6.262.Lec3

# 6.262.Lec3 - DISCRETE STOCHASTIC PROCESSES Lecture 3 Where...

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Lecture 3 - 2/10/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 3 Where we are headed and why Overview: Counting Processes Renewal Processes Poisson Processes Poisson processes Memoryless property of exponential Stationary increments and independent increments in Poisson process Bernoulli Process Memoryless property of geometric distribution Stationary increments and independent increments in Bernoulli process Poisson Process as a Limiting Case of the Bernoulli Process

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Lecture 3 - 2/10/2010 Discrete Stochastic Processes 2 Where we are headed – and why. Approximate Outline and Schedule of Topics Date Lecture Topics Reading 2/3 Lect. 1 Introduction, course overview, probability review Chap. 1 2/5 Lect. 2 Further probability review; Markov, Chebyshev and Chernoff bounds Chap. 1 2/10 Lect. 3 Bernoulli and Poisson processes Chap. 2 2/12 Lect. 4 Poisson Process, symmetry and order statistics Chap. 2 2/17 Lect. 5 Poisson Process, introduction to queues Chap. 2 2/19 Lect. 6 Convergence of random variables, weak & strong laws Chap. 1 2/24 Lect. 7 Finite Markov Chains Chap. 4 2/26 Lect. 8 Renewal processes – strong law, expected number of renewals Chap. 3 Why? Chapters 2 and 4 are probably easier and more familiar than Chapters 1 and 3. We don’t need the weak and strong laws until Chapter 3. A vivid application of renewal processes is to show some interesting properties of Markov chains.
Lecture 3 - 2/10/2010 Discrete Stochastic Processes 3 C HAPTER 2 OF G ALLAGER —P OISSON P ROCESSES C OUNTING P ROCESS Queueing systems and many other systems operate on "customer" (or electron, or photon, or tumor cell, or telephone call, or data packet) arrivals, so here we study arrivals in isolation. A sample point ω ∈Ω must specify the set of times at which arrivals occur. The usual approach to this is to specify, for each t > 0, and for each sample point , the number of arrivals N ( t , w ) up to (and including) time t . For any given t > 0, N ( t , w ) is a function from the sample space to the reals, and thus is a rv , abbreviated as N ( t ). A counting process Nt () ; t 0 {} is a family of non-negative integer valued rv 's, one for each t 0 , where N 00 af = with probability 1 and with the monotonicity property that NN t τ for all t .

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Lecture 3 - 2/10/2010 Discrete Stochastic Processes 4 N OTATION FOR COUNTING PROCESSES : N(t) t SS S X X 1 1 2 2 3 3 X S n = epoch of n th arrival; X n interarrival time from n -1 st arrival to n th arrival. S n = X i i = 1 n ; X n = S n S n 1 The process is characterized by Nt ( ) ; t 0 { } or by X n ; n 1 { } or by S n ; n 1 { } . For example, St N tn n ≤= kp a f lq
Lecture 3 - 2/10/2010 Discrete Stochastic Processes 5 Renewal Processes A renewal process is a special case of a counting process in which the set of interarrival times X n ; n 1 {} is a set of independent, identically distributed (IID) rv 's.

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## This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec3 - DISCRETE STOCHASTIC PROCESSES Lecture 3 Where...

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