3–4): 171–192. doi:10.1016/S0166-5316(99)00005-X. hdl:20.500.11820/13c57018-5854-4f34-a4c9-833262a71b7c.
12. Harrison, P. G. (2003). "Turning back time in Markovian process algebra". Theoretical Computer Science. 290 (3): 1947–2013. doi:10.1016/S0304-3975(02)00375-4. Archived from the original on 2006-10-15. Retrieved 2015-08-29.
13. Marin, A.; Balsamo, S.; Harrison, P. G. (2012). "Analysis of stochastic Petri nets with signals". Performance Evaluation. 69 (11): 551–572. doi:10.1016/j.peva.2012.06.003. hdl:10044/1/14180.
14. Mairesse, J.; Nguyen, H. T. (2009). "Deficiency Zero Petri Nets and Product Form". Applications and Theory of Petri Nets. Lecture Notes in Computer Science. Vol. 5606. p. 103. CiteSeerX 10.1.1.745.1585. doi:10.1007/978-3-642-02424-5_8. ISBN 978-3-642-02423-8.
15. Anderson, D. F.; Craciun, G.; Kurtz, T. G. (2010). "Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks". Bulletin of Mathematical Biology. 72 (8): 1947–1970. arXiv:0803.3042. doi:10.1007/s11538-010-9517-4. PMID 20306147. S2CID 2204856.
16. Gelenbe, Erol (1993). "Learning in the recurrent random neural network". Neural Computation. 5 (1): 154–164. doi:10.1162/neco.1993.5.1.154. S2CID 38667978.
17. Gelenbe, Erol; Mao, Zhi-Hong; Li, Yan-Da (1991). "Function approximation with the random neural network". IEEE Transactions on Neural Networks. 10 (1): 3–9. CiteSeerX 10.1.1.46.7710. doi:10.1109/72.737488. PMID 18252498.
18. Boxma, O. J.; Kelly, F. P.; Konheim, A. G. (January 1984). "The Product Form for Sojourn Time Distributions in Cyclic Exponential Queues". Journal of the ACM. 31 (1): 128–133. doi:10.1145/2422.322419. S2CID 6770615.
19. Reich, Edgar (1957). "Waiting Times when Queues are in Tandem". The Annals of Mathematical Statistics. 28 (3): 768–773. doi:10.1214/aoms/1177706889.
20. Reich, E. (1963). "Note on Queues in Tandem". The Annals of Mathematical Statistics. 34: 338–341. doi:10.1214/aoms/1177704275.
21. Walrand, J.; Varaiya, P. (1980). "Sojourn Times and the Overtaking Condition in Jacksonian Networks". Advances in Applied Probability. 12 (4): 1000–1018. doi:10.2307/1426753. JSTOR 1426753.
22. Mitrani, I. (1985). "Response Time Problems in Communication Networks". Journal of the Royal Statistical Society. Series B (Methodological). 47 (3): 396–406. doi:10.1111/j.2517-6161.1985.tb01368.x. JSTOR 2345774.
23. Chow, We-Min (April 1980). "The Cycle Time Distribution of Exponential Cyclic Queues". Journal of the ACM. 27 (2): 281–286. doi:10.1145/322186.322193. S2CID 14084475.
24. Schassberger, R.; Daduna, H. (1983). "The Time for a Round Trip in a Cycle of Exponential Queues". Journal of the ACM. 30: 146–150. doi:10.1145/322358.322369. S2CID 33401212.
25. Daduna, H. (1982). "Passage Times for Overtake-Free Paths in Gordon-Newell Networks". Advances in Applied Probability. 14 (3): 672–686. doi:10.2307/1426680. JSTOR 1426680.
26. Kelly, F. P.; Pollett, P. K. (1983). "Sojourn Times in Closed Queueing Networks". Advances in Applied Probability. 15 (3): 638–656. doi:10.2307/1426623. JSTOR 1426623.
27. Baynat, B.; Dallery, Y. (1993). "A unified view of product-form approximation techniques for general closed queueing networks". Performance Evaluation. 18 (3): 205–224. doi:10.1016/0166-5316(93)90017-O.
28. Dallery, Y.; Cao, X. R. (1992). "Operational analysis of stochastic closed queueing networks". Performance Evaluation. 14: 43–61. doi:10.1016/0166-5316(92)90019-D.
29. Thomas, Nigel; Harrison, Peter G. (2010). "State-Dependent Rates and Semi-Product-Form via the Reversed Process". Computer Performance Engineering. Lecture Notes in Computer Science. Vol. 6342. p. 207. doi:10.1007/978-3-642-15784-4_14. ISBN 978-3-642-15783-7.
30. Dębicki, K.; Dieker, A. B.; Rolski, T. (2007). "Quasi-Product Forms for Levy-Driven Fluid Networks". Mathematics of Operations Research. 32 (3): 629–647. arXiv:math/0512119. doi:10.1287/moor.1070.0259. S2CID 16150704.
31. Angius, A.; Horváth, A. S.; Wolf, V. (2013). "Approximate Transient Analysis of Queuing Networks by Quasi Product Forms". Analytical and Stochastic Modeling Techniques and Applications. Lecture Notes in Computer Science. Vol. 7984. p. 22. doi:10.1007/978-3-642-39408-9_3. ISBN 978-3-642-39407-2.