Don't Just Go with the Flow: Cautionary Tales of Fluid Flow Approximation

Fluid flow approximation allows efficient analysis of large scale PEPA models. Given a model, this method outputs how the mean, variance, and any other moment of the model's stochastic behaviour evolves as a function of time. We investigate whether the method's results, i.e. moments of the behaviour, are sufficient to capture system's actual dynamics. We ran a series of experiments on a client-server model. For some parametrizations of the model, the model's behaviour can accurately be characterized by the fluid flow approximations of its moments. However, the experiments show that for some other parametrizations, these moments are not sufficient to capture the model's behaviour, highlighting a pitfall of relying only on the results of fluid flow analysis. The results suggest that the sufficiency of the fluid flow method for the analysis of a model depends on the model's concrete parametrization. They also make it clear that the existing criteria for deciding on the sufficiency of the fluid flow method are not robust

A combined process algebraic, agent and fluid flow approach to emergent crowd behaviour

Emergent phenomena occur due to the pattern of non-linear and distributed local interactions between the elements of a system over time. Surprisingly, agent based crowd models in which the movement of each individual follows a limited set of simple rules often re-produce quite closely the emergent behaviour of crowds that can be observed in reality. An example of such phenomena is the spontaneous self-organisation of drinking parties in the squares of cities in Spain, also known as "El Botellon" [22]. We revisit this case study providing an elegant stochastic process algebraic model in Bio-PEPA amenable to several forms of analyses among which simulation and fluid flow analysis. We show that a fluid flow approximation, i.e. a deterministic reading of the average behaviour of the system, can provide an alternative and efficient way to study the same emergent behaviour as that explored in [22] where simulation was used instead. Besides empirical evidence also an analytical justification is provided for the good correspondence found between simulation results and the fluid flow approximation. Scalability features of the fluid flow approach may make it particularly useful when studying models of more complex city topologies with very large populations

A fluid analysis framework for a Markovian process algebra

Markovian process algebras, such as PEPA and stochastic π-calculus, bring a powerful compositional approach to the performance modelling of complex systems. However, the models generated by process algebras, as with other interleaving formalisms, are susceptible to the state space explosion problem. Models with only a modest number of process algebra terms can easily generate so many states that they are all but intractable to traditional solution techniques. Previous work aimed at addressing this problem has presented a fluid-flow approximation allowing the analysis of systems which would otherwise be inaccessible. To achieve this, systems of ordinary differential equations describing the fluid flow of the stochastic process algebra model are generated informally. In this paper, we show formally that for a large class of models, this fluid-flow analysis can be directly derived from the stochastic process algebra model as an approximation to the mean number of component types within the model. The nature of the fluid approximation is derived and characterised by direct comparison with the Chapman–Kolmogorov equations underlying the Markov model. Furthermore, we compare the fluid approximation with the exact solution using stochastic simulation and we are able to demonstrate that it is a very accurate approximation in many cases. For the first time, we also show how to extend these techniques naturally to generate systems of differential equations approximating higher order moments of model component counts. These are important performance characteristics for estimating, for instance, the variance of the component counts. This is very necessary if we are to understand how precise the fluid-flow calculation is, in a given modelling situation

Modelling Non-linear Crowd Dynamics in Bio-PEPA

Emergent phenomena occur due to the pattern of non-linear and distributed local interactions between the elements of a system over time. Surprisingly, agent based crowd models, in which the movement of each individual follows a limited set of simple rules, often re-produce quite closely the emergent behaviour of crowds that can be observed in reality. An example of such phenomena is the spontaneous self-organisation of drinking parties in the squares of cities in Spain, also known as "El Botellon" [20]. We revisit this case study providing an elegant stochastic process algebraic model in Bio-PEPA amenable to several forms of analyses, among which simulation and fluid flow analysis. We show that a fluid flow approximation, i.e. a deterministic reading of the average behaviour of the system, can provide an alternative and efficient way to study the same emergent behaviour as that explored in [20] where simulation was used instead. Besides empirical evidence, also an analytical justification is provided for the good correspondence found between simulation results and the fluid flow approximation

Fluid passage-time calculation in large Markov models

Recent developments in the analysis of large Markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process. So-called fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added. In this work, we show how fluid approximation techniques may be used to extract passage-time measures from performance models. We focus on two types of passage measure: passage-times involving individual components; as well as passage-times which capture the time taken for a population of components to evolve. Specifically, we show that for models of sufficient scale, passage-time distributions can be well approximated by a deterministic fluid-derived passage-time measure. Where models are not of sufficient scale, we are able to generate approximate bounds for the entire cumulative distribution function of these passage-time random variables, using moment-based techniques. Finally, we show that for some passage-time measures involving individual components the cumulative distribution function can be directly approximated by fluid techniques

A new tool for the performance analysis of massively parallel computer systems

We present a new tool, GPA, that can generate key performance measures for very large systems. Based on solving systems of ordinary differential equations (ODEs), this method of performance analysis is far more scalable than stochastic simulation. The GPA tool is the first to produce higher moment analysis from differential equation approximation, which is essential, in many cases, to obtain an accurate performance prediction. We identify so-called switch points as the source of error in the ODE approximation. We investigate the switch point behaviour in several large models and observe that as the scale of the model is increased, in general the ODE performance prediction improves in accuracy. In the case of the variance measure, we are able to justify theoretically that in the limit of model scale, the ODE approximation can be expected to tend to the actual variance of the model

Practical applications of performance modelling of security protocols using PEPA

PhD ThesisTrade-off between security and performance has become an intriguing area in recent years in both the security and performance communities. As the security aspects of security protocol research is fully- edged, this thesis is therefore devoted to conducting a performance study of these protocols. The long term objective is to translate formal de nitions of security protocols to formal performance models automatically, then analysing by relevant techniques. In this thesis, we take a preliminary step by studying five typical security protocols, and exploring the methodology of construction and analysis of their models by using the Markovian process algebra PEPA. Through these case studies, an initial framework of performance analysis of security protocol is established. Firstly, a key distribution centre is investigated. The basic model su ers from the commonly encountered state space explosion problem, and so we apply some efficient solution techniques, which include model reduction techniques and ordinary di fferential equation based fluid flow analysis. Finally, we evaluate a utility function for this secure key exchange model. Then, we explore two non-repudiation protocols. Mean value analysis has been applied here for a class of PEPA models, and it is compared with an ODE approximation. After that, an optimistic nonrepudiation protocol with off-line third trust party is studied. The PEPA model has been formulated using a concept of multi-threaded servers with functional rates. The nal case study is a cross-realm Kerberos protocol. A simplified technique of aggregation with an ODE approximation is performed to do efficient cient analysis. All these modelling and analysis methods are illustrated through numerical examples