A biologically inspired fluid model of the cyclic service system


A deterministic fluid model in the form of nonlinear ordinary differential equations is developed to provide the description for a multichannel service system with service-in-random-order queue discipline, abandonment and re-entry, where servers are treated like enzyme molecules. The parametric analysis of the model’s fixed point is given, particularly, how the arrival rate of new customers affects the steady-state demand. It is also shown that the model implies a saturating clearing function (yield vs. demand) of the Karmarkar type providing the mean service time is much shorter than the characteristic waiting time.

Keyword : fluid queues, multiple server, abandonment, re-entry, random order service, clearing function

How to Cite
Kantarbayeva, A., & Mustafin, A. (2020). A biologically inspired fluid model of the cyclic service system. Mathematical Modelling and Analysis, 25(4), 505-521.
Published in Issue
Oct 13, 2020
Abstract Views
PDF Downloads
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.


D. Armbruster. The production planning problem: Clearing functions, variable lead times, delay equations and partial differential equations. In D. Armbruster and K. G. Kempf(Eds.), Decision Policies for Production Networks, pp. 289–302. Springer, London, 2012.

D. Armbruster, K. Kaneko and A. S. Mikhailov(Eds.). Networks of Interacting Machines: Production Organization in Complex Industrial Systems and Biological Cells, volume 3 of World Scientific Lecture Notes in Complex Systems. World Scientific, Singapore, 2005.

D. Armbruster, D. Marthaler and C. Ringhofer. Kinetic and fluid model hierarchies for supply chains. Multiscale Modeling & Simulation, 2(1):43–61, 2003.

M. Armony, N. Shimkin and W. Whitt. The impact of delay announcements in many-server queues with abandonment. Operations Research, 57(1):66–81, 2009.

R.U. Ayres and U.E. Simonis(Eds.). Industrial Metabolism: Restructuring for Sustainable Development. United Nations University Press, Tokyo, 1994.

M. Bramson. Stability of Queueing Networks. Number 1950 in Lecture Notes in Mathematics. Springer, Berlin; Heidelberg, 2008.

A. Cornish-Bowden. Fundamentals of Enzyme Kinetics. Wiley-Blackwell, Weinheim, 4th edition, 2012.

H. Fromm and J. Cardoso. Foundations. In J. Cardoso, H. Fromm, S. Nickel, G. Satzger, R. Studer and C. Weinhardt(Eds.), Fundamentals of Service Systems, Service Science: Research and Innovations in the Service Economy, pp. 1–32. Springer, New York, NY, 2015.

J. Gadrey. The characterization of goods and services: an alternative approach. Review of Income and Wealth, 46(3):369–387, 2000.

F. Gallouj. Innovation in services and the attendant old and new myths. The Journal of Socio-Economics, 31(2):137–154, 2002.

D. Gamarnik. Fluid models of queueing networks. In J.J. Cochran, L.A. Cox, P. Keskinocak, J.P. Kharoufeh and J.C. Smith(Eds.), Wiley Encyclopedia of Operations Research and Management Science. Wiley, Hoboken, NJ, 2011.

N. Georgescu-Roegen. Some properties of a generalized Leontief model. In Analytical Economics: Issues and Problems, chapter 9, pp. 316–337. Harvard University Press, Cambridge, MA, 1966.

R. Grima, N.G. Walter and S. Schnell. Single-molecule enzymology à la Michaelis–Menten. FEBS Journal, 281(2):518–530, 2014.

D. Helbing, D. Armbruster, A.S. Mikhailov and E. Lefeber. Information and material flows in complex networks. Physica A: Statistical Mechanics and its Applications, 363(1):xi–xvi, 2006.

T.P. Hill. On goods and services. Review of Income and Wealth, 23(4):315–338, 1977.

P. Hochendoner, C. Ogle and W.H. Mather. A queueing approach to multi-site enzyme kinetics. Interface Focus, 4:1–11, 2014.

W.J. Hopp and M.L. Spearman. Factory Physics: Foundations of Manufacturing Management. McGraw-Hill, New York, NY, 3rd edition, 2008.

U.S. Karmarkar. Manufacturing lead times, order release and capacity loading. In S. C. Graves, A. H. G. Rinnooy Kan and P. H. Zipkin(Eds.), Logistics of Production and Inventory, volume 4 of Handbook in Operations Research and Management Science, chapter 6, pp. 287–329. North Holland, Amsterdam, 1993.

W. Klonowski. Simplifying principles for chemical and enzyme reaction kinetics. Biophysical Chemistry, 18(2):73–87, 1983.

E. Koenigsberg. Cyclic queues. Operational Research Quarterly, 9(1):22–35, 1958.

E. Koenigsberg. Twenty five years of cyclic queues and closed queue networks: A review. Journal of the Operational Research Society, 33(7):605–619, 1982.

G.I. Kolesova and I.A. Poletaev. Nekotorye voprosy issledovaniia sistem s limitiruiushchimi faktorami [Selected problems in research of the systems with limiting factors]. In Upravliaemye sistemy [Controllable systems], number 3, pp. 71–80. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 1969. (In Russian)

C. Kuehn. Multiple Time Scale Dynamics, volume 191 of Applied Mathematical Sciences. Springer, New York, NY, 2014.

L.D. Landau and E.M. Lifshitz. Statistical Physics. Part 1, volume 5 of Course of Theoretical Physics. Elsevier Butterworth-Heinemann, Burlington, MA, 3rd edition, 2013.

E. Levine and T. Hwa. Stochastic fluctuations in metabolic pathways. Proceedings of the National Academy of Sciences, 104(22):9224–9229, 2007.

A. Mustafin and A. Kantarbayeva. Opening the Leontief’s black box. Heliyon, 4(5):e00626, 2018.

R.R. Nelson and S.G. Winter. An Evolutionary Theory of Economic Change. Belknap Press of Harvard University Press, Cambridge, MA, 1982.

J. Niyirora and J. Zhuang. Fluid approximations and control of queues in emergency departments. European Journal of Operational Research, 261(3):1110– 1124, 2017.

Yu.M. Romanovskii, N.M. Stepanova and D.S. Chernavskii. Chto takoe matematicheskaia biofizika: Kineticheskie modeli v biofizike [What is mathematical biophysics: Kinetic models in biophysics]. Prosveshchenie, Moscow, 1971. (In Russian)

L.A. Segel and M. Slemrod. The quasi-steady-state assumption: A case study in perturbation. SIAM Review, 31(3):446–477, 1989.

R.W. Shephard. Theory of Cost and Production Functions. Princeton University Press, Princeton, NJ, 2016.

J.F. Shortle, J.M. Thompson, D. Gross and C.M. Harris. Fundamentals of Queueing Theory. Wiley Series in Probability and Statistics. Wiley, Hoboken, NJ, 5th edition, 2018.

J. Taylor and R.R.P. Jackson. An application of the birth and death process to the provision of spare machines. Operational Research Quarterly, 5(4):95–108, 1954.

The World Bank. World development indicators. Table 4.2: Structure of output. The World Bank Group, Washington, DC, 2020. Available from Internet:

W. Whitt. Time-varying queues. Queueing Models and Service Management, 1(2):79–164, 2018.

G.B. Yom-Tov and A. Mandelbaum. Erlang-R: A time-varying queue with reentrant customers, in support of healthcare staffing. Manufacturing & Service Operations Management, 16(2):283–299, 2014.