PMNPRO Stochastic processes

Faculty of Economics and Administration
Spring 2009
Extent and Intensity
2/1. 4 credit(s). Type of Completion: zk (examination).
Teacher(s)
RNDr. Dalibor Moravanský, CSc. (lecturer)
Guaranteed by
prof. Ing. Osvald Vašíček, CSc.
Department of Economics – Faculty of Economics and Administration
Contact Person: Lydie Pravdová
Timetable
Wed 18:00–19:35 S311
  • Timetable of Seminar Groups:
PMNPRO/1: Thu 12:50–14:30 VT105, D. Moravanský
Prerequisites
Mathematics II,Statistics I,Statistics II
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Stochastic processes (PMNPRO). Students will be acquainted with foundations of theory of stochastic processes, the application use of which interferes with many branches of research and practice. They shall also adopt the tools of the analysis of the recurrent events and of classifications of their types. Further, the students will be given basic knowledge on Markov chains, both discrete and continuous. The exposition will be oriented on the classification of states of a Markov chain, on the solution of the Chapman-Kolmogorov equations providing an efficient method for computing n-step transition probabilities, on steady-state equations etc. The detailed attention will be devoted mainly to Poisson and Yule processes, to general process of the birth-and-death and to other important topics. Acquired theoretical knowledge will be consequently applied in relevant application areas such as the theory of queues and theory of renovation and in models of management of inventories. The emphasis will be placed on the basic schemes of queuing theory, both from the point of view of their formulations as well as of the means of their solution and implementation. The explanation will be concentrated on the models with one and the more exponential service channels (servers), and on several (both parallel and serial) systems with exponential, Erlang, and general distributions of the access time, and of the time of the dispatching of the costumer at the server. In the framework of the inventory theory, the attention will be focused on the three basic topics: the renovation of the ageing equipment, the renewal of the failing elements and the reliability models.
Syllabus
  • Fundamentals of the random processes theory. Discrete Markov processes: Basic notions: transition probability matrix, initial probabilities Definition of the Markov chain and relating notions, probability of transition, finite and infinite Markov chains. Examples - spatially homogenous Markov chains. Basic characteristics of states of Markov chains, passage of state probability, or probability of transition to another state, stationary transition probabilities. Probability of first occurrence of state and first transition to another state. First passage times, Average waiting time (to passage), Average time of first return Chapman-Kolmogorov equations for discrete Markov chains Basic classification of states of Markov chain: persistent, recurrent, null, no-null and absorbing states, transient states Mutual accessibility of states. Reducible and irreducible chains. Family of mutually accessible states = communicating states, closed class. Periodicity of the states of Markov chain, periodic/aperiodic states. Classification of states persistent/transient based on behavior of infinity series. Theorems on admissibility of types of states in finite chain. Reducible and irreducible chains. Theorems on admissibility of types of states in irreducible chains. Regular Markovovy chains, steady-state probabilities. Expected average cost per unit time, Absorption states, Absorbing probabilities, Random walk, Probabilities of transition to absorbing states Notion of moment generating function of the series , Laplace transforms. Moment generating function and renewal equation, Abel lemma for power series. Continuous Markov processes. General properties of random processes with continuous time, intensity of passage, transition intensity matrix. Moment generating functions for continuous Markov processes Birth-and-death processes, genetic processes. Chapman-Kolmogorov equations for continuous Markov processes. Probability distribution of continuous Markov process initial distribution and probabilities and intensities of passage, vector of absolute probabilities. Chapman-Kolmogorov equations for transition probabilities. Kolmogorov differential equations for continuous Markov processes. Kolmogorov retrospective and prospective system of equations. Poisson process, Yule process, Linear birth and death process. Fundamentals of the queing theory. Fundamentals of the inventory managing.
Literature
  • JABLONSKÝ, Josef. Operační výzkum :kvantitativní modely pro ekonomické rozhodování. 1. vyd. Praha: Professional publishing, 2002, 323 s. ISBN 80-86419-23-1. info
  • KOŘENÁŘ, Václav. Stochastické procesy. Vyd. 1. Praha: Vysoká škola ekonomická v Praze, 2002, 227 s. ISBN 8024503115. info
  • JABLONSKÝ, J. Operační výzkum. Praha: VŠE Praha, 1999. ISBN 80-7079-597-2. info
  • HILLIER, Frederick S. and Gerald J. LIEBERMAN. Introduction to operations research. 6th ed. New York: McGraw-Hill, 1995, xix, 998. ISBN 0071139893. info
Assessment methods
The final exam consists of the written test lasting 90 minutes. Achieving of 55% success rate (usually at least 11 points of maximum 20 ones) is considered as sufficient. As a rule, the test comprises 7 tasks; 4 of them are of a computable character, the remaining 3 tasks have a reasoning nature. It is then followed by a short oral part of the exam (lasting about 15 minutes).
Language of instruction
Czech
Further Comments
Study Materials
The course is taught annually.
The course is also listed under the following terms Spring 2008.
  • Enrolment Statistics (recent)
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