Abstract
We develop a way of analyzing the behavior of systems modeled using Discrete Time Markov Chains (DTMC). Specifically, we define iLTL, an LTL with linear inequalities on the pmf vectors as atomic propositions. iLTL allows us to express not only properties such as the expected number of jobs or the expected energy consumption of a protocol during a time interval, but also inequalities over such values. We present an algorithm for model checking properties of DTMCs expressed in iLTL. Our model checker differs from existing probabilistic ones in that the latter do not check properties of the transitions on the probability mass function (pmf) itself. Thus, iLTLChecker can check, given an interval estimate of current pmf, whether future pmfs will always satisfy a specification. We believe such properties often arise in distributed systems and networks and may, in particular, be useful in specifying requirements for routing or load balancing protocols. Our algorithm has been implemented in a tool called iLTLChecker and we illustrate the use of the tool by means of some examples.
This research has been supported by the DARPA IXO NEST Award F33615-01-C-1907 and the DARPA/AFOSR MURI Award F49620-02-1-0325.
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
Aziz, A., Singhal, V., Balarin, F.: It usually works: The temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)
Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)
Miner, A.S., Ciardo, G.: Efficient reachability set generation and storage using decision diagrams. In: Donatelli, S., Kleijn, J. (eds.) ICATPN 1999. LNCS, vol. 1639, pp. 6–25. Springer, Heidelberg (1999)
Tanenbaum, A.S.: Computer Networks, 4th edn. Prentice Hall, Englewood Cliffs (2003)
Papoulis, A.: Probability, Random Variables, and Stochastic Processes, 3rd edn. McGraw-Hill, New York (1991)
Baier, C., Katoen, J.-P., Hermanns, H., Haverkort, B.: Simulation for continuous-time markov chains. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 338–354. Springer, Heidelberg (2002)
Luenberger, D.G.: Linear and Nonlinear Programming, 2nd edn. Addison Wesley, Reading (1989)
Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT-Press, Cambridge (2000)
Holzmann, G.J.: The model checker spin. IEEE Transactions on Software Engineering 23, 279–295 (1997)
Ciardo, G., Marie, R.A., Sericola, B., Trivedi, K.S.: Performability analysis using semi-markov reward process. IEEE Transactions on Computers 39, 1251–1264 (1990)
Strang, G.: Linear Algebra and Its Applications, 3rd edn. Harcourt Brace Jovanovich (1988)
Start, H., Woods, J.W.: Probability and Random Processes with Applications to Signal Processing, 3rd edn. Prentice-Hall, Englewood Cliffs (2002)
Hermanns, H., Katoen, J.-P., Meyer-Kayser, J., Siegle, M.: A markov chain model checker. In: Schwartzbach, M.I., Graf, S. (eds.) TACAS 2000. LNCS, vol. 1785, pp. 347–362. Springer, Heidelberg (2000)
Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)
Keilson, J.: Markov Chain Models-Rarity and Exponentiality. Springer, Heidelberg (1979)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, pp. 200–204. Springer, Heidelberg (2002)
Vardi, M.Y.: Probabilistic linear-time model checking: an overview of the automata-theoretic approach. In: Proc. 5th Int. AMAST Workshop Formal Methods for Real-Time and Probabilistic Systems, May 1999, vol. 1601 (1999)
Gerth, R., Peled, D., Vardi, M.Y., Wolper, P.: Simple on-the-fly automatic verification of linear temporal logic. In: IFIP/WG, vol. 6.1, pp. 3–18 (1995)
Andova, S., Hermanns, H., Katoen, J.-P.: Discrete-time rewards model-checked. In: Formal Modeling and Analysis of Timed Systems 2003. LNCS, pp. 88–104. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kwon, Y., Agha, G. (2004). Linear Inequality LTL (iLTL): A Model Checker for Discrete Time Markov Chains. In: Davies, J., Schulte, W., Barnett, M. (eds) Formal Methods and Software Engineering. ICFEM 2004. Lecture Notes in Computer Science, vol 3308. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30482-1_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-30482-1_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23841-6
Online ISBN: 978-3-540-30482-1
eBook Packages: Springer Book Archive