The Stochastic Systems Laboratory within the Maxwell Institute for Mathematical Sciences focuses on the probabilistic, statistical and economic modelling of complex systems. The aim of the Laboratory is to facilitate both research and knowledge transfer. Important areas of application include energy, transport and communications networks.
Such systems contain complex interacting components, whose evolution is typically governed by the actions of many competing agents whose behaviour cannot be deterministically predicted. Mathematical modelling is therefore required in order to understand, control and optimise their behaviour. The laboratory aims to bring together the mathematical and statistics expertise of highly skilled academics, with extensive experience in working with industry.
Projects within the Laboratory are funded both by UK and EU research grants, and by those industrial organisations with whom it works.
Areas of Application
Electrical power grids are complex networked systems. Demand and supply must be balanced on a minute-by-minute basis; there are limited opportunities for large-scale storage, and injections and withdrawals from storage must also be managed on a minute-by-minute basis.
Further, flows in networks are subject to the laws of physics (Kirchoff's voltage law), so that there is very little control over, for example, the routing of flows; generating capacity cannot in general be instantly switched on or off; sources of generation capacity, whether renewable or nuclear, are often located far from the urban and industrial areas they must serve. In today's market the provision of generation capacity, both in its construction and in its short term availability, is typically determined by market forces in which many competing operators each seek to optimize their own returns. Mathematical and statistical modelling is focused on:
- the balancing of supply and demand in a stochastic environment through the use of storage and demand-side response;
- the understanding of market mechanisms that contribute to the optimal overall performance of the systems to benefit of consumers;
- the timely forecasting and prediction of available renewable energy, such as that from the wind, and of demand.
Modern telephonic and computer communications networks, such as the Internet, carry constant streams of high-volume traffic between huge numbers of sources and destinations. This traffic is often very heterogeneous in its characteristics, for example, in the ability of different connection types to tolerate small delays (e.g. file transfers versus streaming in the Internet), and in the natural priorities which should attach to different types of call or connection. Further it is typically the case that neither the individual components of a network, nor the individual users of that network, have any knowledge of network loads other than their own. Finally, while a network may be engineered so that its individual components function cooperatively, users are in general in competition with each other for the network resources. Research issues include:
- stochastic modelling of network loads and behaviour;
- network admission control in order to optimise performance;
- resource allocation and prioritisation between competing users; congestion control;
Many of the characteristics of road and rail networks are analogous to those of computer communications networks. The major distinction is the scarcity of capacity, and the difficulties of increasing this.
Thus issues of pricing and of congestion control come to the fore. A further issue is that fare structures on public transport systems, so as to maximise the use of resources to the benefit of users.
Other complex and stochastic systems in which many users compete for scarce resources include in particular those of the health service, notably hospitals, and also complex industrial production systems.
Members of the Laboratory include those in the Probability and Statistics Group and also other academics from the School of Mathematical and Computer Sciences and the Maxwell Institute, together with economists and engineers from both Heriot-Watt and Edinburgh Universities.
The Laboratory also has a steady stream of visitors from other academic institutions both in the UK and overseas.