Marco Paolieri

Transient analysis of non-Markovian models using stochastic state classes

The method of stochastic state classes approaches the analysis of Generalised Semi Markov Processes (GSMPs) through the symbolic derivation of probability density functions over supports described by Difference Bounds Matrix (DBM) zones. This makes steady state analysis viable, provided that at least one regeneration point is visited by every cyclic behaviour of the model. We extend the approach providing a way to derive transient probabilities. To this end, stochastic state classes are extended with a supplementary timer that enables the symbolic derivation of the distribution of time at which a class can be entered. The approach is amenable to efficient implementation when model timings are given by expolynomial distributions, and it can be applied to perform transient analysis of GSMPs within any given time bound. In the special case of models underlying a Markov Regenerative Process (MRGP), the method can also be applied to the symbolic derivation of local and global kernels, which in turn provide transient probabilities through numerical integration of generalised renewal equations. Since much of the complexity of this analysis is due to the local kernel, we propose a selective derivation of its entries depending on the specific transient measure targeted by the analysis.

Probabilistic Model Checking of Regenerative Concurrent Systems

We consider the problem of verifying quantitative reachability properties in stochastic models of concurrent activities with generally distributed durations. Models are specified as stochastic time Petri nets and checked against Boolean combinations of interval until operators imposing bounds on the probability that the marking process will satisfy a goal condition at some time in the interval [α, β] after an execution that never violates a safety property. The proposed solution is based on the analysis of regeneration points in model executions: a regeneration is encountered after a discrete event if the future evolution depends only on the current marking and not on its previous history, thus satisfying the Markov property. We analyze systems in which multiple generally distributed timers can be started or stopped independently, but regeneration points are always encountered with probability 1 after a bounded number of discrete events. Leveraging the properties of regeneration points in probability spaces of execution paths, we show that the problem can be reduced to a set of Volterra integral equations, and we provide algorithms to compute their parameters through the enumeration of finite sequences of stochastic state classes encoding the joint probability density function (PDF) of generally distributed timers after each discrete event. The computation of symbolic PDFs is limited to discrete events before the first regeneration, and the repetitive structure of the stochastic process is exploited also before the lower bound α, providing crucial benefits for large time bounds. A case study is presented through the probabilistic formulation of Fischers mutual exclusion protocol, a well-known real-time verification benchmark.

Towards the Quantitative Evaluation of Phased Maintenance Procedures Using Non-Markovian Regenerative Analysis

The concept of Phased Mission Systems (PMS) can be used to describe maintenance procedures made of sequential actions that use a set of resources and may severely affect them, for instance operations that require outage of hardware and/or software components to recover from a failure or to perform upgrades, tests, and configuration changes. We propose an approach for modeling and evaluation of this class of maintenance procedures, notably addressing the case of actions with non-exponential and firmly bounded duration. This yields stochastic models that underlie a Markov Regenerative Process (MRP) with multiple concurrent timed events having a general (GEN) distribution over a bounded support, which can be effectively analyzed through the method of stochastic state classes. The approach allows evaluation of transient availability measures, which can be exploited to support the selection of a rejuvenation plan of system resources and the choice among different feasible orderings of actions. The experiments were performed through a new release of the Oris tool based on the Sirio framework.

Non-Markovian Performability Evaluation of ERTMS/ETCS Level 3

The European Rail Traffic Management System/European Train Control System (ERTMS/ETCS) is an innovative standard introduced to enhance reliability, safety, performance, and interoperability of trans-European railways. In Level 3, the standard replaces fixed-block safety mechanisms, in which only one train at a time is allowed to be in each railway block, with moving blocks: a train proceeds as long as it receives radio messages ensuring that the track ahead is clear of other trains. This mechanism increases line capacity, but relies crucially on the communication link: if messages are lost, the train must stop within a safe deadline even if the track ahead is clear. We develop upon results of the literature to propose an approach for the evaluation of transient availability of the communication channel and probability of train stops due to lost messages. We formulate a non-Markovian model of communication availability and system operation, and leverage solution techniques of the ORIS Tool to provide experimental results in the presence of multiple concurrent activities with non-exponential durations.

A Scalable Approach to the Assessment of Storm Impact in Distributed Automation Power Grids

We present models and metrics for the survivability assessment of distribution power grid networks accounting for the impact of multiple failures due to large storms. The analytical models used to compute the proposed metrics are built on top of three design principles: state space factorization, state aggregation, and initial state conditioning. Using these principles, we build scalable models that are amenable to analytical treatment and efficient numerical solution. Our models capture the impact of using reclosers and tie switches to enable faster service restoration after large storms.We have evaluated the presented models using data from a real power distribution grid impacted by a large storm: Hurricane Sandy. Our empirical results demonstrate that our models are able to efficiently evaluate the impact of storm hardening investment alternatives on customer affecting metrics such as the expected energy not supplied until complete system recovery.

Transient analysis of networks of stochastic timed automata using stochastic state classes

Stochastic Timed Automata (STA) associate logical locations with continuous, generally distributed sojourn times. In this paper, we introduce Networks of Stochastic Timed Automata (NSTA), where the components interact with each other by message broadcasts. This results in an underlying stochastic process whose state is made of the vector of logical locations, the remaining sojourn times, and the value of clocks. We characterize this general state space Markov process through transient stochastic state classes that sample the state and the absolute age after each event. This provides an algorithmic approach to transient analysis of NSTA models, with fairly general termination conditions which we characterize with respect to structural properties of individual components that can be checked through straightforward algorithms.

Performance Evaluation of Fischer's Protocol through Steady-State Analysis of Markov Regenerative Processes

Fischers protocol is a well-known timed mechanism through which a set of processes can synchronize access to a critical section without relying on atomic test-and-set operations, as might occur in a distributed environment or on a low-level computing platform. The protocol is based on a deterministic waiting time that can be defined so as to guarantee that possible interference due to concurrent accesses with random bounded delays be resolved with certainty. While protocol correctness descends from firm lower and upper bounds on waiting times and random delays, performance attained in synchronization also depends on continuous distributions of delays. Performance evaluation of a correct implementation thus requires the solution of a non-Markovian model whose underlying stochastic process falls in the class of Markov regenerative processes (MRPs) with multiple concurrent delays with non-exponential duration. Numerical solution of this class of models is to a large extent still an open problem. We provide a twofold contribution. We first introduce a novel method for the steady-state analysis of MRPs where regenerations are reached in a bounded number of discrete events, which enlarges the class amenable to numerical solution by allowing multiple concurrent timers with non-exponential distributions. The proposed technique is then applied to Fischers protocol by characterizing the latency overhead due to synchronization, which comprises the first case where performance of the protocol is quantitatively assessed by jointly accounting for firm bounds and continuous distributions of delays.

Probabilistic Model Checking of Non-Markovian Models with Concurrent Generally Distributed Timers

In the analysis of stochastic concurrent timed models, probabilistic model checking combines qualitative identification of feasible behaviors with quantitative evaluation of their probability. If the stochastic process underlying the model is a Continuous Time Markov Chain (CTMC), the problem can be solved by leveraging on the memoryless property of exponential distributions. However, when multiple generally distributed timers can be concurrently enabled, the underlying process may become a Generalized Semi Markov Process (GSMP) for which simulation is often advocated as the only viable approach to evaluation. The method of stochastic state classes provides a means for the analysis of models belonging to this class, that relies on the derivation of multivariate joint distributions of times to fire supported over Difference Bounds Matrix (DBM) zones. Transient stochastic state classes extend the approach with an additional age clock associating each state with the distribution of the time at which it can be reached. We show how transient stochastic state classes can be used to perform bounded probabilistic model checking also for models with underlying GSMPs, and we characterize the conditions for termination of the resulting algorithm, both in exact and approximate evaluation. We also show how the number of classes enumerated to complete the analysis can be largely reduced through a look-ahead in the non-deterministic state class graph of reachable DBM zones. As notable traits, the proposed technique accepts efficient implementation based on DBM zones without requiring the split of domains in regions, and it expresses the bound in terms of a bilateral constraint on the elapsed time without requiring assumptions on the discrete number of executed transitions. Experimental results based on a preliminary implementation in the Oris tool are reported.

Software rejuvenation impacts on a phased-mission system for Mars exploration

When software contains aging-related faults and the system has a long mission period, phased-mission systems consisting of several software components can suffer from software aging, which is a progressive degradation of the software execution environment. Failures caused by software aging might impact on the mission success probability. In this paper, we present a model for a phased-mission system with software rejuvenation, and analyze the impacts of software rejuvenation on the success probability and completion time distribution of the mission. The mission of Mars exploration rover is considered as an example of phased-mission system. The analysis results show that the mission success probability is improved by software rejuvenation at the cost of the mission completion time.

Exploiting Non-deterministic Analysis in the Integration of Transient Solution Techniques for Markov Regenerative Processes

Transient analysis of Markov Regenerative Processes (MRPs) can be performed through the solution of Markov renewal equations defined by global and local kernels, which respectively characterize the occurrence of regenerations and transient probabilities between them. To derive kernels from stochastic models (e.g., stochastic Petri nets), existing methods exclusively address the case where at most one generally-distributed timer is enabled in each state, or where regenerations occur in a bounded number of events. In this work, we analyze the state space of the underlying timed model to identify epochs between regenerations and apply distinct methods to each epoch depending on the satisfied conditions. For epochs not amenable to existing methods, we propose an adaptive approximation of kernel entries based on partial exploration of the state space, leveraging heuristics that permit to reduce the error on transient probabilities. The case study of a polling system with generally-distributed service times illustrates the effect of these heuristics and how the approach extends the class of models that can be analyzed.