Abstract
We introduce a stochastic process algebra, PEPA, as a high-level modelling paradigm for continuous time Markov chains (CTMC). Process algebras are mathematical theories which model concurrent systems by their algebra and provide apparatus for reasoning about the structure and behaviour of the model. Recent extensions of these algebras, associating random variables with actions, make the models also amenable to Markovian analysis. A compositional structure is inherent in the PEPA language. As well as the clear advantages that this offers for model construction, we demonstrate how this compositionality may be exploited to reduce the state space of the CTMC. This leads to an exact aggregation based on lumpability.
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
M. Ajmone Marsan, G. Conte and G. Balbo, A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems, ACM Trans. on Computer Systems 2(2), 1984.
M. Bernardo, R. Gorrieri and L. Donatiello, MPA: A Stochastic Process Algebra, Technical Report UBLCS-94–10, University of Bologna, 1994.
P. Buchholz, Hierarchical Markovian Models - Symmetries and Reduction, in Computer Performance Evaluation: Modelling Techniques and Tools, R.J. Pooley and J. Hillston (Eds), EUP, 1993.
P. Buchholz, On a Markovian Process Algebra, Technical Report 500/1994, University of Dortmund, 1994.
G. Ciardo, J. Muppala and K.S. Trivedi, On the Solution of GSPN Reward Models, Performance Evaluation, 12, 1991.
S. Donatelli, Superposed Generalized Stochastic Petri Nets: Definition and Efficient Solution, in Proc. of 15th Int. Conf. on Application and Theory of Petri Nets, M. Silva (Ed), Springer-Verlag, 1994.
H. Hermanns and M. Rettelbach, Markovian Processes go Algebra, Technical Report IMMD7–10–1994, University of Erlangen, 1994.
J. Hillston, A Compositional Approach to Performance Modelling, PhD Thesis (CST-107–94), University of Edinburgh, 1994.
J. Hillston, The Nature of Synchronisation, in Proc. of PAPM’94, U. Herzog and M. Rettelbach (Eds), University of Erlangen, 1994.
R. Howard, Dynamic Probabilistic Systems: Semi-Markov and Decision Systems, Vol.II, Wiley, 1971.
C-C. Jou and S.A. Smolka, Equivalences, Congruences and Complete Axiomatizations of Probabilistic Processes, in CONCUR’90, J.C.M. Baeten and J.W. Klop (Eds), Springer-Verlag, 1990.
J.G. Kemeny and J.L. Snell, Finite Markov Chains, Van Nostrand, 1960.
K. Larsen and A. Skou, Bisimulation Through Probabilistic Testing, Information and Computation 94(1), 1991.
V. Nicola, Lumping in Markov Reward Processes, Technical Report RC14719, IBM, Yorktown Heights, NY, 1989.
X. Nicollin and J. Stifakis, An Overview and Synthesis on Timed Process Algebras, in Real-Time: Theory in Practice, J.W. de Bakker et al. (Eds), Springer-Verlag, 1991.
G.D. Plotkin, A Structured Approach to Operational Semantics, Technical Report DAIMI FM-19, Aarhus University, 1981.
W.H. Sanders and J.F. Meyer, Reduced Base Model Construction Methods for Stochastic Activity Networks, IEEE JSAC 9(1), 1991.
W.J. Stewart, K. Atif and B. Plateau, The Numerical Solution of Stochastic Automata Network, Technical Report 6, LMC-IMAG, 1993.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media New York
About this paper
Cite this paper
Hillston, J. (1995). Compositional Markovian Modelling Using a Process Algebra. In: Stewart, W.J. (eds) Computations with Markov Chains. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2241-6_12
Download citation
DOI: https://doi.org/10.1007/978-1-4615-2241-6_12
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5943-2
Online ISBN: 978-1-4615-2241-6
eBook Packages: Springer Book Archive