Approximation Algorithm and Perfect Sampler for Closed Jackson Networks with Single Servers
Abstract
In this paper, we propose the first fully polynomial-time randomized approximation scheme (FPRAS) for closed Jackson networks with single servers. Our algorithm is based on the Markov chain Monte Carlo (MCMC) method, and our scheme returns an approximate solution, for which the size of error satisfies a given error rate. We propose two Markov chains: one is for approximate sampling, and the other is for perfect sampling based on the monotone coupling from the past algorithm.
MSC codes
Keywords
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
D. Aldous, Random walks on finite groups and rapidly mixing Markov chains, in Séminarie de Probabilitiés XVII 1981/1982, D. Dold and B. Eckmann, eds., Lecture Notes in Math. 986, Springer-Verlag, New York, 1983, pp. 243–297.
2.
F. Baskett, K. M. Chandy, R. R. Muntz, and F. G. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. ACM, 22 (1975), pp. 248–260.
3.
R. Bubley, Randomized Algorithms: Approximation, Generation, and Counting, Springer-Verlag, New York, 2001.
4.
R. Bubley and M. Dyer, Path coupling: A technique for proving rapid mixing in Markov chains, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1997, pp. 223–231.
5.
J. P. Buzen, Computational algorithms for closed queueing networks with exponential servers, Comm. ACM, 16 (1973), pp. 527–531.
6.
W. Chen and C. A. O'Cinneide, Towards a polynomial-time randomized algorithm for closed product-form networks, ACM Trans. Modeling Comput. Simul., 8 (1998), pp. 227–253.
7.
M. Dyer and C. Greenhill, Polynomial-time counting and sampling of two-rowed contingency tables, Theoret. Comput. Sci., 246 (2000), pp. 265–278.
8.
J. Fill, An interruptible algorithm for perfect sampling via Markov chains, Ann. Appl. Probab., 8 (1998), pp. 131–162.
9.
J. Fill, M. Machida, D. Murdoch, and J. Rosenthal, Extension of Fill's perfect rejection sampling algorithm to general chains, Random Structures Algorithms, 17 (2000), pp. 290–316.
10.
E. Gelenbe and G. Pujolle, Introduction to Queueing Networks, 2nd ed., John Wiley & Sons, New York, 1998.
11.
W. J. Gordon and G. F. Newell, Closed queueing systems with exponential servers, Oper. Res., 15 (1967), pp. 254–265.
12.
O. Häggström, Finite Markov Chains and Algorithmic Application, London Math. Soc. Stud. Texts 52, Cambridge University Press, Cambridge, UK, 2002.
13.
J. R. Jackson, Networks of waiting lines, J. Oper. Res. Soc. Amer., 5 (1957), pp. 518–521.
14.
J. R. Jackson, Jobshop-like queueing systems, Management Sci., 10 (1963), pp. 131–142.
15.
M. Jerrum, Counting, Sampling and Integrating: Algorithms and Complexity, Lectures in Math., ETH Zürich, Birkhäuser Verlag, Basel, 2003.
16.
M. Jerrum and A. Sinclair, The Markov chain Monte Carlo method: An approach to approximate counting and integration, in Approximation Algorithm for NP-hard Problems, D. Hochbaum, ed., PWS, Boston, MA, 1996, pp. 482–520.
17.
S. Kijima and T. Matsui, Approximate counting scheme for $m \times n$ contingency tables, IEICE Trans. Inform. Systems, E87-D (2004), pp. 308–314.
18.
S. Kijima and T. Matsui, Polynomial time perfect sampling algorithm for two-rowed contingency tables, Random Structures Algorithms, 29 (2006), pp. 243–256.
19.
S. Kijima and T. Matsui, Rapidly mixing chain and perfect sampler for logarithmic separable concave distributions on simplex, in 2005 International Conference on Analysis of Algorithms, Discrete Math. Theor. Comput. Sci. Proc. AD, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, France, 2005, pp. 369–380.
20.
S. Kijima and T. Matsui, Randomized Approximation Scheme and Perfect Sampler for Closed Jackson Networks with Multiple Servers, Mathematical Engineering Technical Report 2006-34, University of Tokyo, Tokyo, Japan, 2006. Available online at http://www.keisu.t.u-tokyo.ac.jp/Research/techrep.0.html.
21.
T. Matsui and S. Kijima, Polynomial time perfect sampler for discretized Dirichlet distribution, in The Grammar of Technology Development, H. Tsubaki, K. Nishina, and S. Yamada, eds., Springer Tokyo, Tokyo, 2008, pp. 179–199.
22.
T. Matsui, M. Motoki, and N. Kamatani, Polynomial time approximate sampler for discretized Dirichlet distribution, in Algorithms and Computation, Lecture Notes in Comput. Sci. 2906, Springer-Verlag, Berlin, 2003, pp. 676–685.
23.
T. Ozawa, Perfect simulation of a closed Jackson network, in Proceedings of the ORSJ Queueing Symposium, Hikone, Japan, 2004.
24.
J. Propp and D. Wilson, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures Algorithms, 9 (1996), pp. 223–252.
25.
D. Wilson, How to couple from the past using a read-once source of randomness, Random Structures Algorithms, 16 (2000), pp. 85–113.
Information & Authors
Information
Published In
Copyright
Copyright © 2008 Society for Industrial and Applied Mathematics.
History
Submitted: 13 January 2006
Accepted: 28 May 2008
Published online: 19 September 2008
MSC codes
Keywords
Authors
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Deterministic Random Walks for Rapidly Mixing ChainsSIAM Journal on Discrete Mathematics, Vol. 32, No. 3 | 30 August 2018
- Efficiency of simulation in monotone hyper-stable queueing networksQueueing Systems, Vol. 76, No. 1 | 20 April 2013
- Randomness in algorithm designJournal of Physics: Conference Series, Vol. 473 | 16 December 2013
- Probability and Computation2011 Second International Conference on Networking and Computing | 1 Nov 2011
- On the efficiency of perfect simulation in monotone queueing networksACM SIGMETRICS Performance Evaluation Review, Vol. 39, No. 2 | 15 September 2011
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].