Spatial modelling formalisms

In the past many of the modelling formalisms mentioned in Sections 2.2 and 2.4.7 have also been considered for spatial modelling scenarios. Spatial Stochastic automata [e.g. 70] models for instance, which are commonly analysed using discrete event simulation techniques, were among the �rst modelling tools used to study spatial processes. However, as the example in [14] shows, Petri nets are also an excellent choice for representing graph location models, especially when o�ering di�erent views on the system by having a high-level Petri net where places refer to actual physical locations. An early idea for a spatial version of the PEPA process algebra was its extension to PEPANets [88], which is essentially a Petri net formalism where individual tokens are PEPA processes. Later, a much simpler spatial extension to PEPA [83] featured a location pre�x to add a notion of space to the description of processes.

The above extensions all provide basic capabilities to represent spatially heterogeneous features of systems in models, however, none of the papers truly considers the appropriateness of certain represent-ations with regards to the level of realism that could be derived in the subsequent model analysis. One of the �rst papers in the formal modelling literature that touches on this point is the one by Ciocchetta et al. [59] where the authors present a spatial extension to the BioPEPA language, akin to the spatial PEPA extension proposed in [83]. While the extension itself is straightforward, the authors illustrate that the main challenge is to provide a modelling formalism that strikes a balance between model analysis cost and realism with regards to the size of the volume that individuals interact in, the size of the population inside the volume and the nature and the radius of the perception function that governs

intra- and inter-population interactions. Another example inspired by applications from the �eld of systems biology is the work by Versari et al. [202], who use an adaptation of stochastic -calculus to study biological systems with varying volume/location sizes, although such extensions appear to limit the analysis to stochastic simulation.

One of the �rst population formalisms for large spatial communication networks was presented by Gribaudo in [93]. Markovian Agent Models (MAMs) describe the evolution of single agents as partially de�ned CTMCs, where some transitions only occur due to interaction with other agents in the form of a broadcast message exchange. Given the de�nition of a Markovian Agent, a MAM further requires a notion of space and the de�nition of a perception function. In the model, the perception function is interpreted as a broadcast message exchange between agents. Any agent can send a broadcast message while sojourning in a particular state or as part of making a state transition. Other agents in the vicinity of the broadcasting agent receive the message with some probability provided they are listening for it. In their paper on the PALOMA stochastic process algebra for discrete-space MAMs, Feng et al. [75] show that the communication dynamics can be captured by a PSMP. Our own attempt to formalise MAMs as PCTMCs using the Markovian Agent spatial stochastic algebra (MASSPA) [2] forced us to change the communication dynamics from broadcast to unicast. While it is generally possible to represent discrete-space MAMs with �xed population sizes as PCTMC models, the broadcast nature causes the number of classes in the reaction class set C to grow with the number of agents n in the system as every single message can theoretically reach up n 1 agents, depending on the range of the perception function. As a result, for static WSNs with moderate neighbourhood sizes of 4 8 wireless sensor nodes, any numerical higher-order moment approximation would be highly inaccurate, not to mention extremely costly due to the emerging highly non-linear reaction rates (see Section 5.5). While a coarse grid could decrease the number of populations drastically and thereby reduce the ODE-analysis cost, such a representation of space does not allow for a precise representation of the perception function.

The discussion shows that the stochastic e�ects of broadcast communication in population models are rather hard to analyse, as their potential for causing multiple, simultaneous agent-state changes leads to complex dependencies.

3.5. Conclusions

In this chapter we reviewed the application of high-level spatial stochastic population models in the literature. Our discussion emphasises that there is no single best way to capture the notion of space.

Instead, the choice of spatial representation is largely governed by the challenge of �nding an appropriate level of abstraction to maintain the right balance between the accuracy of the perception function and the e�ciency of the analysis. Often this comes down to the question as to whether the modeller tries to describe the evolution of the system in terms of microscopic behaviour, e.g. continuous-space representations with highly accurate perception functions or if a macroscopic view of spatial interaction is possible, e.g. aggressive discretisation of space and perception function. Whenever the relationship between space and perception function is obscured, stochastic simulation or experimentation should be used as a �rst tool to investigate the impact of spatial abstraction on the accuracy of the population model. Furthermore, it is advisable to consider the following key points prior to choosing a mathematical representation for a process:

1. Nature of agent interaction and perception function 2. Area to be modelled and spatial features to be kept 3. Distribution of agents in space

4. Level of realism required

5. Available computational resources

Despite the di�culty of deciding on the right level of spatial abstraction, the variety of multi-disciplinary examples of high-level, spatial population model applications highlights the potential of this modelling paradigm. Yet, for large models, the analysis methods of choice have largely been numerical population-mean approximations and population-mean-�eld analysis, which underlines the need to develop models that also facilitate numerical second-order population-moment analysis. Given the apparent limitations of purely numerical solution techniques, it seems that some form of hybrid analysis is more likely to live up to this challenge if the dependency on simulation techniques is seeked to be reduced. Aside from choosing an appropriate analysis method, our case studies in Chapters 6 and 7 illustrate that hybrid-space representations such as in [14, 207], where the authors use discrete locations and continuous paths connecting them, also help to address this challenge.

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