Vladimir Kalashnikov

Geometric Sums: Bounds for Rare Events with Applications

Preface. Glossary of Notation. 1. Introduction. 2. Miscellaneous Probability Topics. 3. Generalized Renyi Theorem. 4. Two-Sided Bounds. 5. Metric Bounds. 6. Ruin Probability. 7. Reliability Regenerative Models. References. Index.

Submicron-sized actuators based on enhanced shape memory composite material fabricated by FIB-CVD

An enhanced scheme for a functional bilayered composite material with shape memory effect has been successfully applied on the microscale to fabricate a thermally controlled microactuator. Fabrication of cantilever-type microactuators from melt spun ribbon of TiNiCu shape memory alloy included electro-chemical polishing followed by focused ion beam milling and ion-assisted chemical vapor deposition of Pt elastic layer. The smallest working microactuator had a volume of 0:9 m 3 . The structure and thermal stability of the Pt layer have been investigated. The fabricated actuator has been proposed for use as micromechanical nanotweezers for manipulation of submicron- and nano-sized objects. Manipulation of a carbon nanotube bunch has been demonstrated. (Some figures may appear in colour only in the online journal)

Spatial No-Waiting Stations with Moving Customers

We analyze spatial MAP/G/∞-, spatial MAP/G/c/01 and spatial Cox/G/∞-stations with group arrivals over some Polish space X (with Borel σ-algebra X), including the aspect of customer motion in space. For models with MAP-input, characteristic differential equations are set up that describe the dynamics of phase dependent random functions Qr;ij(u,t;S′), where Qr;ij(u,t;S′) is the probability to observe, at time u≤t, the number r of those customers in some source set S′∈X, who will be in a destination set S∈X at time t. For Cox/G/∞-stations, i.e., infinite server stations with doubly stochastic input, the arrival intensities as well as service times may depend on some general stochastic process (J′t)t≥0 with countable state space. For that case we obtain explicit expressions for space–time distributions as well as stationary and non-stationary characteristics.

Two-Sided Bounds

This chapter contains various two-sided bounds of the d.f. W q obtained by further evolving of the method accepted in Chapter 3. We obtain also lower estimates of W q by the so-called test function method. The bounds are derived for several particular cases (where the summands satisfy the Cramer condition, have heavy tails, etc.) and they are stated in the form ready for numerical calculations, revealing tail behaviour of W q (x) and giving explicit estimates of the rate of convergence to the limiting law. Various examples illustrate specific features of the bounds and their accuracy.

Miscellaneous Probability Topics

The present chapter provides a quick reference to the probabilistic facts that are often used throughout the book. For more detailed treatment of these topics readers may consult the commentaries to this chapter and the works cited therein. The most statements below are given for granted and only a few of them that cannot be found easily in the literature are equipped with proofs. The topics reviewed are not related to each other directly. Because of this, readers may skip this chapter and return to it when necessary. For the purposes of this book, it is quite sufficient to assume that the r.v.’s under consideration take their values in a complete separable metric space. We assume this throughout the book without additional commentaries. Moreover, typically we restrict ourselves with real or vector-valued r.v.’s.

Reliability Regenerative Models

In this chapter we are concerned with problems of application of the bounds derived in the preceding chapters to the stochastic models described as regenerative processes. Such processes play a noticeable role in the theory of random processes and have many applications in biology, queueing, reliability, Markov chains, risk theory, simulation, etc. Typically, we study rare events taking reliability regenerative models as an example where such events can be viewed as failures. But all the constructions remain valid for any regenerative model and can be used in storage, queueing, etc. The problem lays in the fact that both probability q and d.f. F are not defined a priori but should be calculated from existing input data. With this in mind, the contents of the chapter are designed to solve this problem.

Generalized Rényi Theorem

In the present chapter we evolve a constructive method that provides limiting results in the form of a generalized Renyi theorem. This allows us to consider the case where the d.f. F of summands in the underlying geometric sum may vary together with parameter q of the corresponding geometric distribution. Although the limiting results are qualitative, they can easily be stated in the form of quantitative bounds. This is partly done in this chapter but generally this problem will be solved in the following chapter. Applications to the heavy traffic regime in queueing and to rare excursions of general Markov chains are considered; they provide practice and help to illuminate the concepts and methods. Applications to insurance and reliability will be discussed in Chapters 6 and 7 correspondingly.