The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-pro t purposes provided that:• a full bibliographic reference is made to the original source• a link is made to the metadata record in Durham E-Theses• the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. April 1997 AbstractThe work of this thesis is concerned with the following problem and its derivatives.Consider the problem of searching for a target which moves randomly between n sites . The movement is modelled with an n state Markov chain.One of the sites is searched at each time t = 1, 2,... until the target is found.Associated with each search of site i is an overlook probability a; and a cost d.Our aim is to determine the policy that will find the target with the minimal average cost.Notably in the two site case we examine the conjecture that if we let p denote the probability that the target is at site 1, an optimal policy can be defined in terms of a threshold probability P* such that site 1 is searched if and only if p > P*. We show this conjecture to be correct (i) for general Ci ^ C2 when the overlook probabilities Qfj are small and (ii) for general and Ci for a large range of transition laws for the movement. We also derive some properties of the optimal policy for the problem on n sites in the no-overlook case and for the case where each site has the same a, and Ci.We also examine related problems such as ones in which we have the ability to divide available search resources between different regions, and a couple of machine replacement problems. Preface : Motivating ExampleImagine you are a Coast Guard team leader, and you have just received a distress call from a sinking ship. You know that there is a liferaft somewhere on the (finite) ocean you patrol. Based on tidal charts and metrological information, you can build up some model for the possible motion of that liferaft. At your disposal you have certain resources -a boat and 2 helicopters, all of which can be used to search for this missing raft. You realise that even if you look in the right place , there is a chance that you won't see the liferaft, as it is small in comparison to large waves. Your aim is to find the liferaft as quickly as possible. How best should you allocate your resources in order to achieve this aim?The above is a motivational example of the work in this thesis and gives an idea of one of the many applications of search theory We consider only the simplest cases and develop theory which will hopefully be of use in future research in the field. Acknowledgements
We study a model of a polling system, that is, a collection of d 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.
Summary A Galton‐Watson process in varying environments (Zn), with essentially constant offspring means, i.e. E(Zn)/mn→α∈(0, ∞), and exactly two rates of growth is constructed. The underlying sample space Ω can be decomposed into parts A and B such that (Zn)n grows like 2non A and like mnon B (m > 4).
We study the first exit time τ from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on Z d (d ≥ 2) with mean drift that is asymptotically zero. Specifically, if the mean drift at x ∈ Z d is of magnitude O( x −1 ), we show that τ < ∞ a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude x −β , β ∈ (0, 1), we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on 2nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model. Key words and phrases: Asymptotic direction; exit from cones; inhomogeneous random walk; perturbed random walk; random walk in random environment.
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 $s$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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP519 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org
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