9 research outputs found
Proportional lumpability and proportional bisimilarity
3noIn this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.openopenMarin A.; Piazza C.; Rossi S.Marin, A.; Piazza, C.; Rossi, S
On approximating the stochastic behaviour of Markovian process algebra models
Markov chains offer a rigorous mathematical framework to describe systems that exhibit
stochastic behaviour, as they are supported by a plethora of methodologies to
analyse their properties. Stochastic process algebras are high-level formalisms, where
systems are represented as collections of interacting components. This compositional
approach to modelling allows us to describe complex Markov chains using a compact
high-level specification.
There is an increasing need to investigate the properties of complex systems, not
only in the field of computer science, but also in computational biology. To explore
the stochastic properties of large Markov chains is a demanding task in terms of computational
resources. Approximating the stochastic properties can be an effective way
to deal with the complexity of large models. In this thesis, we investigate methodologies
to approximate the stochastic behaviour of Markovian process algebra models.
The discussion revolves around two main topics: approximate state-space aggregation
and stochastic simulation. Although these topics are different in nature, they are both
motivated by the need to efficiently handle complex systems.
Approximate Markov chain aggregation constitutes the formulation of a smaller
Markov chain that approximates the behaviour of the original model. The principal
hypothesis is that states that can be characterised as equivalent can be adequately represented
as a single state. We discuss different notions of approximate state equivalence,
and how each of these can be used as a criterion to partition the state-space
accordingly. Nevertheless, approximate aggregation methods typically require an explicit
representation of the transition matrix, a fact that renders them impractical for
large models. We propose a compositional approach to aggregation, as a means to
efficiently approximate complex Markov models that are defined in a process algebra
specification, PEPA in particular.
Regarding our contributions to Markov chain simulation, we propose an accelerated
method that can be characterised as almost exact, in the sense that it can be
arbitrarily precise. We discuss how it is possible to sample from the trajectory space
rather than the transition space. This approach requires fewer random samples than a
typical simulation algorithm. Most importantly, our approach does not rely on particular
assumptions with respect to the model properties, in contrast to otherwise more
efficient approaches
Stochastic abstraction of programs: towards performance-driven development
Distributed computer systems are becoming increasingly prevalent, thanks to modern
technology, and this leads to significant challenges for the software developers of these
systems. In particular, in order to provide a certain service level agreement with users,
the performance characteristics of the system are critical. However, developers today
typically consider performance only in the later stages of development, when it may be
too late to make major changes to the design. In this thesis, we propose a performance driven
approach to development — based around tool support that allows developers
to use performance modelling techniques, while still working at the level of program
code.
There are two central themes to the thesis. The first is to automatically relate performance
models to program code. We define the Simple Imperative Remote Invocation
Language (SIRIL), and provide a probabilistic semantics that interprets a program
as a Markov chain. To make such an interpretation both computable and efficient,
we develop an abstract interpretation of the semantics, from which we can derive a
Performance Evaluation Process Algebra (PEPA) model of the system. This is based
around abstracting the domain of variables to truncated multivariate normal measures.
The second theme of the thesis is to analyse large performance models by means
of compositional abstraction. We use two abstraction techniques based on aggregation
of states — abstract Markov chains, and stochastic bounds — and apply both of
them compositionally to PEPA models. This allows us to model check properties in
the three-valued Continuous Stochastic Logic (CSL), on abstracted models. We have
implemented an extension to the Eclipse plug-in for PEPA, which provides a graphical
interface for specifying which states in the model to aggregate, and for performing the
model checking
Scalable Performance Analysis of Massively Parallel Stochastic Systems
The accurate performance analysis of large-scale computer and communication systems is directly
inhibited by an exponential growth in the state-space of the underlying Markovian performance
model. This is particularly true when considering massively-parallel architectures
such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative
performance measures such as passage-time distributions from performance models of
these systems is critical for providers of these services. Indeed, without such an ability, they
remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have
any confidence of honouring. Additionally, this must be possible in a short enough period of
time to allow many different parameter combinations in a complex system to be tested. If we
can achieve this rapid performance analysis goal, it will enable service providers and engineers
to determine the cost-optimal behaviour which satisfies the SLAs.
In this thesis, we develop a scalable performance analysis framework for the grouped PEPA
stochastic process algebra. Our approach is based on the approximation of key model quantities
such as means and variances by tractable systems of ordinary differential equations (ODEs).
Crucially, the size of these systems of ODEs is independent of the number of interacting entities
within the model, making these analysis techniques extremely scalable. The reliability of our
approach is directly supported by convergence results and, in some cases, explicit error bounds.
We focus on extracting passage-time measures from performance models since these are very
commonly the language in which a service level agreement is phrased. We design scalable analysis
techniques which can handle passages defined both in terms of entire component populations
as well as individual or tagged members of a large population.
A precise and straightforward specification of a passage-time service level agreement is as important
to the performance engineering process as its evaluation. This is especially true of
large and complex models of industrial-scale systems. To address this, we introduce the unified
stochastic probe framework. Unified stochastic probes are used to generate a model augmentation
which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis,
we deploy these probes to define many detailed and derived performance measures that can
be automatically and directly analysed using rapid ODE techniques. In this way, we tackle
applicable problems at many levels of the performance engineering process: from specification
and model representation to efficient and scalable analysis
Studying the effects of adding spatiality to a process algebra model
We use NetLogo to create simulations of two models of disease transmission originally expressed in WSCCS. This allows us to introduce spatiality into the models and explore the consequences of having different contact structures among the agents. In previous work, mean field equations were derived from the WSCCS models, giving a description of the aggregate behaviour of the overall population of agents. These results turned out to differ from results obtained by another team using cellular automata models, which differ from process algebra by being inherently spatial. By using NetLogo we are able to explore whether spatiality, and resulting differences in the contact structures in the two kinds of models, are the reason for this different results. Our tentative conclusions, based at this point on informal observations of simulation results, are that space does indeed make a big difference. If space is ignored and individuals are allowed to mix randomly, then the simulations yield results that closely match the mean field equations, and consequently also match the associated global transmission terms (explained below). At the opposite extreme, if individuals can only contact their immediate neighbours, the simulation results are very different from the mean field equations (and also do not match the global transmission terms). These results are not surprising, and are consistent with other cellular automata-based approaches. We found that it was easy and convenient to implement and simulate the WSCCS models within NetLogo, and we recommend this approach to anyone wishing to explore the effects of introducing spatiality into a process algebra model
Scalable analysis of stochastic process algebra models
The performance modelling of large-scale systems using discrete-state approaches is
fundamentally hampered by the well-known problem of state-space explosion, which
causes exponential growth of the reachable state space as a function of the number
of the components which constitute the model. Because they are mapped onto
continuous-time Markov chains (CTMCs), models described in the stochastic process
algebra PEPA are no exception. This thesis presents a deterministic continuous-state
semantics of PEPA which employs ordinary differential equations (ODEs) as the underlying
mathematics for the performance evaluation. This is suitable for models consisting
of large numbers of replicated components, as the ODE problem size is insensitive
to the actual population levels of the system under study. Furthermore, the ODE is
given an interpretation as the fluid limit of a properly defined CTMC model when the
initial population levels go to infinity. This framework allows the use of existing results
which give error bounds to assess the quality of the differential approximation. The
computation of performance indices such as throughput, utilisation, and average response
time are interpreted deterministically as functions of the ODE solution and are
related to corresponding reward structures in the Markovian setting.
The differential interpretation of PEPA provides a framework that is conceptually
analogous to established approximation methods in queueing networks based on meanvalue
analysis, as both approaches aim at reducing the computational cost of the analysis
by providing estimates for the expected values of the performance metrics of interest.
The relationship between these two techniques is examined in more detail in
a comparison between PEPA and the Layered Queueing Network (LQN) model. General
patterns of translation of LQN elements into corresponding PEPA components are
applied to a substantial case study of a distributed computer system. This model is
analysed using stochastic simulation to gauge the soundness of the translation. Furthermore,
it is subjected to a series of numerical tests to compare execution runtimes
and accuracy of the PEPA differential analysis against the LQN mean-value approximation
method.
Finally, this thesis discusses the major elements concerning the development of a
software toolkit, the PEPA Eclipse Plug-in, which offers a comprehensive modelling environment
for PEPA, including modules for static analysis, explicit state-space exploration,
numerical solution of the steady-state equilibrium of the Markov chain, stochastic
simulation, the differential analysis approach herein presented, and a graphical
framework for model editing and visualisation of performance evaluation results
Computing bounds for the performance indices of quasi-lumpable Stochastic Well-Formed Nets
Structural symmetries in stochastic well-formed colored Petri nets (SWN's) lead to behavioral symmetries that can be exploited by using the symbolic reachability graph (SRG) construction algorithm. The SRC allows one to compute an aggregated reachability graph (RG) and a \u201clumped\u201d continuous time Markov chain (CTMC) that contain all the information needed to study the qualitative properties and the performance of the modeled system, respectively. Some models exhibit qualitative behavioral symmetries that are not completely reflected at the CTMC level. We call them quasi-lumpable SWN models. In these cases, exact performance indices can be obtained by avoiding the aggregation of those markings that are qualitatively, but not quantitatively, equivalent. An alternative approach consists of aggregating all the qualitatively equivalent states and computing approximated performance indices. In this paper, a technique is proposed to compute bounds on the performance of SWN models of this kind, using the results we have presented elsewhere. The technique is based on the Courtois and Semal bounded aggregation metho