Getting the Priorities Right: Saturation for Prioritised Petri Nets

Prioritised Petri net is a powerful modelling language that often constitutes the core of even more expressive modelling languages such as GSPNs (Generalized Stochastic Petri nets). The saturation state space traversal algorithm has proved to be efficient for non-prioritised concurrent models. Previous works showed that priorities may be encoded into the transition relation, but doing so defeats the main idea of saturation by spoiling the locality of transitions. This paper presents an extension of saturation to natively handle priorities by considering the priority-related enabledness of transitions separately, adopting the idea of constrained saturation. To encode the highest priority of enabled transitions in every state we introduce edge-valued interval decision diagrams. We show that in case of Petri nets, this data structure can be constructed offline. According to preliminary measurements, the proposed solution scales better than previously known matrix decision diagram-based approaches, paving the way towards efficient stochastic analysis of GSPNs and the model checking of prioritised models

Improving BDD Based Symbolic Model Checking with Isomorphism Exploiting Transition Relations

Symbolic model checking by using BDDs has greatly improved the applicability of model checking. Nevertheless, BDD based symbolic model checking can still be very memory and time consuming. One main reason is the complex transition relation of systems. Sometimes, it is even not possible to generate the transition relation, due to its exhaustive memory requirements. To diminish this problem, the use of partitioned transition relations has been proposed. However, there are still systems which can not be verified at all. Furthermore, if the granularity of the partitions is too fine, the time required for verification may increase. In this paper we target the symbolic verification of asynchronous concurrent systems. For such systems we present an approach which uses similarities in the transition relation to get further memory reductions and runtime improvements. By applying our approach, even the verification of systems with an previously intractable transition relation becomes feasible.Comment: In Proceedings GandALF 2011, arXiv:1106.081

Towards faster numerical solution of Continuous Time Markov Chains stored by symbolic data structures

This work considers different aspects of model-based performance- and dependability analysis. This research area analyses systems (e.g. computer-, telecommunication- or production-systems) in order to quantify their performance and reliability. Such an analysis can be carried out already in the planning phase, without a physically existing system. All aspects treated in this work are based on finite state spaces (i.e. the models only have finitely many states) and a representation of the state graphs by Multi-Terminal Binary Decision Diagrams (MTBDDs). Currently, there are many tools that transform high-level model specifications (e.g. process algebra or Petri-Net) to low-level models (e.g. Markov chains). Markov chains can be represented by sparse matrices. For complex models very large state spaces may occur (this phenomenon is called state space explosion in the literature) and accordingly very large matrices representing the state graphs. The problem of building the model from the specification and storing the state graph can be regarded as solved: There are heuristics for compactly storing the state graph by MTBDD or Kronecker data structure and there are efficient algorithms for the model generation and functional analysis. For the quantitative analysis there are still problems due to the size of the underlying state space. This work provides some methods to alleviate the problems in case of MTBDD-based storage of the state graph. It is threefold: 1. For the generation of smaller state graphs in the model generation phase (which usually are easier to solve) a symbolic elimination algorithm is developed. 2. For the calculation of steady-state probabilities of Markov chains a multilevel algorithm is developed which allows for faster solutions. 3. For calculating the most probable paths in a state graph, the mean time to the first failure of a system and related measures, a path-based solver is developed

MULTIMODAL PERFORMANCE AND RELIABILITY ANALYSIS OF THE PIERRE AUGER OBSERVATORY NORTH SITE

The Pierre Auger Cosmic Ray Observatory North site employs a large array of surface detector stations (tanks) to detect the secondary particle showers generated by ultra-high energy cosmic rays. Due to the rare nature of ultra-high energy cosmic rays, it is important to have a high reliability on tank communications, ensuring no valuable data is lost. The Auger North site employs a peer-to-peer paradigm, the Wireless Architecture for Hard Real-Time Embedded Networks (WAHREN), designed specifically for highly reliable message delivery over fixed networks, under hard real-time deadlines. The WAHREN design included two retransmission protocols, Micro- and Macro- retransmission. To fully understand how each retransmission protocol increased the reliability of communications, this analysis evaluated the system without using either retransmission protocol (Case-0), both Micro- and Macro-retransmission individually (Micro and Macro), and Micro- and Macro-retransmission combined. This thesis used a multimodal modeling methodology to prove that a performance and reliability analysis of WAHREN was possible, and provided the results of the analysis. A multimodal approach was necessary because these processes were driven by different mathematical models. The results from this analysis can be used as a framework for making design decisions for the Auger North communication system

Constructing Matrix Exponential Distributions by Moments and Behavior around Zero

This paper deals with moment matching of matrix exponential (ME) distributions used to approximate general probability density functions (pdf). A simple and elegant approach to this problem is applying Padé approximation to the moment generating function of the ME distribution. This approach may, however, fail if the resulting ME function is not a proper probability density function; that is, it assumes negative values. As there is no known, numerically stable method to check the nonnegativity of general ME functions, the applicability of Padé approximation is limited to low-order ME distributions or special cases. In this paper, we show that the Padé approximation can be extended to capture the behavior of the original pdf around zero and this can help to avoid representations with negative values and to have a better approximation of the shape of the original pdf. We show that there exist cases when this extension leads to ME function whose nonnegativity can be verified, while the classical approach results in improper pdf. We apply the ME distributions resulting from the proposed approach in stochastic models and show that they can yield more accurate results