Model Checking Markov Chains with Actions and State Labels

In the past, logics of several kinds have been proposed for reasoning about discrete-time or continuous-time Markov chains. Most of these logics rely on either state labels (atomic propositions) or on transition labels (actions). However, in several applications it is useful to reason about both state properties and action sequences. For this purpose, we introduce the logic as CSL which provides a powerful means to characterize execution paths of Markov chains with actions and state labels. asCSL can be regarded as an extension of the purely state-based logic CSL (continuous stochastic logic). In asCSL, path properties are characterized by regular expressions over actions and state formulas. Thus, the truth value of path formulas depends not only on the available actions in a given time interval, but also on the validity of certain state formulas in intermediate states. We compare the expressive power of CSL and asCSL and show that even the state-based fragment of asCSL is strictly more expressive than CSL if time intervals starting at zero are employed. Using an automaton-based technique, an asCSL formula and a Markov chain with actions and state labels are combined into a product Markov chain. For time intervals starting at zero, we establish a reduction of the model checking problem for asCSL to CSL model checking on this product Markov chain. The usefulness of our approach is illustrated with an elaborate model of a scalable cellular communication system, for which several properties are formalized by means of asCSL formulas and checked using the new procedure

[1]  Bernhard Walke,et al.  Mobile Radio Networks , 1999 .

[2]  Adnan Aziz,et al.  It Usually Works: The Temporal Logic of Stochastic Systems , 1995, CAV.

[3]  William H. Sanders,et al.  State-Space Support for Path-Based Reward Variables , 1999, Perform. Evaluation.

[4]  H. Hermanns,et al.  Syntax , Semantics , Equivalences , and Axioms for MTIPP y , 1994 .

[5]  Christel Baier,et al.  On the Logical Characterisation of Performability Properties , 2000, ICALP.

[6]  Prakash Panangaden,et al.  Continuous stochastic logic characterizes bisimulation of continuous-time Markov processes , 2003, J. Log. Algebraic Methods Program..

[7]  Rance Cleaveland,et al.  A Theory of Testing for Markovian Processes , 2000, CONCUR.

[8]  Joël Ouaknine,et al.  State/Event-Based Software Model Checking , 2004, IFM.

[9]  Bengt Jonsson,et al.  A logic for reasoning about time and reliability , 1990, Formal Aspects of Computing.

[10]  Joachim Meyer-Kayser Automatische Verifikation stochastischer Systeme , 2004 .

[11]  Kishor S. Trivedi,et al.  SPNP: stochastic Petri net package , 1989, Proceedings of the Third International Workshop on Petri Nets and Performance Models, PNPM89.

[12]  Christel Baier,et al.  Model Checking Continuous-Time Markov Chains by Transient Analysis , 2000, CAV.

[13]  Rocco De Nicola,et al.  Three logics for branching bisimulation , 1990, [1990] Proceedings. Fifth Annual IEEE Symposium on Logic in Computer Science.

[14]  Christel Baier,et al.  Model Checking pathCSL , 2003 .

[15]  Richard E. Ladner,et al.  Propositional Dynamic Logic of Regular Programs , 1979, J. Comput. Syst. Sci..

[16]  P. Buchholz Exact and ordinary lumpability in finite Markov chains , 1994, Journal of Applied Probability.

[17]  Edmund M. Clarke,et al.  Model Checking , 1999, Handbook of Automated Reasoning.

[18]  Christel Baier,et al.  Model checking action- and state-labelled Markov chains , 2004, International Conference on Dependable Systems and Networks, 2004.

[19]  Boudewijn R. Haverkort,et al.  On the efficient sequential and distributed generation of very large Markov chains from stochastic Petri nets , 1999, Proceedings 8th International Workshop on Petri Nets and Performance Models (Cat. No.PR00331).

[20]  Jane Hillston,et al.  A compositional approach to performance modelling , 1996 .

[21]  Joost-Pieter Katoen,et al.  Implementing a Model Checker for Performability Behaviour , 2001 .

[22]  Rocco De Nicola,et al.  Three Logics for Branching Bisimulation (Extended Abstract) , 1990, LICS 1990.

[23]  Robert K. Brayton,et al.  Verifying Continuous Time Markov Chains , 1996, CAV.

[24]  Joost-Pieter Katoen,et al.  Towards Model Checking Stochastic Process Algebra , 2000, IFM.

[25]  Christel Baier,et al.  Model-Checking Algorithms for Continuous-Time Markov Chains , 2002, IEEE Trans. Software Eng..

[26]  Pierre Wolper,et al.  Specification and synthesis of communicating processes using an extended temporal logic: (preliminary version) , 1982, POPL '82.

[27]  Raimo Kantola,et al.  Performance evaluation of GSM handover traffic in a GPRS/GSM network , 2003, Proceedings of the Eighth IEEE Symposium on Computers and Communications. ISCC 2003.

[28]  B. Nordstrom FINITE MARKOV CHAINS , 2005 .

[29]  Stefania Gnesi,et al.  Model checking for action-based logics , 1994, Formal Methods Syst. Des..

[30]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .