PEPA Models Research Articles

Performance Modelling of the Impact of Cyber Attacks on a Web-based Sales System

In this paper we present two performance models of a web-based sales system, one without the presence of an attack and the other with the presence of a denial of service attack. Models are formulated using the PEPA formalism. The PEPA eclipse plug-in is used to support the creation of the PEPA models for the web-based sales system and the automatic calculation of the performance measures identified to evaluate the models. The evaluation of the models illustrates how the performance of the warehouse's sale is negatively affected by denial of service attack through preventing some or all customers' orders from being fulfilled. The resultant delay on selling perishable products would result on products being discarded.

A Case Study in Capacity Planning for PEPA Models with the PEPA Eclipse Plug-in

We report on the addition of Capacity Planning facilities to the PEPA Eclipse Plug-in, a software tool for analysing performance models written in the PEPA language. The PEPA language allows the compositional description of complex systems consisting of different kinds of processes. The capacity planning addition allows modellers to automatically search for the populations of processes that allows for an optimal trade-off between the performance of the system and the cost of acquiring or operating the components of the system under the modeller's control.

Component aggregation for PEPA models: An approach based on approximate strong equivalence

Approximate aggregation for PEPA components involves the construction of a smaller component that approximates the behaviour of the original one. Such an approximation at the component level can be very efficient and it can also result in a considerable reduction of the state-space for the underlying continuous-time Markov chain. We propose an approximate PEPA component aggregation strategy that relies on an approximate form of strong equivalence. The notion of strong equivalence captures behavioural similarity between components of different size. This quality renders approximate strong equivalence appropriate as a criterion to aggregate the state-space of PEPA components. We compare our newly proposed approach with previous work on component aggregation, where only a part of the component behaviour has been used as a criterion for aggregation. Our method requires fewer assumptions regarding the form of the components, and is therefore readily applicable to a larger family of PEPA models.

Tackling continuous state-space explosion in a Markovian process algebra

Fluid or mean-field methods are approximate analytical techniques which have proven effective in tackling the infamous state-space explosion problem which typically arises when modelling large-scale concurrent systems based on interleaving semantics. These methods are particularly suitable in situations which present large populations of simple interacting objects characterised by small local state spaces, since they require the analysis of a problem which is insensitive to the population sizes but is dependent only on the size of the local state spaces. This paper studies the case when the replicated objects are best described as composites which consist of smaller simple objects. A congenial formal modelling framework for situations of this kind may be given by stochastic process algebra. Using PEPA as a representative case, we find that fluid models with replicated copies of composite processes do not scale well with increasing population sizes, thus rendering intractable the analysis of the underlying system of ordinary differential equations (ODEs). We call this problem continuous state-space explosion, by analogy with its counterpart phenomenon in discrete state spaces. The main contribution of this paper is a result of equivalence that simplifies, in an exact way, the potentially massive ODE system arising in those circumstances to one whose size is independent from all the multiplicities in the model. As a byproduct, we find that these simplified ODEs turn out to characterise the fluid behaviour of a family of PEPA models whose elements cannot be related to each other through any known equivalence relation. A substantial numerical assessment investigates the relationship between the different underlying Markov chains and their unique fluid limit, demonstrating its generally good accuracy for all practical purposes.

Mean value analysis for a class of PEPA models

In this paper, a class of closed queueing networks is modelled in the Markovian process algebra PEPA and solved using the classical mean value analysis. This approach is attractive as it negates the need to derive the entire state space, and so certain metrics from large models can be obtained with little computational effort. The class of models considered includes models that are not obviously classical closed queueing models. This paper extends previous results by deriving exact solutions for models with multiple copies of queue stations. Due to the cost of deriving an exact solution, a simplifying approximation is derived, which is shown to be highly accurate when the population is large. The approach is illustrated with three examples.

Continuous approximation of PEPA models and Petri nets

Modelling very large systems that consist of many similar components can lead to a state space explosion. A continuous approximation of the system can be used to avoid this problem. In the stochastic process algebra, PEPA, models with large numbers of identical components can be approximated in a continuous fashion by a set of coupled ordinary differential equations (ODEs). Similarly, timed continuous Petri nets can be used to approximate behaviour via ODEs where there are many servers. These two approaches are compared and infinite and finite server semantics are considered.

Relating continuous and discrete PEPA models of signalling pathways

PEPA and its semantics have recently been extended to model biological systems. In order to cope with massive quantities of processes (as is usually the case when considering biological reactions) the model is interpreted in terms of a small set of coupled ordinary differential equations (ODEs) instead of a large state space continuous time Markov chain (CTMC). So far the relationship between these two semantics of PEPA had not been established. This is the goal of the present paper. After introducing a new extension of PEPA, denoted PEPA+Π, that allows models to capture both mass action law and bounded capacity law cooperations, the relationship between these two semantics is demonstrated. The result relies on Kurtz’s Theorem that expresses that a set of ODEs can be, in some sense, considered as the limit of pure jump Markov processes.

PEPA Queues: Capturing Customer Behaviour in Queueing Networks

Queueing network formalisms are very good at describing the spatial movement of customers, but typically poor at describing how customers change as they move through the network. We present the PEPA Queues formalism, which uses the popular stochastic process algebra PEPA to represent the individual state and behaviour of customers and servers. We offer a formal semantics for PEPA Queues, plus a direct translation to PEPA, allowing access to the existing tools for analysing PEPA models. Finally, we use the ipc/DNAmaca tool-chain to provide passage-time analysis of a dual Web server example.

Approximate solution of PEPA models using component substitution

Performance models specified using compositional algebra suffer the well-known state space explosion problem, where a relatively small definition leads to a Markov chain with a large state space that is problematic to solve. As a result it is widely recognised that the development of techniques to solve performance models efficiently is of particular practical importance. Recently the notion of behavioural independence was introduced to exploit the structure of Markovian process algebra models in order to solve models in a compositional manner. The opposite property, namely control, is now used to solve models by substituting components in the model with simpler versions. The approach is validated through two examples and by deriving a variety of performance measures.

Product form solution for a class of PEPA models

The advantages of the compositional structure within the Markovian process algebra PEPA for model construction and simplification have already been demonstrated. In this paper we show that for some PEPA models this structure may also be used to advantage during the solution of the model. Several papers offering product form solutions of stochastic Petri nets have been published during the last 10 years. In [R. Boucherie, A characterisation of independence for competing Markov chains with applications to stochastic Petri nets, IEEE Trans. Software Engrg. 20 (7) (1994) 536–544], Boucherie showed that these solutions were a special case of a simple exclusion mechanism for the product process of a collection of Markov chains. The results presented in this paper take advantage of his observation. In particular we show that PEPA models that generate such processes may be readily identified and show how the product form solution may be obtained. Although developed here in the context of PEPA the results presented can be easily generalised to any of the other Markovian process algebra languages.