This book is an introductionary course in stochastic ordering and dependence in the field of applied probability for readers with some background in mathematics. It is based on lectures and senlinars I have been giving for students at Mathematical Institute of Wroclaw University, and on a graduate course a.t Industrial Engineering Department of Texas A&M University, College Station, and addressed to a reader willing to use for example Lebesgue measure, conditional expectations with respect to sigma fields, martingales, or compensators as a common language in this field. In Chapter 1 a selection of one dimensional orderings is presented together with applications in the theory of queues, some parts of this selection are based on the recent literature (not older than five years). In Chapter 2 the material is centered around the strong stochastic ordering in many dimenยญ sional spaces and functional spaces. Necessary facts about conditioning, Markov processes an"d point processes are introduced together with some classical results such as the product formula and Poissonian departure theorem for Jackson networks, or monotonicity results for some reยญ newal processes, then results on stochastic ordering of networks, rẽ̃ment policies and single server queues connected with Markov renewal processes are given. Chapter 3 is devoted to dependence and relations between dependence and ordering, exemยญ plified by results on queueing networks and point processes among others
CONTENT
1 Univariate Ordering -- 1.1 Construction of iid random variables -- 1.2 Strong ordering -- 1.3 Convex ordering -- 1.4 Conditional orderings -- 1.5 Relative inverse function orderings -- 1.6 Dispersive ordering -- 1.7 Compounding -- 1.8 Integral orderings for queues -- 1.9 Relative inverse orderings for queues -- 1.10 Loss systems -- 2 Multivariate Ordering -- 2.1 Strassenโ{128}{153}s theorem -- 2.2 Coupling constructions -- 2.3 Conditioning -- 2.4 Markov processes -- 2.5 Point processes on R, martingales -- 2.6 Markovian queues and Jackson networks -- 2.7 Poissonian flows and product formula -- 2.8 Stochastic ordering of Markov processes -- 2.9 Stochastic ordering of point processes -- 2.10 Renewal processes -- 2.11 Comparison of replacement policies -- 2.12 Stochastically monotone networks -- 2.13 Queues with MR arrivals -- 3 Dependence -- 3.1 Association -- 3.2 MTP2 -- 3.3 A general theory of positive dependence -- 3.4 Multivariate orderings and dependence -- 3.5 Negative association -- 3.6 Independence via uncorrelatedness -- 3.7 Association for Markov processes -- 3.8 Dependencies in Markovian networks -- 3.9 Dependencies in Markov renewal queues -- 3.10 Associated point processes -- A -- A.1 Probability spaces -- A.2 Distribution functions -- A.3 Examples of distribution functions -- A.4 Other characteristics of probability measures -- A.5 Random variables equal in distribution -- A.6 Bibliography
