Markov Regenerative Stochastic Petri Nets Research Articles

Modeling dynamic scenarios for safety, reliability, availability, and maintainability analysis

Safety analysis uses probability combinatorial models like fault tree and/or event tree. Such methods have static basic events and do not consider complex scenarios of dynamic reliability, leading to conservative results. Reliability, availability, and maintainability (RAM) analysis using reliability block diagram (RBD) experience the same limitations. Continuous Markov chains model dynamic reliability scenarios but suffer from other limitations like states explosion and restriction of exponential life distribution only. Markov Regenerative Stochastic Petri Nets oblige complex mathematical formalism and still subject to state explosions for large systems. In the design of complex systems, distinct teams make safety and RAM analyses, each one adopting tools better fitting their own needs. Teams using different tools turns obscure the detection of problems and their correction is even harder. This work aims to improve design quality, reduce design conservatism, and ensure consistency by proposing a single and powerful tool to perform any probabilistic analysis. The suggested tool is the Stochastic Colored class of Petri Nets, which supplies hierarchical organization, a set of options for life distributions, dynamic reliability scenarios and simple and easy construction for large systems. This work also proposes more quality rules to assure model consistency. Such method for probabilistic analysis may have the effect of shifting systems design from “redundancy, segregation and independency” approach to “maintainability, maintenance and contingency procedures” approach. By modeling complex human and automated interventional scenarios, this method reduces capital costs and keeps safety and availability of systems.

A Case Study on Reliability Evaluation with Markov Regenerative Stochastic Petri Nets

A Case Study on Reliability Evaluation with Markov Regenerative Stochastic Petri Nets

MRSPN analysis of Semi-Markovian finite source retrial queues

In this paper, the analysis of Semi-Markovian single server retrial queues by means of Markov Regenerative Stochastic Petri Nets (MRSPN) is considered. We propose MRSPN models for the two retrial queues M/G/1/N/N and M/G/1/N/N with orbital search. By inspecting the reduced reachability graph of both MRSPN models, the qualitative analysis is obtained. The quantitative analysis is carried out after constructing their one step transition probability matrix and computing the steady state probability distribution of each tangible marking. As an example, the queue $$M/Hypo_{2}/1/2/2$$ is treated in order to illustrate the functionality of the MRSPN approach. The exact performance measures (mean number of customers in the system, mean response time, mean waiting time,...) are computed for different parameters of the two systems by an algorithm elaborated in Matlab environment.

Transient Analysis of Software Rejuvenation Policies in Virtualized System: Phase-Type Expansion Approach

HeThis paper presents a transient analysis of software rejuvenation with virtualization. In particular, we apply the phase-type expansion to analyze the pointwise availability of virtual-machine (VM) based software rejuvenation with two policies; cold-VM and warm-VM rejuvenation policies. The performance measures can be derived from the stochastic models described by labeled Markov regenerative stochastic Petri Nets (MRSPNs).

Mission reliability analysis of fault-tolerant multiple-phased systems

Fault-tolerant multiple-phased systems (FTMPS) are defined as systems whose critical components are independently replicated and whose operational life can be partitioned into a set of disjoint periods, called “phases”. Because of their deployment in critical applications, their mission reliability analysis is a task of primary relevance to validate the designs. This paper is focused on the reliability analysis of FTMPS with random phase durations, non-exponentially distributed repair activities and different repair policies. For self-repairable FTMPS with a component-level reconfiguration architecture, we derive several efficient formulations from the underlying structure characteristics for their intraphase behavior analysis. We also present a uniform solution framework of the mission reliability for FTMPS with generally distributed phase durations. Compared with existing methods based on deterministic and stochastic Petri nets or Markov regenerative stochastic Petri nets, our approach is more simple in concept and powerful in computation. Two examples of FTMPS are analyzed to illustrate the advantages of our approach.

Performance Analysis of the M/G/c Retrial Queueing Systems Using the Theory of Markov Regenerative Stochastic Petri nets

In this paper, we exhibit a method of modeling and analyzing the M/G/c retrial queueing system with the help of Markov Regenerative Stochastic Petrinets (MRSPN). We have indicated as to how this representation could be used to obtain the transient and the steady state probabilities of the retrial queueing system. We illustrate the methodology with a simple example.

Markov regenerative stochastic petri nets to model and evaluate phased mission systems dependability

This study deals with model-based dependability transient analysis of phased mission systems. A review of the studies in the literature showed that several aspects of multiphased systems pose challenging problems to the dependability evaluation methods and tools. To attack the weak points of the state-of-the-art we propose a modeling methodology that exploits the power of the class of Markov regenerative stochastic Petri net models. By exploiting the techniques available in the literature for the analysis of the Markov Regenerative Processes, we obtain an analytical solution technique with a low computational complexity, basically dominated by the cost of the separate analysis of the system inside each phase. Last, the existence of analytical solutions allows us to derive the sensitivity functions of the dependability measures, thus providing the dependability engineer with additional means for the study of phased mission systems.

Transient analysis of Age-MRSPNs by the method of supplementary variables

Abstract In order to assist the performance evaluation of complex stochastic models, automatic program tools were developed since a long time. Stochastic Petri nets (SPN) are applied as an effective model description language supported by several analytical and simulation tools. The analytical description and the numerical analysis of non-Markovian stochastic Petri net models gained attention recently. There are different theoretical approaches and numerical methods considered in recent works, such as the Markov renewal theory and the supplementary variable approach, but to find the most effective way of the analysis of such models is still an open research problem. The supplementary variable approach was successfully applied to the transient and steady state analysis of Markov regenerative stochastic Petri nets (MSRPN) when the preemption policy associated with the Petri net (PN) transitions was preemptive repeat different (prd), but it was not applicable with other preemption mechanisms. In this paper we extend the applicability of the supplementary variable approach to a class of MRSPNs in which preemptive resume (prs) policy can also be assigned to the transitions of the PN.

Markov regenerative stochastic Petri nets with general execution policies: supplementary variable analysis and a prototype tool

Stochastic Petri nets (SPNs) with general firing time distributions are considered. Generally timed transitions can have general execution policies: the preemption policy may be preemptive repeat different (prd) or preemptive resume (prs) and the firing time distribution can be marking-dependent. A stationary analysis method covering all possible combinations is presented by means of supplementary variables. The method is implemented in a prototype tool SPNica which is based on Mathematica. The use of the general execution policies is illustrated by a WWW server model.

A modeling framework to implement preemption policies in non-Markovian SPNs

Petri nets represent a useful tool for performance, dependability, and performability analysis of complex systems. Their modeling power can be increased even more if nonexponentially distributed events are considered. However, the inclusion of nonexponential distributions destroys the memoryless property and requires to specify how the marking process is conditioned upon its past history. We consider, in particular, the class of stochastic Petri nets whose marking process can be mapped into a Markov regenerative process. An adequate mathematical framework is developed to deal with the considered class of Markov Regenerative Stochastic Petri Nets (MRSPN). A unified approach for the solution of MRSPNs where different preemption policies can be defined in the same model is presented. The solution is provided both in steady-state and in transient condition. An example concludes the paper.

Petri nets with k simultaneously enabled generally distributed timed transitions

Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations. In this paper we extend the class of Markov regenerative stochastic Petri nets ∗ (MRSPNs ∗), removing the restriction that at most one generally distributed timed transition can be enabled in any marking. This new class of Petri nets, which we call concurrent generalized Petri nets (CGPNs), allows simultaneous enabling of immediate, exponentially and generally distributed timed transitions, under the hypothesis that the latter are all enabled at the same instant. The stochastic process underlying a CGPN is shown to be still an MRGP. We evaluate the kernel distribution of the underlying MRGP and define the steps required to generate it automatically. The methodology described is used to assess the behavior of a system in both steady-state and transient functioning conditions.

Markov regenerative stochastic Petri nets

Stochastic Petri nets of various types (SPN, GSPN, ESPN, DSPN etc.) are recognized as useful modeling tools for analyzing the performance and reliability of systems. The analysis of such Petri nets proceeds by utilizing the underlying continuous-time stochastic processes — continuous-time Markov chains for SPN and GSPN, semi-Markov processes for a subset of ESPNs and Markov regenerative processes for DSPN. In this paper, we introduce a new class of stochastic Petri nets, called Markov Regenerative Stochastic Petri Nets (MRSPNs), that can be analyzed by means of Markov regenerative processes and constitutes a true generalization of all the above classes. The MRSPNs allow immediate transitions, exponentially distributed timed transitions and generally distributed timed transitions. With a restriction that at most one generally distributed timed transition be enabled in each marking, the transient and steady state analysis of MRSPNs can be carried out analytically-numerically rather than by simulation. Equations for the solution of MRSPNs are developed in this paper, and are applied to an example.