Iain M. MacPhee

A Markov chain model of a polling system with parameter regeneration

We study a model of a polling system i.e.\ a collection of

Nonparametric predictive inference for system reliability with redundancy allocation

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Nonparametric predictive reliability of series of voting systems

queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.

Nonparametric predictive system reliability with redundancy allocation following component testing

This paper presents lower and upper probabilities for the reliability of k-out-of-m systems, which include series and parallel systems, and of series systems with independent k i -out-of-m i subsystems, for which optimal redundancy allocation is also presented in case of zero-failure testing. First, attention is restricted to k-out-of-m systems with exchangeable components. The lower and upper probabilities for successful functioning of the system are based on the nonparametric predictive inferential (NPI) approach for Bernoulli data. In this approach, it is assumed that test data are available on the components, and that the future components to be used in the system are exchangeable with these. Thereafter, systems are considered that consist of a series of independent subsystems, with subsystem i a k i -out-of-m i system consisting of exchangeable components. For such systems, an algorithm for optimal redundancy allocation after zero-failure testing is presented. A particularly attractive feature of NPI in reliability, with lower and upper probabilities, is that data containing zero failures can be dealt with in an attractive manner.

Periodicity in the transient regime of exhaustive polling systems

Nonparametric Predictive Inference (NPI) for system reliability reflects the dependence of reliabilities of similar components due to limited knowledge from testing. NPI has recently been presented for reliability of a single voting system consisting of multiple types of components. The components are all assumed to play the same role within the system, but with regard to their reliability components of different types are assumed to be independent. The information from tests is available per type of component. This paper presents NPI for systems with subsystems in a series structure, where all subsystems are voting systems and components of the same type can be in different subsystems. As NPI uses only few modelling assumptions, system reliability is quantified by lower and upper probabilities, reflecting the limited information in the test data. The results are illustrated by examples, which also illustrate important aspects of redundancy and diversity for system reliability.

Polling systems with parameter regeneration, the general case

In a recent paper, Coolen-Schrijner, Coolen, and MacPhee [

Moments of exit times from wedges for non-homogeneous random walks with asymptotically zero drifts

We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability.

Passage-time moments and hybrid zones for the exclusion-voter model

We consider a polling model with multiple stations, each with Poisson arrivals and a queue of infinite capacity. The service regime is exhaustive and there is Jacksonian feedback of served customers. What is new here is that when the server comes to a station it chooses the service rate and the feedback parameters at random; these remain valid during the whole stay of the server at that station. We give criteria for recurrence, transience and existence of the

Nonparametric predictive inference for reliability of a k-out-of-m:G system with multiple component types

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Dynamics of the Non-Homogeneous Supermarket Model

th moment of the return time to the empty state for this model. This paper generalizes the model, when only two stations accept arriving jobs, which was considered in [Ann. Appl. Probab. 17 (2007) 1447--1473]. Our results are stated in terms of Lyapunov exponents for random matrices. From the recurrence criteria it can be seen that the polling model with parameter regeneration can exhibit the unusual phenomenon of null recurrence over a thick region of parameter space.