Abstract
We analyze a timed Petri net model of an emergency call center which processes calls with different levels of priority. The counter variables of the Petri net represent the cumulated number of events as a function of time. We show that these variables are determined by a piecewise linear dynamical system. We also prove that computing the stationary regimes of the associated fluid dynamics reduces to solving a polynomial system over a tropical (min-plus) semifield of germs. This leads to explicit formulæ expressing the throughput of the fluid system as a piecewise linear function of the resources, revealing the existence of different congestion phases. Numerical experiments show that the analysis of the fluid dynamics yields a good approximation of the real throughput.
The three authors were partially supported by the programme “Concepts, Systèmes et Outils pour la Sécurité Globale” of the French National Agency of Research (ANR), project “DEMOCRITE”, number ANR-13-SECU-0007-01. The first and last authors were partially supported by the programme “Ingènierie Numérique & Sécurité” of ANR, project “MALTHY”, number ANR-13-INSE-0003.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Baccelli, F., Cohen, G., Olsder, G., Quadrat, J.: Synchronization and Linearity. Wiley (1992)
Heidergott, G., Olsder, G.J., van der Woude, J.: Max Plus at work. Princeton University Press (2006)
Cottenceau, B., Hardouin, L., Boimond, J.: Modeling and control of weight-balanced timed event graphs in dioids. IEEE Trans. Autom. Control 59(5) (2014)
Cohen, G., Gaubert, S., Quadrat, J.: Asymptotic throughput of continuous timed Petri nets. In: 34th Conference on Decision and Control (1995)
Cohen, G., Gaubert, S., Quadrat, J.: Algebraic system analysis of timed Petri nets. In: Gunawardena, J. (ed.) Idempotency. Publications of the Isaac Newton Institute, pp. 145–170. Cambridge University Press (1998)
Libeaut, L.: Sur l’utilisation des dioïdes pour la commande des systèmes à événements discrets. Thèse, École Centrale de Nantes (1996)
Plus, M.: Max-plus-times linear systems. In: Open Problems in Mathematical Systems and Control Theory. Springer (1999)
Farhi, N., Goursat, M., Quadrat, J.P.: Piecewise linear concave dynamical systems appearing in the microscopic traffic modeling. Linear Algebra and Appl. (2011)
Berthomieu, B., Diaz, M.: Modeling and verification of time dependent systems using time Petri nets. IEEE Transactions on Software Engineering 17(3) (1991)
Abdulla, P.A., Nylén, A.: Timed Petri nets and BQOs. In: Colom, J.-M., Koutny, M. (eds.) ICATPN 2001. LNCS, vol. 2075, pp. 53–70. Springer, Heidelberg (2001)
Gardey, G., Roux, O.H., Roux, O.F.: Using zone graph method for computing the state space of a time Petri net. In: Larsen, K.G., Niebert, P. (eds.) FORMATS 2003. LNCS, vol. 2791. Springer, Heidelberg (2004)
Jacobsen, L., Jacobsen, M., Møller, M.H., Srba, J.: Verification of timed-arc Petri nets. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 46–72. Springer, Heidelberg (2011)
Berthomieu, B., Vernadat, F.: Time Petri nets analysis with TINA. In: QEST 2006. IEEE (2006)
Lime, D., Roux, O.H., Seidner, C., Traonouez, L.-M.: Romeo: a parametric model-checker for Petri nets with stopwatches. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 54–57. Springer, Heidelberg (2009)
Byg, J., Jørgensen, K.Y., Srba, J.: TAPAAL: editor, simulator and verifier of timed-arc Petri nets. In: Liu, Z., Ravn, A.P. (eds.) ATVA 2009. LNCS, vol. 5799, pp. 84–89. Springer, Heidelberg (2009)
Bérard, B., Cassez, F., Haddad, S., Lime, D., Roux, O.H.: Comparison of the expressiveness of timed automata and time Petri nets. In: Pettersson, P., Yi, W. (eds.) FORMATS 2005. LNCS, vol. 3829, pp. 211–225. Springer, Heidelberg (2005)
Srba, J.: Comparing the expressiveness of timed automata and timed extensions of Petri nets. In: Cassez, F., Jard, C. (eds.) FORMATS 2008. LNCS, vol. 5215, pp. 15–32. Springer, Heidelberg (2008)
Raclot, S., Reboul, R.: Analysis of the new call center organization at PP and BSPP. Personal communication to the authors (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Allamigeon, X., Bœuf, V., Gaubert, S. (2015). Performance Evaluation of an Emergency Call Center: Tropical Polynomial Systems Applied to Timed Petri Nets. In: Sankaranarayanan, S., Vicario, E. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2015. Lecture Notes in Computer Science(), vol 9268. Springer, Cham. https://doi.org/10.1007/978-3-319-22975-1_2
Download citation
DOI: https://doi.org/10.1007/978-3-319-22975-1_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-22974-4
Online ISBN: 978-3-319-22975-1
eBook Packages: Computer ScienceComputer Science (R0)