In this set of notes we study a notion of a random process assoc- ted with a point process. The presented theory was inSpired by q- ueing problems. However it seems to be of interest in other branches of applied probability, as for example reliability or dam theory. Using developed tools, we work out known, aswell as new results from queueing or dam theory. Particularly queues which cannot be treated by standard techniques serve as illustrations of the theory. In Chapter 1 the preliminaries are given. We acquaint the reader with the main ideas of these notes, introduce some useful notations, concepts and abbreviations. He also recall basic facts from ergodic theory, an important mathematical tool employed in these notes. Finally some basic notions from queues are reviewed. Chapter 2 deals with discrete time theory. It serves two purposes. The first one is to let the reader get acquainted with the main lines of the theory needed in continuous time without being bothered by techยญ nical details. However the discrete time theory also seems to be of interest itself. There are examples which have no countẽ in continuous time. Chapter 3 deals with continuous time theory. It also contains many basic results from queueing or dam theory. Three applications of the continuous time theory are given in Chapter 4. We show how to use the theory in order to get some useful bounds for the stationary distribution of a random process
CONTENT
1 Preliminaries -- 1 Introduction -- 2 Some notations and conventions -- 3 Ergodic theory โ{128}{148} discrete parameter -- 4 Ergodic theory โ{128}{148} continuous parameter -- 5 Stationary random elements -- 6 Loynesโ{128}{153} lemma -- 7 Queues -- Notes -- 2 Discrete time r.p.@ p.p. -- 1 Basic concepts -- 2 Construction of stationary processes -- 3 First and second type relations -- 4 Applications in queueing theory -- Notes -- 3 Continuous time r.p.@ m.p.p. -- 1 Random process associated with marked point process -- 2 Construction of stationary r.p.@ m.p.p.โ{128}{153}s -- 3 Ergodic theorems -- 4 Relations of the first and second type -- 5 A rate conservative principle approach -- Notes -- 4 Miscellaneous examples -- 1 Inequalities and identities -- 2 Kopoci?skaโ{128}{153}s model -- 3 Equivalence of distributions of embedded chains in the queue size process -- Notes -- 5 Application to single server queues -- 1 Introductory remarks -- 2 Single server queue with periodic input -- 3 Fagging queueing systems -- 4 $$\overrightarrow G /\overrightarrow G /1$$ queue with work-conserving normal discipline -- 5 Takรกcs relation in G/G/1; FIFO. queues -- 6 Takรกcs relation in GI/GI/1; FIFO queues -- Notes -- References
