# Product-form solution

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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

$\text{P}(x_1,x_2,x_3,\ldots,x_n) = B \prod_{i=1}^n \text{P}(x_i)$

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 showed this to not be the case.[4] Product-form solutions are sometimes described as "stations are independent in equilibrium".[5] Product form solutions also exist in networks of bulk queues.[6]

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"[5] More recently, product-form solutions have been published for Markov process algebras (e.g. RCAT in PEPA[7][8]) and stochastic petri nets.[9][10] Martin Feinberg's deficiency zero theorem gives a sufficient condition for chemical reaction networks to exhibit a product-form stationary distribution.[11]

## 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.[12] In 1957 Reich showed the result for two M/M/1 queues in tandem,[13] later extending this to n M/M/1 queues in tandem[14] and it has been shown to apply to overtake–free paths in Jackson networks.[15] 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.[15] Mitrani offers exact solutions to some simple networks with overtaking, showing that none of these exhibit product-form sojourn time distributions.[16]

For closed networks, Chow showed a result to hold for two service nodes,[17] which was later generalised to a cycle of queues[18] and to overtake–free paths in Gordon–Newell networks.[19][20]

## Extensions

• Approximate product-form solutions are computed assuming independent marginal distributions, which can give a good approximation to the stationary distribution under some conditions.[21][22]
• 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.[23]
• 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[24] or
• approximate form for transient probability distributions which allows transient moments to be approximated.[25]

## 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: 463–492. doi:10.1007/BF02033315.
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: 250–263. doi:10.1145/322003.322009.
4. ^ Gelenbe, Erol (1993). "G-Networks with triggered customer movement". Journal of Applied Probability 30 (3): 742–748. doi:10.2307/3214781.
5. ^ a b 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. doi:10.1214/aoap/1177005704.
6. ^ Henderson, W.; Taylor, P. G. (1990). "Product form in networks of queues with batch arrivals and batch services". Queueing Systems 6: 71. doi:10.1007/BF02411466. edit
7. ^ Hillston, J.; Thomas, N. (1999). "Product form solution for a class of PEPA models". Performance Evaluation 35 (3–4): 171. doi:10.1016/S0166-5316(99)00005-X. edit
8. ^ 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. edit
9. ^ Marin, A.; Balsamo, S.; Harrison, P. G. (2012). "Analysis of stochastic Petri nets with signals". Performance Evaluation. doi:10.1016/j.peva.2012.06.003. edit
10. ^ Mairesse, J.; Nguyen, H. T. (2009). "Deficiency Zero Petri Nets and Product Form". Applications and Theory of Petri Nets. Lecture Notes in Computer Science 5606. p. 103. doi:10.1007/978-3-642-02424-5_8. ISBN 978-3-642-02423-8. edit
11. ^ 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. doi:10.1007/s11538-010-9517-4. PMID 20306147. edit
12. ^ 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). doi:10.1145/2422.322419.
13. ^ Reich, Edgar (1957). "Waiting Times when Queues are in Tandem". The Annals of Mathematical Statistics 28 (3): 768–773. doi:10.1214/aoms/1177706889. edit
14. ^ Reich, E. (1963). "Note on Queues in Tandem". The Annals of Mathematical Statistics 34: 338. doi:10.1214/aoms/1177704275. edit
15. ^ a b 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. edit
16. ^ Mitrani, I. (1985). "Response Time Problems in Communication Networks". Journal of the Royal Statistical Society. Series B (Methodological) 47 (3): 396–406. JSTOR 2345774. edit
17. ^ Chow, We-Min (April 1980). "The Cycle Time Distribution of Exponential Cyclic Queues". Journal of the ACM 27 (2). doi:10.1145/322186.322193.
18. ^ Schassberger, R.; Daduna, H. (1983). "The Time for a Round Trip in a Cycle of Exponential Queues". Journal of the ACM 30: 146. doi:10.1145/322358.322369. edit
19. ^ 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. edit
20. ^ 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. edit
21. ^ Baynat, B.; Dallery, Y. (1993). "A unified view of product-form approximation techniques for general closed queueing networks". Performance Evaluation 18 (3): 205. doi:10.1016/0166-5316(93)90017-O. edit
22. ^ Dallery, Y.; Cao, X. R. (1992). "Operational analysis of stochastic closed queueing networks". Performance Evaluation 14: 43. doi:10.1016/0166-5316(92)90019-D. edit
23. ^ 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 6342. p. 207. doi:10.1007/978-3-642-15784-4_14. ISBN 978-3-642-15783-7. edit
24. ^ Debicki, K.; Dieker, A. B.; Rolski, T. (2007). "Quasi-Product Forms for Levy-Driven Fluid Networks". Mathematics of Operations Research 32 (3): 629. arXiv:math/0512119. doi:10.1287/moor.1070.0259. edit
25. ^ 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 7984. p. 22. doi:10.1007/978-3-642-39408-9_3. ISBN 978-3-642-39407-2. edit