Passage time densities and quantiles are important performance and quality of service metrics, but their numerical derivation is, in general, computationally expensive. We present an iterative algorithm for the calculation of passage time densities in semi-Markov models, along with a theoretical analysis and empirical measurement of its convergence behaviour. In order to implement the algorithm efficiently in parallel, we use hypergraph partitioning to minimise communication between processors and to balance workloads. This enables the analysis of models with very large state spaces which could not be held within the memory of a single machine. We produce passage time densities and quantiles for very large semi-Markov models with over 15 million states and validate the results against simulation.

Probability distributions of response times are important in the design and analysis of transaction processing systems and computercommunication systems. We present a general technique for deriving such distributions from high-level modelling formalisms whose state spaces can be mapped onto finite Markov chains. We use a load-balanced, distributed implementation to find the Laplace transform of the first passage time density and its derivatives at arbitrary values of the transform parameter s. Setting s=0 yields moments while the full passage time distribution is obtained using a novel distributed Laplace transform inverter based on the Laguerre method. We validate our method against a variety of simple densities, cycle time densities in certain overtake-free (tree-like) queueing networks and a simulated Petri net model. Our implementation is thereby rigorously validated and has already been applied to substantial Markov chains with over 1 million states. Corresponding theoretical results for semi-Markov chains are also presented.

Generalised Stochastic Petri nets (GSPNs) have been widely used to analyse the performance of hardware and software systems. This paper presents a novel technique for the numerical determination of response time densities in GSPN models. The technique places no structural restrictions on the models that can be analysed, and allows for the high-level specification of multiple source and destination markings, including any combination of tangible and vanishing markings. The technique is implemented using a scalable parallel Laplace transform inverter that employs a modified Laguerre inversion technique. We present numerical results, including a study of the full distribution of end-to-end response time in a GSPN model of the Courier communication protocol software. The numerical results are validated against simulation. 1.

Passage time densities and quantiles are important performance metrics which are increasingly used in specifying service level agreements (SLAs) and benchmarks. PEPA is a popular stochastic process algebra and a powerful formalism for describing performance models of communication and computer systems. We present a case study passage time analysis of an 82,944 state PEPA model using the HYDRA tool. HYDRA specialises in passage time analysis of large Markov systems based on stochastic Petri nets. By using the new Imperial PEPA compiler (ipc), we can construct a HYDRA model from a PEPA model and obtain passage time densities based on the original PEPA description.

Many physical or computer systems can be modelled as Markov chains. A range of solution techniques exist to address the state-space explosion problem, encountered while analysing such Markov models.

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Petri nets are a widely used formalism for the analysis of concurrent systems and as such there are a plethora of existing tools which allow users to edit, animate and analyse a range of Petri net classes. These tools are essentially limited, however, to the functionality incorporated into them when they are written. The main aim of the project was therefore to produce a basic Petri net editor/animator tool which could be arbitrarily extended by the user. This was achieved by designing an architecture which would allow the program to load user-designed modules about which nothing is known until runtime. The project also provided an opportunity to design a tool which offered new features not present in existing pieces of software and which also corrected known flaws in these tools. In particular, the animator designed as part of Medusa incorporates a novel backwards animator which allows the user to step backwards through the sequence of transitions which they have fired. Two modules were produced as part of the project. The first of these was

In this paper, we present parallel multilevel algorithms for the hypergraph partitioning problem. In particular, we describe schemes for parallel coarsening, parallel greedy k-way refinement and parallel multi-phase refinement. Using an asymptotic theoretical performance model, we derive the isoefficiency function for our algorithms and hence show that they are technically scalable when the maximum vertex and hyperedge degrees are small. We conduct experiments on hypergraphs from six different application domains to investigate the empirical scalability of our algorithms both in terms of runtime and partition quality. Our findings confirm that the quality of partition produced by our algorithms is stable as the number of processors is increased while being competitive with those produced by a state-of-the-art serial multilevel partitioning tool. We also validate our theoretical performance model through an isoefficiency study. Finally, we evaluate the impact of introducing parallel multi-phase refinement into our parallel multilevel algorithm in terms of the trade off between improved partition quality and higher runtime cost.

Software library for the Markov analysis of complex queuing models Abstract—This paper presents a new version of OLYMP (Object Oriented LibrarY for modeling Markovian Processes) which delivers objects allowing to build and solve the queuing models of computer systems using the Markow chain method with continuous time and discrete state space. The previous version of this software was available only for the Solaris system. The new one is completely rebuilt and compiled by g++ (freely redistributable C++ compiler) that can be portable for all platforms including super computers which is very important due to the numerical complexity of the Markov analysis. Index Terms—Markov chains, queuing models, state space exploration technique, state vector, transition matrix. M I.

Abstract. Response time quantiles reflect user-perceived quality of service more accurately than mean or average response time measures. Consequently, on-line transaction processing benchmarks, telecommunications Service Level Agreements and emergency services legislation all feature stringent 90th percentile response time targets. This chapter describes a range of techniques for extracting response time densities and quantiles from large-scale Markov and semi-Markov models of real-life systems. We describe a method for the computation of response time densities or cumulative distribution functions which centres on the calculation and subsequent numerical inversion of their Laplace transforms. This can be applied to both Markov and semi-Markov models. We also review the use of uniformization to calculate such measures more efficiently in purely Markovian models. We demonstrate these techniques by using them to generate response time quantiles in a semi-Markov model of a high-availability web-server. We show how these techniques can be used to analyse models with state spaces of O 10 7 states and above. 1. Introduction. A