The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points

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June 2013, 8(2): 529-540. doi: 10.3934/nhm.2013.8.529

The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points

Jean-Yves Le Boudec ^1,

1.	EPFL, I&C, CH-1015 Lausanne, Switzerland

Received July 2012 Revised October 2012 Published May 2013

Abstract
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Under a Creative Commons license

Assume that a stochastic process can be approximated, when some scale parameter gets large, by a fluid limit (also called "mean field limit", or "hydrodynamic limit"). A common practice, often called the "fixed point approximation" consists in approximating the stationary behaviour of the stochastic process by the stationary points of the fluid limit. It is known that this may be incorrect in general, as the stationary behaviour of the fluid limit may not be described by its stationary points. We show however that, if the stochastic process is reversible, the fixed point approximation is indeed valid. More precisely, we assume that the stochastic process converges to the fluid limit in distribution (hence in probability) at every fixed point in time. This assumption is very weak and holds for a large family of processes, among which many mean field and other interaction models. We show that the reversibility of the stochastic process implies that any limit point of its stationary distribution is concentrated on stationary points of the fluid limit. If the fluid limit has a unique stationary point, it is an approximation of the stationary distribution of the stochastic process.

Keywords: reversible, hydrodynamic limit, stationary points, fluid limit, fixed points., Mean field.

Mathematics Subject Classification: Primary: 37A60, 82C22, 60K35, 60K30; Secondary: 60K2.

Citation: Jean-Yves Le Boudec. The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points. Networks and Heterogeneous Media, 2013, 8 (2) : 529-540. doi: 10.3934/nhm.2013.8.529

References:

[1]	M. Benaïm, Recursive algorithms, urn processes and chaining number of chain recurrent sets, Ergodic Theory and Dynamical System, 18 (1998), 53-87. doi: 10.1017/S0143385798097557.
[2]	M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, 65 (2008), 823-838.
[3]	M. Benaïm and J. Weibull, Deterministic approximation of stochastic evolution, Econometrica, 71 (2003), 873-904. doi: 10.1111/1468-0262.00429.
[4]	M. Benaïm, Dynamics of stochastic approximation algorithms, in "Séminaire de Probabilités XXXIII," Lecture Notes in Math., 1709, Springer, Berlin, (1999), 1-68. doi: 10.1007/BFb0096509.
[5]	G. Bianchi, IEEE 802.11-Saturation throughput analysis, IEEE Communications Letters, 2 (1998), 318-320. doi: 10.1109/4234.736171.
[6]	C. Bordenave, D. McDonald and A. Proutière, A particle system in interaction with a rapidly varying environment: Mean field limits and applications, Networks and Heterogeneous Media, 5 (2010), 31-62. doi: 10.3934/nhm.2010.5.31.
[7]	J. A. M Borghans, R. J. De Boer, E. Sercarz and V. Kumar, T cell vaccination in experimental autoimmune encephalomyelitis: A mathematical model, The Journal of Immunology, 161 (1998), 1087-1093.
[8]	L. Bortolussi, J.-Y. Le Boudec, D. Latella and M. Massink, Revisiting the limit behaviour of "El Botellon," Technical Report EPFL-REPORT-179935, EPFL, 2012. Available from: https://infoscience.epfl.ch/record/179935.
[9]	V. Capasso and D. Bakstein, "An Introduction to Continuous-Time Markov Processes. Theory, Models, and Applications to Finance, Biology, and Medicine," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, Inc., Boston, MA, 2005.
[10]	J.-W. Cho, J.-Y. Le Boudec and Y. Jiang, On the asymptotic validity of the fixed point equation and decoupling assumption for analyzing the 802.11 MAC protocol, IEEE Transactions on Information Theory, 58 (2012), 6879-6893. doi: 10.1109/TIT.2012.2208582.
[11]	J.-P. Crametz and P. J. Hunt, A limit result respecting graph structure for a fully connected loss network with alternative routing, The Annals of Applied Probability, 1 (1991), 436-444. doi: 10.1214/aoap/1177005876.
[12]	S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.
[13]	C. Graham and S. Méléard, Propagation of chaos for a fully connected loss network with alternate routing, Stochastic Processes and Their Applications, 44 (1993), 159-180. doi: 10.1016/0304-4149(93)90043-4.
[14]	F. P. Kelly, "Reversibility and Stochastic Networks," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1979.
[15]	F. P. Kelly, Loss networks, The Annals of Applied Probability, 1 (1991), 319-378. doi: 10.1214/aoap/1177005872.
[16]	A. Kumar, E. Altman, D. Miorandi and M. Goyal, New insights from a fixed-point analysis of single cell ieee 802.11 wlans, IEEE/ACM Transactions on Networking, 15 (2007), 588-601.
[17]	T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, Journal of Applied Probability, 7 (1979), 49-58. doi: 10.2307/3212147.
[18]	Thomas G. Kurtz, "Approximation of Population Processes," CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.
[19]	J.-Y. Le Boudec, D. McDonald and J. Mundinger, A generic mean field convergence result for systems of interacting objects, in "Fourth International Conference on the Quantitative Evaluation of Systems" (QEST 2007), IEEE, (2007), 3-18. doi: 10.1109/QEST.2007.8.
[20]	J.-Y. Le Boudec, "Performance Evaluation of Computer and Communication Systems," EPFL Press, Lausanne, Switzerland, 2010. Available from: http://perfeval.epfl.ch.
[21]	J.-Y. Le Boudec, Interinput and interoutput time distribution in classical product-form networks, IEEE Transactions on Software Engineering, 6 (1987), 756-759.
[22]	M. Massink, D. Latella, A. Bracciali and J. Hillston, Modelling non-linear crowd dynamics in bio-PEPA, in "Fundamental Approaches to Software Engineering," Lecture Notes in Computer Science, 6603, Springer Berlin Heidelberg, (2011), 96-110. doi: 10.1007/978-3-642-19811-3_8.
[23]	R. Merz, J.-Y. Le Boudec and S. Vijayakumaran, Effect on network performance of common versus private acquisition sequences for impulse radio UWB networks, in "IEEE International Conference on Ultra-Wideband" (ICUWB 2006), IEEE, (2006), 375-380. doi: 10.1109/ICU.2006.281579.
[24]	J. E. Rowe and R. Gomez, El Botellón: Modeling the movement of crowds in a city, Complex Systems, 14 (2003), 363-370.
[25]	W. H. Sandholm, "Population Games and Evolutionary Dynamics," Economic Learning and Social Evolution, MIT press, Cambridge, MA, 2010.

show all references

References:

[1]	M. Benaïm, Recursive algorithms, urn processes and chaining number of chain recurrent sets, Ergodic Theory and Dynamical System, 18 (1998), 53-87. doi: 10.1017/S0143385798097557.
[2]	M. Benaïm and J.-Y. Le Boudec, A class of mean field interaction models for computer and communication systems, Performance Evaluation, 65 (2008), 823-838.
[3]	M. Benaïm and J. Weibull, Deterministic approximation of stochastic evolution, Econometrica, 71 (2003), 873-904. doi: 10.1111/1468-0262.00429.
[4]	M. Benaïm, Dynamics of stochastic approximation algorithms, in "Séminaire de Probabilités XXXIII," Lecture Notes in Math., 1709, Springer, Berlin, (1999), 1-68. doi: 10.1007/BFb0096509.
[5]	G. Bianchi, IEEE 802.11-Saturation throughput analysis, IEEE Communications Letters, 2 (1998), 318-320. doi: 10.1109/4234.736171.
[6]	C. Bordenave, D. McDonald and A. Proutière, A particle system in interaction with a rapidly varying environment: Mean field limits and applications, Networks and Heterogeneous Media, 5 (2010), 31-62. doi: 10.3934/nhm.2010.5.31.
[7]	J. A. M Borghans, R. J. De Boer, E. Sercarz and V. Kumar, T cell vaccination in experimental autoimmune encephalomyelitis: A mathematical model, The Journal of Immunology, 161 (1998), 1087-1093.
[8]	L. Bortolussi, J.-Y. Le Boudec, D. Latella and M. Massink, Revisiting the limit behaviour of "El Botellon," Technical Report EPFL-REPORT-179935, EPFL, 2012. Available from: https://infoscience.epfl.ch/record/179935.
[9]	V. Capasso and D. Bakstein, "An Introduction to Continuous-Time Markov Processes. Theory, Models, and Applications to Finance, Biology, and Medicine," Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, Inc., Boston, MA, 2005.
[10]	J.-W. Cho, J.-Y. Le Boudec and Y. Jiang, On the asymptotic validity of the fixed point equation and decoupling assumption for analyzing the 802.11 MAC protocol, IEEE Transactions on Information Theory, 58 (2012), 6879-6893. doi: 10.1109/TIT.2012.2208582.
[11]	J.-P. Crametz and P. J. Hunt, A limit result respecting graph structure for a fully connected loss network with alternative routing, The Annals of Applied Probability, 1 (1991), 436-444. doi: 10.1214/aoap/1177005876.
[12]	S. N. Ethier and T. G. Kurtz, "Markov Processes. Characterization and Convergence," Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. doi: 10.1002/9780470316658.
[13]	C. Graham and S. Méléard, Propagation of chaos for a fully connected loss network with alternate routing, Stochastic Processes and Their Applications, 44 (1993), 159-180. doi: 10.1016/0304-4149(93)90043-4.
[14]	F. P. Kelly, "Reversibility and Stochastic Networks," Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Ltd., Chichester, 1979.
[15]	F. P. Kelly, Loss networks, The Annals of Applied Probability, 1 (1991), 319-378. doi: 10.1214/aoap/1177005872.
[16]	A. Kumar, E. Altman, D. Miorandi and M. Goyal, New insights from a fixed-point analysis of single cell ieee 802.11 wlans, IEEE/ACM Transactions on Networking, 15 (2007), 588-601.
[17]	T. G. Kurtz, Solutions of ordinary differential equations as limits of pure jump Markov processes, Journal of Applied Probability, 7 (1979), 49-58. doi: 10.2307/3212147.
[18]	Thomas G. Kurtz, "Approximation of Population Processes," CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.
[19]	J.-Y. Le Boudec, D. McDonald and J. Mundinger, A generic mean field convergence result for systems of interacting objects, in "Fourth International Conference on the Quantitative Evaluation of Systems" (QEST 2007), IEEE, (2007), 3-18. doi: 10.1109/QEST.2007.8.
[20]	J.-Y. Le Boudec, "Performance Evaluation of Computer and Communication Systems," EPFL Press, Lausanne, Switzerland, 2010. Available from: http://perfeval.epfl.ch.
[21]	J.-Y. Le Boudec, Interinput and interoutput time distribution in classical product-form networks, IEEE Transactions on Software Engineering, 6 (1987), 756-759.
[22]	M. Massink, D. Latella, A. Bracciali and J. Hillston, Modelling non-linear crowd dynamics in bio-PEPA, in "Fundamental Approaches to Software Engineering," Lecture Notes in Computer Science, 6603, Springer Berlin Heidelberg, (2011), 96-110. doi: 10.1007/978-3-642-19811-3_8.
[23]	R. Merz, J.-Y. Le Boudec and S. Vijayakumaran, Effect on network performance of common versus private acquisition sequences for impulse radio UWB networks, in "IEEE International Conference on Ultra-Wideband" (ICUWB 2006), IEEE, (2006), 375-380. doi: 10.1109/ICU.2006.281579.
[24]	J. E. Rowe and R. Gomez, El Botellón: Modeling the movement of crowds in a city, Complex Systems, 14 (2003), 363-370.
[25]	W. H. Sandholm, "Population Games and Evolutionary Dynamics," Economic Learning and Social Evolution, MIT press, Cambridge, MA, 2010.

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