
-
Previous Article
Numerical investigation of a neural field model including dendritic processing
- JCD Home
- This Issue
-
Next Article
An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling
Continuous approximation of $ M_t/M_t/ 1 $ distributions with application to production
1. | Arizona State University, School of Mathematical and Statistical Sciences, Tempe, AZ 85257-1804, USA |
2. | University of Mannheim, Department of Mathematics, 68131 Mannheim, Germany |
A single queueing system with time-dependent exponentially distributed arrival processes and exponential machine processes (Kendall notation $ M_t/M_t/1 $) is analyzed. Modeling the time evolution for the discrete queue-length distribution by a continuous drift-diffusion process a Smoluchowski equation on the half space is derived approximating the forward Kolmogorov equations. The approximate model is analyzed and validated, showing excellent agreement for the probabilities of all queue lengths and for all queuing utilizations, including ones that are very small and some that are significantly larger than one. Having an excellent approximation for the probability of an empty queue generates an approximation of the expected outflow of the queueing system. Comparisons to several well-established approximations from the literature show significant improvements in several numerical examples.
References:
[1] |
D. Armbruster, D. Marthaler and C. Ringhofer,
Kinetic and fluid model hierarchies for supply chains, Multiscale Model. Simul., 2 (2003), 43-61.
doi: 10.1137/S1540345902419616. |
[2] |
D. Armbruster and R. Uzsoy, Continuous dynamic models, clearing functions, and discrete-event simulation in aggregate production planning, INFORMS TutORials in Operations Research, (2014).
doi: 10.1287/educ.1120.0102. |
[3] |
D. Armbruster and M. Wienke,
Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model, Kinet. Relat. Models, 12 (2019), 177-193.
doi: 10.3934/krm.2019008. |
[4] |
N. Bellomo, C. Bianca and V. Coscia,
On the modeling of crowd dynamics: An overview and research perspectives, SeMA J., 54 (2011), 25-46.
doi: 10.1007/bf03322586. |
[5] |
H. Chen and D. D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization, vol. 46, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-5301-1. |
[6] |
R. M. Colombo, M. Herty and M. Mercier,
Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[7] |
J.-M. Coron and Z. Wang,
Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201.
doi: 10.1016/j.jde.2011.08.042. |
[8] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[9] |
P. Degond and C. Ringhofer,
Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79.
doi: 10.1137/060674302. |
[10] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[11] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, 4th edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2008.
doi: 10.1002/9781118625651. |
[12] |
C. Grossmann and H.-G. Roos, Numerical Treatment of Partial Differential Equations, Universitext, Springer, Berlin, 2007, Translated and revised from the 3rd (2005) German edition by Martin Stynes.
doi: 10.1007/978-3-540-71584-9. |
[13] |
A. Keimer and L. Pflug,
Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.
doi: 10.1016/j.jde.2017.05.015. |
[14] |
K. G. Kempf, P. Keskinocak and R. Uzsoy, Planning Production and Inventories in the Extended Enterprise, vol. 2, Springer, 2011. Google Scholar |
[15] |
Y. M. Ko and N. Gautam,
Critically loaded time-varying multiserver queues: Computational challenges and approximations, INFORMS J. Comput., 25 (2013), 285-301.
doi: 10.1287/ijoc.1120.0502. |
[16] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer,
Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[17] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. |
[18] |
G. Lamm and K. Schulten,
Extended brownian dynamics. Ⅱ. reactive, nonlinear diffusion, J. Chem. Phys, 78 (1983), 2713-2734.
doi: 10.1063/1.445002. |
[19] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[20] |
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.
doi: 10.1098/rspa.1955.0089. |
[21] |
A. Mandelbaum and W. A. Massey,
Strong approximations for time-dependent queues, Math. Oper. Res., 20 (1995), 33-64.
doi: 10.1287/moor.20.1.33. |
[22] |
A. Mandelbaum, W. A. Massey and M. I. Reiman,
Strong approximations for Markovian service networks, Queueing Systems Theory Appl., 30 (1998), 149-201.
doi: 10.1023/A:1019112920622. |
[23] |
W. A. Massey and J. Pender,
Gaussian skewness approximation for dynamic rate multi-server queues with abandonment, Queueing Syst., 75 (2013), 243-277.
doi: 10.1007/s11134-012-9340-8. |
[24] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅰ. The transition through saturation, J. Appl. Probability, 5 (1968), 436-451.
doi: 10.2307/3212264. |
[25] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅱ. The maximum queue and the return to equilibrium, J. Appl. Probability, 5 (1968), 579-590.
doi: 10.1017/S0021900200114421. |
[26] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅲ. A mild rush hour, J. Appl. Probability, 5 (1968), 591-606.
doi: 10.1017/S0021900200114433. |
[27] |
J. Pender,
A Poisson-Charlier approximation for nonstationary queues, Oper. Res. Lett., 42 (2014), 293-298.
doi: 10.1016/j.orl.2014.05.001. |
[28] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[29] |
K. L. Rider,
A simple approximation to the average queue size in the time-dependent $M/M/1$ queue, J. Assoc. Comput. Mach., 23 (1976), 361-367.
doi: 10.1145/321941.321955. |
[30] |
M. v. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, part ⅰ and part ⅱ, Physik. Z., 17 (1916), 557–571 (part Ⅰ); 585–599 (part Ⅱ). Google Scholar |
[31] |
M. v. Smoluchowski, Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Physik, 48 (1916), 1103-1112. Google Scholar |
[32] |
W. A. Strauss, Partial Differential Equations, 2nd edition, John Wiley & Sons, Ltd., Chichester, 2008. |
[33] |
W.-P. Wang, D. Tipper and S. Banerjee, A Simple Approximation for Modeling Nonstationary Queues, in Proceedings of IEEE INFOCOM '96. Conference on Computer Communications, IEEE Comput. Soc. Press, 1996. Google Scholar |
[34] |
W. Whitt, Time-varying queues, Queueing Models and Service Management, 1 (2018), 079-164. Google Scholar |
[35] |
M. Wienke, An Aggregate Second Order Continuum Model for Transient Production Planning, PhD thesis, Arizona State University, 2015, https://repository.asu.edu/attachments/162150/content/Wienke_asu_0010E_15448.pdf |
show all references
References:
[1] |
D. Armbruster, D. Marthaler and C. Ringhofer,
Kinetic and fluid model hierarchies for supply chains, Multiscale Model. Simul., 2 (2003), 43-61.
doi: 10.1137/S1540345902419616. |
[2] |
D. Armbruster and R. Uzsoy, Continuous dynamic models, clearing functions, and discrete-event simulation in aggregate production planning, INFORMS TutORials in Operations Research, (2014).
doi: 10.1287/educ.1120.0102. |
[3] |
D. Armbruster and M. Wienke,
Kinetic models and intrinsic timescales: Simulation comparison for a 2nd order queueing model, Kinet. Relat. Models, 12 (2019), 177-193.
doi: 10.3934/krm.2019008. |
[4] |
N. Bellomo, C. Bianca and V. Coscia,
On the modeling of crowd dynamics: An overview and research perspectives, SeMA J., 54 (2011), 25-46.
doi: 10.1007/bf03322586. |
[5] |
H. Chen and D. D. Yao, Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization, vol. 46, Springer-Verlag, New York, 2001.
doi: 10.1007/978-1-4757-5301-1. |
[6] |
R. M. Colombo, M. Herty and M. Mercier,
Control of the continuity equation with a non local flow, ESAIM Control Optim. Calc. Var., 17 (2011), 353-379.
doi: 10.1051/cocv/2010007. |
[7] |
J.-M. Coron and Z. Wang,
Controllability for a scalar conservation law with nonlocal velocity, J. Differential Equations, 252 (2012), 181-201.
doi: 10.1016/j.jde.2011.08.042. |
[8] |
E. Cristiani, B. Piccoli and A. Tosin, Multiscale Modeling of Pedestrian Dynamics, vol. 12 of MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[9] |
P. Degond and C. Ringhofer,
Stochastic dynamics of long supply chains with random breakdowns, SIAM J. Appl. Math., 68 (2007), 59-79.
doi: 10.1137/060674302. |
[10] |
M. Garavello and B. Piccoli, Traffic Flow on Networks, AIMS Series on Applied Mathematics, 1. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2006. |
[11] |
D. Gross, J. F. Shortle, J. M. Thompson and C. M. Harris, Fundamentals of Queueing Theory, 4th edition, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2008.
doi: 10.1002/9781118625651. |
[12] |
C. Grossmann and H.-G. Roos, Numerical Treatment of Partial Differential Equations, Universitext, Springer, Berlin, 2007, Translated and revised from the 3rd (2005) German edition by Martin Stynes.
doi: 10.1007/978-3-540-71584-9. |
[13] |
A. Keimer and L. Pflug,
Existence, uniqueness and regularity results on nonlocal balance laws, J. Differential Equations, 263 (2017), 4023-4069.
doi: 10.1016/j.jde.2017.05.015. |
[14] |
K. G. Kempf, P. Keskinocak and R. Uzsoy, Planning Production and Inventories in the Extended Enterprise, vol. 2, Springer, 2011. Google Scholar |
[15] |
Y. M. Ko and N. Gautam,
Critically loaded time-varying multiserver queues: Computational challenges and approximations, INFORMS J. Comput., 25 (2013), 285-301.
doi: 10.1287/ijoc.1120.0502. |
[16] |
M. La Marca, D. Armbruster, M. Herty and C. Ringhofer,
Control of continuum models of production systems, IEEE Trans. Automat. Control, 55 (2010), 2511-2526.
doi: 10.1109/TAC.2010.2046925. |
[17] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translated from the Russian by S. Smith. Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1968. |
[18] |
G. Lamm and K. Schulten,
Extended brownian dynamics. Ⅱ. reactive, nonlinear diffusion, J. Chem. Phys, 78 (1983), 2713-2734.
doi: 10.1063/1.445002. |
[19] |
R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.
doi: 10.1017/CBO9780511791253.![]() ![]() |
[20] |
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.
doi: 10.1098/rspa.1955.0089. |
[21] |
A. Mandelbaum and W. A. Massey,
Strong approximations for time-dependent queues, Math. Oper. Res., 20 (1995), 33-64.
doi: 10.1287/moor.20.1.33. |
[22] |
A. Mandelbaum, W. A. Massey and M. I. Reiman,
Strong approximations for Markovian service networks, Queueing Systems Theory Appl., 30 (1998), 149-201.
doi: 10.1023/A:1019112920622. |
[23] |
W. A. Massey and J. Pender,
Gaussian skewness approximation for dynamic rate multi-server queues with abandonment, Queueing Syst., 75 (2013), 243-277.
doi: 10.1007/s11134-012-9340-8. |
[24] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅰ. The transition through saturation, J. Appl. Probability, 5 (1968), 436-451.
doi: 10.2307/3212264. |
[25] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅱ. The maximum queue and the return to equilibrium, J. Appl. Probability, 5 (1968), 579-590.
doi: 10.1017/S0021900200114421. |
[26] |
G. F. Newell,
Queues with time-dependent arrival rates. Ⅲ. A mild rush hour, J. Appl. Probability, 5 (1968), 591-606.
doi: 10.1017/S0021900200114433. |
[27] |
J. Pender,
A Poisson-Charlier approximation for nonstationary queues, Oper. Res. Lett., 42 (2014), 293-298.
doi: 10.1016/j.orl.2014.05.001. |
[28] |
P. I. Richards,
Shock waves on the highway, Operations Res., 4 (1956), 42-51.
doi: 10.1287/opre.4.1.42. |
[29] |
K. L. Rider,
A simple approximation to the average queue size in the time-dependent $M/M/1$ queue, J. Assoc. Comput. Mach., 23 (1976), 361-367.
doi: 10.1145/321941.321955. |
[30] |
M. v. Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen, part ⅰ and part ⅱ, Physik. Z., 17 (1916), 557–571 (part Ⅰ); 585–599 (part Ⅱ). Google Scholar |
[31] |
M. v. Smoluchowski, Über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung, Ann. Physik, 48 (1916), 1103-1112. Google Scholar |
[32] |
W. A. Strauss, Partial Differential Equations, 2nd edition, John Wiley & Sons, Ltd., Chichester, 2008. |
[33] |
W.-P. Wang, D. Tipper and S. Banerjee, A Simple Approximation for Modeling Nonstationary Queues, in Proceedings of IEEE INFOCOM '96. Conference on Computer Communications, IEEE Comput. Soc. Press, 1996. Google Scholar |
[34] |
W. Whitt, Time-varying queues, Queueing Models and Service Management, 1 (2018), 079-164. Google Scholar |
[35] |
M. Wienke, An Aggregate Second Order Continuum Model for Transient Production Planning, PhD thesis, Arizona State University, 2015, https://repository.asu.edu/attachments/162150/content/Wienke_asu_0010E_15448.pdf |











moderate cyclic case | strong cyclic case | very strong cyclic case | |
25 | 1.4876 | 6.8495 | 6.5022 |
10 | 1.7911 | 8.8031 | 11.6854 |
5 | 2.2439 | 12.0694 | 17.2880 |
2 | 3.1356 | 17.1375 | 25.8903 |
1 | 3.2942 | 18.1220 | 30.9673 |
moderate cyclic case | strong cyclic case | very strong cyclic case | |
25 | 1.4876 | 6.8495 | 6.5022 |
10 | 1.7911 | 8.8031 | 11.6854 |
5 | 2.2439 | 12.0694 | 17.2880 |
2 | 3.1356 | 17.1375 | 25.8903 |
1 | 3.2942 | 18.1220 | 30.9673 |
[1] |
Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106 |
[2] |
Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043 |
[3] |
Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319 |
[4] |
Elvio Accinelli, Humberto Muñiz. A dynamic for production economies with multiple equilibria. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021002 |
[5] |
Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021011 |
[6] |
Yanjun He, Wei Zeng, Minghui Yu, Hongtao Zhou, Delie Ming. Incentives for production capacity improvement in construction supplier development. Journal of Industrial & Management Optimization, 2021, 17 (1) : 409-426. doi: 10.3934/jimo.2019118 |
[7] |
Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020 doi: 10.3934/mcrf.2020049 |
[8] |
Pan Zheng. Asymptotic stability in a chemotaxis-competition system with indirect signal production. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1207-1223. doi: 10.3934/dcds.2020315 |
[9] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020451 |
[10] |
Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 |
[11] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[12] |
Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020458 |
[13] |
Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 |
[14] |
Tuoc Phan, Grozdena Todorova, Borislav Yordanov. Existence uniqueness and regularity theory for elliptic equations with complex-valued potentials. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1071-1099. doi: 10.3934/dcds.2020310 |
[15] |
Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051 |
[16] |
Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 |
[17] |
Lan Luo, Zhe Zhang, Yong Yin. Simulated annealing and genetic algorithm based method for a bi-level seru loading problem with worker assignment in seru production systems. Journal of Industrial & Management Optimization, 2021, 17 (2) : 779-803. doi: 10.3934/jimo.2019134 |
[18] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
[19] |
Jann-Long Chern, Sze-Guang Yang, Zhi-You Chen, Chih-Her Chen. On the family of non-topological solutions for the elliptic system arising from a product Abelian gauge field theory. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3291-3304. doi: 10.3934/dcds.2020127 |
[20] |
Van Duong Dinh. Random data theory for the cubic fourth-order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (2) : 651-680. doi: 10.3934/cpaa.2020284 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]