"Probability theory is a mathematical framework for the study of randomness," he says. "The simplest example of why you would want to use probability theory is tossing a coin. It is actually chaotic so it is difficult to predict whether it will come up heads or tails."
Devising a physical model for this chaotic system would be impossibly complicated but doing so using probability theory is "quite simple", he says. "That carries over into many application areas where the underlying dynamics of the thing are very complicated."
He mentions predicting traffic flows, weather, communications networks and queues as examples where probability theory can be used to model performance. All tend towards randomness and are a challenge to model and predict.
"A probability model can be very effective in describing what is happening in these systems," says O'Connell.
Modelling the to and fro of communications traffic over the internet is a prime example of what probability theory can do, he says. The internet has become enormously complex and traffic moves in a random way between network nodes.
Maths can be used to describe both the traffic, which is inherently unpredictable, and also how the network will perform, despite the complexity of its structure.
"It is very complicated but in many situations a probability model can be very good at describing what is happening," O'Connell states.
His particular research focus is at the point of contact between randomness and symmetry, and he cites as an example the mathematics needed to describe how queues will form.
This is no small consideration for companies dependent on customer goodwill. Banks and supermarkets must strike a balance between the high cost of more ATMs or cashiers and annoyed customers who feel they have queued too long.
Variables here are the random arrival and accumulation of customers at, say, an ATM and how quickly the ATM responds to the customer. It is the same whether it is a supermarket queue or when looking at random traffic over a communications network.
Service providers want long queues to be a rare event, although the opposite often seems to apply. O'Connell uses mathematical tools such as queuing theory and large deviation theory to help him model queue behaviour.
"A lot of my research has been in that area, understanding large deviation theory, the study of rare events," he explains. "There is a whole theory of queuing networks. There are surprising mathematical results in queuing theory and a lot of them have to do with symmetry."
He can use large deviation or queuing theory to study the chaotic side of queuing, but for the symmetry he needs algebra. Here he depends on concepts such as representation theory which allow him to connect the chaotic and the organised.
"The connection between queuing theory and representation theory is quite a fruitful link," he says. "It gives you a pathway between the two areas of maths."
Big Queues is published by Springer and is written for mathematicians. It describes how large deviation theory can be applied to queuing problems
© The Irish Times
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