Christoph Lindemann

Analysis of stochastic Petri nets by the method of supplementary variables

Abstract This paper presents a unified numerical solution framework for stochastic Petri nets in which transition firing is immediate, exponentially distributed, or generally distributed. The proposed solution approach is based on the method of supplementary variables and extends the class of nets for which a numerical analysis is possible. Deterministic and Stochastic Petri Nets (DSPN) are taken as the basic modeling formalism which is extended by other non-exponential firing times and concurrently enabled deterministic transition firings. Under the assumption that the non-exponential distributions belong to the class of polynomial distributions, computational formulas for numerically solving the state equations are given. In case of concurrently enabled deterministic transitions the solution approach yields a system of partial differential equations which we propose to solve numerically by discretization.

DSPNexpress: a software package for the efficient solution of deterministic and stochastic Petri nets

Abstract This paper describes the analysis tool DSPNexpress which has been developed at the Technische Universitat Berlin since 1991. The development of DSPNexpress has been motivated by the lack of a powerful software package for the numerical solution of deterministic and stochastic Petri nets (DSPNs) and the complexity requirements imposed by evaluating memory consistency models for multicomputer systems. The development of DSPNexpress has gained by the authors experience with the version 1.4 of the software package GreatSPN. However, opposed to GreatSPN, the software architecture of DSPNexpress is particularly tailored to the numerical evaluation of DSPNs. Furthermore, DSPNexpress contains a graphical interface running under the X11 window system. To the best of the authors knowledge, DSPNexpress is the first software package which contains an efficient numerical algorithm for computing steady-state solutions of DSPNs.

An improved numerical algorithm for calculating steady-state solutions of deterministic and stochastic Petri net models

Abstract This paper introduces an efficient algorithm for calculating the time-dependent quantities required by the numerical solution process of Deterministic and Stochastic Petri Nets (DSPNs). The described method employs the randomization technique and a stable calculation of Poisson probabilities. A complete re-design and re-implementation of the appropriate components implemented in the version 1.4 of the software package GreatSPN has lead to significant savings in both computation time and memory space. These benefits are illustrated by DSPN models of an E r /D/1/K queueing system and a fault-tolerant clock synchronization system. These examples show that steady-state solutions of DSPNs are calculated with significantly less computational effort by the algorithm described in this paper than by the method implemented in the version 1.4 of the software package GreatSPN.

Performance Evaluation: Origins and Directions

Position Paper.- Performance Evaluation in Industry: A Personal Perspective.- Topical Area Papers.- Mainframe Systems.- Performance Analysis of Storage Systems.- Ad Hoc, Wireless, Mobile Networks: The Role of Performance Modeling and Evaluation.- Trace-Driven Memory Simulation: A Survey.- Performance Issues in Parallel Processing Systems.- Measurement-Based Analysis of Networked System Availability.- Performance of Client/Server Systems.- Performance Characteristics of the World Wide Web.- Parallel Job Scheduling: A Performance Perspective.- Scheduling of Real-Time Tasks with Complex Constraints.- Software Performance Evaluation by Models.- Performance Analysis of Database Systems.- Performance Analysis of Concurrency Control Methods.- Numerical Analysis Methods.- Product Form Queueing Networks.- Stochastic Modeling Formalisms for Dependability, Performance and Performability.- Analysis and Application of Polling Models.- Discrete-Event Simulation in Performance Evaluation.- Workload Characterization Issues and Methodologies.- Personal Accounts of Key Contributors.- From the Central Server Model to BEST/1(c).- Mean Value Analysis: A Personal Account.- The Early Days of GSPNs.- The Discovery of Self-Similar Traffic.

Numerical analysis of deterministic and stochastic Petri nets with concurrent deterministic transitions

This paper introduces an efficient numerical algorithm for transient analysis of deterministic and stochastic Petri nets (DSPNs) and other discrete-event stochastic systems with exponential and deterministic events. The proposed approach is based on the analysis of a general state space Markov chain (GSSMC) whose state equations constitute a system of multidimensional Fredholm integral equations. Key contributions of this paper constitute the observations that the transition kernel of this system of Fredholm equations is piece-wise continuous and separable. Due to the exploitation of these properties, the GSSMC approach shows great promise for being effectively applicable for the transient analysis of large DSPNs with concurrent deterministic transitions. Moreover, for DSPNs without concurrent deterministic transitions the proposed GSSMC approach requires three orders of magnitude less computational effort than the previously known approach based on the method of supplementary variables.

Modeling discrete event systems with state-dependent deterministic service times

In this paper extensions to the numerical solution method of deterministic and stochastic Petri nets (DSPNs) are introduced in order to cope with deterministic transitions with marking-dependent firing delays. The basic idea lies in scaling each row of the generator matrix of the Markov chain subordinated to a deterministic transition with marking-dependent firing delay by the delay value corresponding to this marking. Computational formulas of this solution method are implemented in the software package DSPNexpress which completely automates the solution process of DSPNs. The extended modeling power of DSPNs is illustrated by a single-server queuing system with Poisson arrivals, degradable deterministic service requirements, and finite capacity.

Exploiting Isomorphisms and Special Structures in the Analysis of Markov Regenerative Stochastic Petri Nets

We introduce a refined algorithm for determining the transition probability matrix of the embedded Markov chain underlying a Markov Regenerative Stochastic Petri Net (MRSPN), which continues previous work on improving the efficiency of the numerical solution method for MRSPNs. By observing that the digraph corresponding to the Markov chains subordinated to timed transitions with non-exponentially distributed firing delays of a MRSPN constitute a forest, we show how isomorphisms between its connected components can be exploited in order to reduce the computation time of their transient analysis. Furthermore, special structures for subordinated Markov chains are introduced for which the computational effort of the transient analysis can also be reduced. The computational benefit of the proposed approach is illustrated by four MRSPNs taken from the literature.

Employing the Randomization Technique for Solving Stochastic Petri Nets Models

In this paper we propose to employ the randomization technique improved by a numerically stable calculation of Poisson probabilities for computing transient solutions of Markov chains underlying stochastic Petri net models. It is shown how to employ this numerical method for calculating the time-dependent quantities required by the solution process of DSPN models. The benefit of the described method is illustrated by stochastic Petri net models for two queueing systems. The evaluation of the transient behavior of the M/M/l/K queue is performed by means of a GSPN model. The steady-state solution of the E10/D/1/K queue is obtained using a DSPN model. The presented results show that the model solutions are calculated with significantly less computational effort and a better error control by the refined randomization method than by an adaptive matrix exponentiation method implemented in the version 1.4 of the software package GreatPN.

Performance Modeling Using DSPNexpress

This paper recalls the numerical solution method for Deterministic and Stochastic Petri Nets (DSPNs) and describes the graphical, interactive analysis tool DSPNexpress which has been developed at the Technische Universitat Berlin since 1991. The software package DSPNexpress allows the employment of complex DSPNs for evaluating the performance and dependability of discrete-event dynamic systems such as computer systems, communication networks, or automated manufacturing systems. A DSPN of a memory consistency model for a multicomputer system with virtually shared memory is presented to illustrate the modeling power of DSPNs.

Modeling Relaxed Memory Consistency Protocols

This paper presents a modeling approach based on deterministic and stochastic Petri nets (DSPNs) for analyzing memory consistency protocols for multiprocessors with Distributed Shared Memory (DSM). DSPNs are a numerically solvable modeling formalism with a graphical representation. The modeling approach addresses in particular the performance degradation due to the amount of message exchange, which is an important issue in software implementations of DSM with shared pages. DSPNs are employed for a comparative performance study of an adaptive memory consistency protocol, an eager-invalidate, and a delayed-invalidate release consistency protocol in a software implementation of DSM.