Mark Fackrell

Characterization of Matrix-Exponential Distributions

The class of matrix-exponential distributions can be equivalently defined as the class of all distributions with rational Laplace–Stieltjes transform. An immediate question that arises is: when does a rational Laplace–Stieltjes transform correspond to a matrix-exponential distribution? For a rational Laplace–Stieltjes transform that has a pole of maximal real part that is real and negative, we give a geometric description of all admissible numerator polynomials that give rise to matrix-exponential distributions. Using this approach we give a complete characterization for all matrix-exponential distributions of order three.

Fitting with Matrix-Exponential Distributions

Abstract It is well known that general phase-type distributions are considerably overparameterized, that is, their representations often require many more parameters than is necessary to define the distributions. In addition, phase-type distributions, even those defined by a small number of parameters, may have representations of high order. These two problems have serious implications when using phase-type distributions to fit data. To address this issue we consider fitting data with the wider class of matrix-exponential distributions. Representations for matrix-exponential distributions do not need to have a simple probabilistic interpretation, and it is this relaxation which ensures that the problems of overparameterization and high order do not present themselves. However, when using matrix-exponential distributions to fit data, a problem arises because it is unknown, in general, when their representations actually correspond to a distribution. In this paper we develop a characterization for matrix-exponential distributions and use it in a method to fit data using maximum likelihood estimation. The fitting algorithm uses convex semi-infinite programming combined with a nonlinear search.

Two Issues Surrounding Parrondo’s Paradox

In the original version of Parrondo’s paradox, two losing sequences of games of chance are combined to form a winning sequence. The games in the first sequence depend on a single parameter p, while those in the second depend on two parameters p 1 and p 2. The paradox is said to occur because there exist choices of p, p 1 and p 2 such that the individual sequences of games are losing but a sequence constructed by choosing randomly between the games at each step is winning.

An EM algorithm for the model fitting of Markovian binary trees

Markovian binary trees form a class of continuous-time branching processes where the lifetime and reproduction epochs of individuals are controlled by an underlying Markov process. An Expectation-Maximization (EM) algorithm is developed to estimate the parameters of the Markov process from the continuous observation of some populations, first with information about which individuals reproduce or die (the distinguishable case), and second without this information (the indistinguishable case). The performance of the EM algorithm is illustrated with some numerical examples. Fits resulting from the distinguishable case are shown not to be significantly better than fits resulting from the indistinguishable case using some goodness of fit measures.

Modelling modal gating of ion channels with hierarchical Markov models

Many ion channels spontaneously switch between different levels of activity. Although this behaviour known as modal gating has been observed for a long time it is currently not well understood. Despite the fact that appropriately representing activity changes is essential for accurately capturing time course data from ion channels, systematic approaches for modelling modal gating are currently not available. In this paper, we develop a modular approach for building such a model in an iterative process. First, stochastic switching between modes and stochastic opening and closing within modes are represented in separate aggregated Markov models. Second, the continuous-time hierarchical Markov model, a new modelling framework proposed here, then enables us to combine these components so that in the integrated model both mode switching as well as the kinetics within modes are appropriately represented. A mathematical analysis reveals that the behaviour of the hierarchical Markov model naturally depends on the properties of its components. We also demonstrate how a hierarchical Markov model can be parametrized using experimental data and show that it provides a better representation than a previous model of the same dataset. Because evidence is increasing that modal gating reflects underlying molecular properties of the channel protein, it is likely that biophysical processes are better captured by our new approach than in earlier models.

A semi-infinite programming approach to identifying matrix-exponential distributions

The Laplace–Stieltjes transform of a matrix-exponential (ME) distribution is a rational function where at least one of its poles of maximal real part is real and negative. The coefficients of the numerator polynomial, however, are more difficult to characterise. It is known that they are contained in a bounded convex set that is the intersection of an uncountably infinite number of linear half-spaces. In order to determine whether a given vector of numerator coefficients is contained in this set (i.e. the vector corresponds to an ME distribution) we present a semi-infinite programming algorithm that minimises a convex distance function over the set. In addition, in the event that the given vector does not correspond to an ME distribution, the algorithm returns a closest vector which does correspond to one.

An alternative characterization for matrix exponential distributions

A necessary condition for a rational Laplace–Stieltjes transform to correspond to a matrix exponential distribution is that the pole of maximal real part is real and negative. Given a rational Laplace–Stieltjes transform with such a pole, we present a method to determine whether or not the numerator polynomial admits a transform that corresponds to a matrix exponential distribution. The method relies on the minimization of a continuous function of one variable over the nonnegative real numbers. Using this approach, we give an alternative characterization for all matrix exponential distributions of order three.

IMPROVED RESULTS ON POISSON PROCESS APPROXIMATION IN JACKSON NETWORKS

Melamed (1979) proved that for an open migration process, a necessary and sufficient condition for the equilibrium flow along a link to be Poissonian is the absence of loops: no customer can travel along the link more than once. Barbour and Brown (1996) quantified the statement by allowing the customers a small probability of travelling along the link more than once and proved Poisson process approximation theorems analogous to Melameds Theorem. Amongst the three bounds presented in Barbour and Brown (1996), the one in terms of the Wasserstein metric is of particular interest since it reveals more insightful information about the closeness between the process of flows and an approximating Poisson process, and it is small when the parameter of the system is small, except a logarithmic factor in terms of time in which the flows are considered. The bound was later improved by Brown, Weinberg and Xia (2000) who showed that the logarithmic factor in terms of time can be lifted at the cost of an extra parameter being introduced into the bound. In this paper, we present a new bound which simplifies and sharpens the bounds in the above-mentioned two papers and compare the performance of these bounds for a simple open migration process.

A sequential stochastic mixed integer programming model for tactical master surgery scheduling

Abstract In this paper, we develop a stochastic mixed integer programming model to optimise the tactical master surgery schedule (MSS) in order to achieve a better patient flow under downstream capacity constraints. We optimise the process over several scheduling periods and we use various sequences of randomly generated patients’ length of stay scenario realisations to model the uncertainty in the process. This model has the particularity that the scenarios are chronologically sequential, not parallel. We use a very simple approach to enhance the non-anticipative feature of the model, and we empirically demonstrate that our approach is useful in achieving the desired objective. We use simulation to show that the most frequently optimal schedule is the best schedule for implementation. Furthermore, we analyse the effect of varying the penalty factor, an input parameter that decides the trade-off between the number of cancellations and occupancy level, on the patient flow process. Finally, we develop a robust MSS to maximise the utilisation level while keeping the number of cancellations within acceptable limits.

Allocation in a Vertical Rotary Car Park

We consider a vertical rotary car park consisting of l levels with c parking spaces per level. Cars arrive at the car park according to a Poisson process, and if there are parking spaces available, they are parked according to some allocation rule. If the car park is full, arrivals are lost. Cars remain in the car park for an exponentially distributed length of time, after which they leave. We develop an allocation algorithm that specifies where to allocate a newly-arrived car that minimises the expected cumulative imbalance of the car park. We do this by modelling the working of the car park as a Markov decision process, and deriving an optimal allocation policy. We simulate the operation of some car parks when the policy decision making protocol is used, and compare the results with those observed when a heuristic allocation algorithm is used.