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December  2020, 7(2): 243-269. doi: 10.3934/jcd.2020010

## 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

Received  October 2019 Published  July 2020

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.

Citation: Dieter Armbruster, Simone Göttlich, Stephan Knapp. Continuous approximation of $M_t/M_t/ 1$ distributions with application to production. Journal of Computational Dynamics, 2020, 7 (2) : 243-269. doi: 10.3934/jcd.2020010
##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] P. I. Richards, Shock waves on the highway, Operations Res., 4 (1956), 42-51.  doi: 10.1287/opre.4.1.42.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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  Google Scholar
Graphical representation of a single queue service unit
Comparison of queue-length distribution with exact ODE system and continuous approximation in the moderate ramp up case, (a) probability $p_0(t)$, (b) $p_3(t)$, (c) supremum error
Comparison of queue-length distribution with exact ODE system and continuous approximation in the strong ramp up case, (a) probability $p_0(t)$, (b) $p_3(t)$, (c) supremum error
Comparison of queue-length distribution with exact ODE system and continuous approximation in the very strong ramp up case, (a) probability $p_0(t)$, (b) $p_{10}(t)$, (c) $p_{40}(t)$, (d) supremum error
Comparison of queue-length distribution with exact ODE system and continuous approximation in the moderate ramp down case, (a) probability $p_0(t)$, (b) $p_3(t)$, (c) supremum error
Comparison of queue-length distribution with exact ODE system and continuous approximation in the strong ramp down case, (a) probability $p_0(t)$, (b) $p_3(t)$, (c) supremum error
Comparison of queue-length distribution with exact ODE system and continuous approximation in the moderate cyclic inflow rate case
Comparison of queue-length distribution with exact ODE system and continuous approximation in the strong cyclic inflow rate case
Comparison of queue-length distribution with exact ODE-system and continuous approximation in the very strong cyclic inflow rate case
Comparison of the expected outflow using approximations from the literature (K, GVA, GSA) and our approach (A), where the inflow and processing rates are taken from [29], see (b)
Comparison of the expected outflow using approximations from the literature (K, GVA, GSA) and our approach (A), where the inflow and processing rates are highly transient and periodic, see (b)
Error of the continuous approximation for different periods in the moderate, strong and very strong cyclic case
 $T_{\text{Per}}$ $\max_j \|\epsilon^A(t_j)\|_\infty \;[10^{-3}]$ 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
 $T_{\text{Per}}$ $\max_j \|\epsilon^A(t_j)\|_\infty \;[10^{-3}]$ 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
Error between the approximations and the ODE system result for inflow from [29]
 $\overline{Out}^\text{A}(t)$ $\overline{Out}^\text{K}(t)$ $\overline{Out}^\text{GVA}(t)$ $\overline{Out}^\text{GSA}(t)$ $\|\cdot\|_\infty$ $0.0457$ $0.6830$ $7.0672$ $1.0314$ $\|\cdot\|_{L^1}$ $0.3522$ $3.9450$ $3.8772$ $3.4053$
 $\overline{Out}^\text{A}(t)$ $\overline{Out}^\text{K}(t)$ $\overline{Out}^\text{GVA}(t)$ $\overline{Out}^\text{GSA}(t)$ $\|\cdot\|_\infty$ $0.0457$ $0.6830$ $7.0672$ $1.0314$ $\|\cdot\|_{L^1}$ $0.3522$ $3.9450$ $3.8772$ $3.4053$
Error between the approximations and the ODE system result for cyclic inflow
 $\overline{Out}^\text{A}(t)$ $\overline{Out}^\text{K}(t)$ $\overline{Out}^\text{GVA}(t)$ $\overline{Out}^\text{GSA}(t)$ $\|\cdot\|_\infty$ $0.0196$ $0.9041$ $0.5876$ $0.2901$ $\|\cdot\|_{L^1}$ $0.2622$ $10.5651$ $4.8328$ $2.8259$
 $\overline{Out}^\text{A}(t)$ $\overline{Out}^\text{K}(t)$ $\overline{Out}^\text{GVA}(t)$ $\overline{Out}^\text{GSA}(t)$ $\|\cdot\|_\infty$ $0.0196$ $0.9041$ $0.5876$ $0.2901$ $\|\cdot\|_{L^1}$ $0.2622$ $10.5651$ $4.8328$ $2.8259$
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