Abstract
Concavely-priced probabilistic timed automata, an extension of probabilistic timed automata, are introduced. In this paper we consider expected reachability, discounted, and average price problems for concavely-priced probabilistic timed automata for arbitrary initial states. We prove that these problems are EXPTIME-complete for probabilistic timed automata with two or more clocks and PTIME-complete for automata with one clock. Previous work on expected price problems for probabilistic timed automata was restricted to expected reachability for linearly-priced automata and integer valued initial states. This work uses the boundary region graph introduced by Jurdziński and Trivedi to analyse properties of concavely-priced (non-probabilistic) timed automata.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alur, R., Dill, D.: A theory of timed automata. Theoretical Computer Science 126 (1994)
Alur, R., La Torre, S., Pappas, G.: Optimal paths in weighted timed automata. In: Di Benedetto, M.D., Sangiovanni-Vincentelli, A.L. (eds.) HSCC 2001. LNCS, vol. 2034, p. 49. Springer, Heidelberg (2001)
Beauquier, D.: Probabilistic timed automata. Theoretical Computer Science 292(1) (2003)
Behrmann, G., David, A., Larsen, K., Möller, O., Pettersson, P., Yi, W.: Uppaal - present and future. In: Proc. CDC 2001, vol. 3. IEEE, Los Alamitos (2001)
Berendsen, J., Chen, T., Jansen, D.: Undecidability of cost-bounded reachability in priced probabilistic timed automata. In: Chen, T., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 128–137. Springer, Heidelberg (2009)
Berendsen, J., Jansen, D., Katoen, J.-P.: Probably on time and within budget - on reachability in priced probabilistic timed automata. In: Proc. QEST 2006. IEEE, Los Alamitos (2006)
Bouyer, P., Brinksma, E., Larsen, K.: Optimal infinite scheduling for multi-priced timed automata. Formal Methods in System Design 32(1) (2008)
Boyd, S., Vandenberghe, L.: Convex Optimization. CUP, Cambridge (2004)
de Alfaro, L.: Computing minimum and maximum reachability times in probabilistic systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, p. 66. Springer, Heidelberg (1999)
Dynkin, E., Yushkevich, A.: Controlled Markov Processes. Springer, Heidelberg (1979)
Falk, J., Horowitz, J.: Critical path problems with concave cost-time curves. Management Science 19(4) (1972)
Feng, Y., Xiao, B.: A Continuous-Time Yield Management Model with Multiple Prices and Reversible Price Changes. Management Science 46(5) (2000)
Henzinger, T., Manna, Z., Pnueli, A.: What good are digital clocks? In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 545–558. Springer, Heidelberg (1992)
Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic verification of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)
Jensen, H.: Model checking probabilistic real time systems. In: Proc. 7th Nordic Workshop on Programming Theory, Report 86, pp. 247–261. Chalmers University of Technology (1996)
Jurdzinski, M., Kwiatkowska, M., Norman, G., Trivedi, A.: Concavely-priced probabilistic timed automata. Technical Report RR-09-06. Oxford University Computing Laboratory (2009)
Jurdziński, M., Sproston, J., Laroussinie, F.: Model checking probabilistic timed automata with one or two clocks. Logical Methods in Computer Science 4(3) (2008)
Jurdziński, M., Trivedi, A.: Concavely-priced timed automata. In: Cassez, F., Jard, C. (eds.) FORMATS 2008. LNCS, vol. 5215, pp. 48–62. Springer, Heidelberg (2008)
Kwiatkowska, M., Norman, G., Parker, D., Sproston, J.: Performance analysis of probabilistic timed automata using digital clocks. Formal Methods in System Design 29 (2006)
Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Automatic verification of real-time systems with discrete probability distributions. Theoretical Computer Science 282 (2002)
Kwiatkowska, M., Norman, G., Sproston, J., Wang, F.: Symbolic model checking for probabilistic timed automata. Information and Computation 205(7) (2007)
Laroussinie, F., Markey, N., Schnoebelen, P.: Model checking timed automata with one or two clocks. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 387–401. Springer, Heidelberg (2004)
Laroussinie, F., Sproston, J.: State explosion in almost-sure probabilistic reachability. Information Processing Letters 102(6) (2007)
Larsen, K., Behrmann, G., Brinksma, E., Fehnker, A., Hune, T., Pettersson, P., Romijn, J.: As cheap as possible: Efficient cost-optimal reachability for priced timed automata. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, p. 493. Springer, Heidelberg (2001)
Larsen, K., Rasmussen, J.: Optimal reachability for multi-priced timed automata. Theoretical Computer Science 390(2-3) (2008)
Papadimitriou, C., Tsitsiklis, J.: The complexity of Markov decision processes. Mathematics of Operations Research 12 (1987)
Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley, Chichester (1994)
Torres, E., Dominguez, J., Valdes, L., Aza, R.: Passenger waiting time in an airport and expenditure carried out in the commercial area. Journal of Air Transport Management 11(6) (2005)
Tripakis, S.: Verifying progress in timed systems. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, p. 299. Springer, Heidelberg (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Jurdziński, M., Kwiatkowska, M., Norman, G., Trivedi, A. (2009). Concavely-Priced Probabilistic Timed Automata. In: Bravetti, M., Zavattaro, G. (eds) CONCUR 2009 - Concurrency Theory. CONCUR 2009. Lecture Notes in Computer Science, vol 5710. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04081-8_28
Download citation
DOI: https://doi.org/10.1007/978-3-642-04081-8_28
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04080-1
Online ISBN: 978-3-642-04081-8
eBook Packages: Computer ScienceComputer Science (R0)