Neil O'Connell

Large deviations and overflow probabilities for the general single-server queue, with applications

We consider queueing systems where the workload process is assumed to have an associated large deviation principle with arbitrary scaling: there exist increasing scaling functions (at,vt,teE+) and a rate function / such that if (Wt,teR+) denotes the workload process, then limi;^ 1 \ogP(Wt/at > w) = — I(w)

Entropy of ATM traffic streams: a tool for estimating QoS parameters

For the purposes of estimating quality-of-service parameters, it is enough to know the large deviation rate-function of an ATM traffic stream; modeling procedures can be bypassed if we can estimate the rate-function directly, exploiting the analogy between the rate-function and thermodynamic entropy. We show that this proposal is soundly based on statistical sampling theory. Experiments on the Fairisle ATM network at the University of Cambridge have established that it is feasible to collect the required data in real time. >

Brownian analogues of Burke's theorem

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burkes theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.

Littelmann paths and brownian paths

We study some path transformations related to Littelmann path model and their applications to representation theory and Brownian motion in a Weyl chamber.

Tropical Combinatorics and Whittaker functions

We establish a fundamental connection between the geometric Robinson–Schensted–Knuth (RSK) correspondence and GL(N,R)-Whittaker functions, analogous to the well-known relationship between the RSK correspondence and Schur functions. This gives rise to a natural family of measures associated with GL(N,R)-Whittaker functions which are the analogues in this setting of the Schur measures on integer partitions. The corresponding analogue of the Cauchy–Littlewood identity can be seen as a generalization of an integral identity for GL(N,R)-Whittaker functions due to Bump and Stade. As an application, we obtain an explicit integral formula for the Laplace transform of the law of the partition function associated with a 1-dimensional directed polymer model with log-gamma weights recently introduced by one of the authors.

Conditioned random walks and the RSK correspondence

We consider the stochastic evolution of three variants of the RSK algorithm, giving both analytic descriptions and probabilistic interpretations. Symmetric functions play a key role, and the probabilistic interpretations are obtained by elementary Doob–Hunt theory. In each case, the evolution of the shape of the tableau obtained via the RSK algorithm can be interpreted as a conditioned random walk. This is intuitively appealing, and can be used for example to obtain certain relationships between orthogonal polynomial ensembles. In a certain scaling limit, there is a continous version of the RSK algorithm which inherits much of the structure exhibited in the discrete settings. Intertwining relationships between conditioned and unconditioned random walks are also given. In the continuous limit, these are related to the Harish-Chandra/Itzyksen–Zuber integral.

A path-transformation for random walks and the Robinson-Schensted correspondence

The author and Marc Yor recently introduced a path-transformation G((k)) with the property that, for X belonging to a certain class of random walks on Z(+)(k), the transformed walk G((k))( X) has the same law as the original walk conditioned never to exit the Weyl chamber {x : x(1) less than or equal to...less than or equal to x(k)}. In this paper, we show that G((k)) is closely related to the Robinson-Schensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of X and G((k))( X). The corresponding results for the Brownian model are recovered by Donskers theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the path-transformation G((k)) and the Robinson-Schensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the Robinson-Schensted algorithm and, moreover, extends easily to a continuous setting.

A large deviation principle with queueing applications

In this paper, we present a large deviation principle for partial sums processes indexed by the half line, which is particularly suited to queueing applications. The large deviation principle is established in a topology that is finer than the topology of uniform convergence on compacts and in which the queueing map is continuous. Consequently, a large deviation principle for steady-state queue lengths can be obtained immediately via the contraction principle.

Weighted occupation time for branching particle systems and a representation for the supercritical superprocess

We obtain a representation for the supercritical Dawson-Watanabe superprocess in terms of a subcritical superprocess with immigration, where the immigration at a given time is governed by the state of an underlying branching particle system. The proof requires a new result on the laws of weighted occupation times for branching particle systems. American Mathematical Society 1980 subject classifications: Primary 60G57, 60J80. Secondary 60J25.

Complexity analysis of a decentralised graph colouring algorithm

Colouring a graph with its chromatic number of colours is known to be NP-hard. Identifying an algorithm in which decisions are made locally with no information about the graphs global structure is particularly challenging. In this article we analyse the complexity of a decentralised colouring algorithm that has recently been proposed for channel selection in wireless computer networks.