December  2011, 4(4): 1081-1096. doi: 10.3934/krm.2011.4.1081

Averaged kinetic models for flows on unstructured networks

1. 

RWTH Aachen, Department of Mathematics, Templergraben 55, 52056 Aachen, Germany

2. 

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804

Received  July 2011 Revised  September 2011 Published  November 2011

We derive a kinetic equation for flows on general, unstructured networks with applications to production, social and transportation networks. This model allows for a homogenization procedure, yielding a macroscopic transport model for large networks on large time scales.
Citation: Michael Herty, Christian Ringhofer. Averaged kinetic models for flows on unstructured networks. Kinetic and Related Models, 2011, 4 (4) : 1081-1096. doi: 10.3934/krm.2011.4.1081
References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920. doi: 10.1137/040604625.

[2]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.

[3]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks, Journal of Economic Dynamics and Control, 31 (2007), 2061-2084. doi: 10.1016/j.jedc.2007.01.004.

[4]

D. Brockmann, Anomalous diffusion and the structure of human transportation networks, Eur. Phys. J. Special Topics, 157 (2008), 173-189. doi: 10.1140/epjst/e2008-00640-0.

[5]

C. Cercignani, I. Gamba and D. Levermore, A drift-collision balance for a Boltzmann- Poisson system in bounded domains, SIAM J. Appl. Math., 61 (2001), 1932-1958. doi: 10.1137/S0036139999360465.

[6]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.

[7]

F. Della Rossa, C. D'Angelo and A. Quarteroni, A distributed model of traffic flows on extended regions, Networks and Heterogeneous Media 5 (2010), 525-544.

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79. doi: 10.1137/060674302.

[9]

A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted aver- ages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256. doi: 10.1093/imanum/drm050.

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.

[11]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks, European Journal of Operational Research, 202 (2010), 456-465. doi: 10.1016/j.ejor.2009.05.028.

[12]

S. Göttlich, M. Herty and C. Ringhofer, Time-dependent order and distribution policies in supply networks, in "Progress in Industrial Mathematics at ECMI 2008," Mathematics in Industry, Vol. 15 (eds, J. Norbury, H. Ockendon and E. Wilson), 521-526, Springer, 2010.

[13]

W. H. S. C. Graves and D. B. Kletter, A dynamic model for requirements planning with application to supply chain optimization, Operations Research, 46 (1998), 35-49. doi: 10.1287/opre.46.3.S35.

[14]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[15]

E. W. Larsen, A generalized Boltzmann equation for "non-classical" particle transport, in "Joint International Topical Meeting on Mathematics and Computation and Supercomputing in Nuclear Applications" (M & C + SNA 2007), Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2007.

[16]

E. W. Larsen and R. Vasques, A generalized linear Boltzmann equation for non-classical particle transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 2010. doi: 10.1016/j.jqsrt.2010.07.003.

[17]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.

[18]

H. Missbauer, Aggregate order release planning for time varying demand, Int. J. Production Research, 40 (2002), 699-718. doi: 10.1080/00207540110090939.

[19]

K. Nagel, Particle hopping models and traffic flow theory, Physical Review E, 3 (1996), 4655-–4672. doi: 10.1103/PhysRevE.53.4655.

[20]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372. doi: 10.1002/zamm.19920720813.

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.

show all references

References:
[1]

D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains, SIAM J. Appl. Math., 66 (2006), 896-920. doi: 10.1137/040604625.

[2]

A.-L. Barabási and R. Albert, Emergence of scaling in random networks, Science, 286 (1999), 509-512. doi: 10.1126/science.286.5439.509.

[3]

S. Battiston, D. Delli Gatti, M. Gallegati, B. Greenwald and J. E. Stiglitz, Credit chains and bankruptcy propagation in production networks, Journal of Economic Dynamics and Control, 31 (2007), 2061-2084. doi: 10.1016/j.jedc.2007.01.004.

[4]

D. Brockmann, Anomalous diffusion and the structure of human transportation networks, Eur. Phys. J. Special Topics, 157 (2008), 173-189. doi: 10.1140/epjst/e2008-00640-0.

[5]

C. Cercignani, I. Gamba and D. Levermore, A drift-collision balance for a Boltzmann- Poisson system in bounded domains, SIAM J. Appl. Math., 61 (2001), 1932-1958. doi: 10.1137/S0036139999360465.

[6]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, 106, Springer-Verlag, New York, 1994.

[7]

F. Della Rossa, C. D'Angelo and A. Quarteroni, A distributed model of traffic flows on extended regions, Networks and Heterogeneous Media 5 (2010), 525-544.

[8]

P. Degond and C. Ringhofer, Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Applied Mathematics, 68 (2007), 59-79. doi: 10.1137/060674302.

[9]

A. Ern, A. Stephansen and P. Zunino, A discontinuous Galerkin method with weighted aver- ages for advection-diffusion equations with locally small and anisotropic diffusivity, IMA J. Numer. Anal., 29 (2009), 235-256. doi: 10.1093/imanum/drm050.

[10]

M. Garavello and B. Piccoli, "Traffic Flow on Networks. Conservation Laws Models," AIMS Series on Applied Mathematics, Vol. 1, American Institute of Mathematical Sciences, Springfield, MO, 2006.

[11]

S. Göttlich, M. Herty and C. Ringhofer, Optimization of order policies in supply networks, European Journal of Operational Research, 202 (2010), 456-465. doi: 10.1016/j.ejor.2009.05.028.

[12]

S. Göttlich, M. Herty and C. Ringhofer, Time-dependent order and distribution policies in supply networks, in "Progress in Industrial Mathematics at ECMI 2008," Mathematics in Industry, Vol. 15 (eds, J. Norbury, H. Ockendon and E. Wilson), 521-526, Springer, 2010.

[13]

W. H. S. C. Graves and D. B. Kletter, A dynamic model for requirements planning with application to supply chain optimization, Operations Research, 46 (1998), 35-49. doi: 10.1287/opre.46.3.S35.

[14]

D. Helbing, Traffic and related self-driven many-particle systems, Rev. Mod. Phys., 73 (2001), 1067-1141. doi: 10.1103/RevModPhys.73.1067.

[15]

E. W. Larsen, A generalized Boltzmann equation for "non-classical" particle transport, in "Joint International Topical Meeting on Mathematics and Computation and Supercomputing in Nuclear Applications" (M & C + SNA 2007), Monterey, California, April 15-19, 2007, on CD-ROM, American Nuclear Society, LaGrange Park, IL, 2007.

[16]

E. W. Larsen and R. Vasques, A generalized linear Boltzmann equation for non-classical particle transport, Journal of Quantitative Spectroscopy and Radiative Transfer, 2010. doi: 10.1016/j.jqsrt.2010.07.003.

[17]

M. J. Lighthill and G. B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads, Proc. Roy. Soc. London. Ser. A., 229 (1955), 317-345.

[18]

H. Missbauer, Aggregate order release planning for time varying demand, Int. J. Production Research, 40 (2002), 699-718. doi: 10.1080/00207540110090939.

[19]

K. Nagel, Particle hopping models and traffic flow theory, Physical Review E, 3 (1996), 4655-–4672. doi: 10.1103/PhysRevE.53.4655.

[20]

F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech., 72 (1992), 359-372. doi: 10.1002/zamm.19920720813.

[21]

D. J. Watts and S. H. Strogatz, Collective dynamics of 'small-world' networks, Nature, 393 (1998), 440-442. doi: 10.1038/30918.

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