6.262.Lec9

# 6.262.Lec9 - DISCRETE STOCHASTIC PROCESSES Lecture 9...

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Lecture 9 - 3/3/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 9 Renewal Processes - Chapter 3 Strong Law for Renewal Processes Time Averages vs. Ensemble Averages Central Limit Theorem for Renewal Processes Pictorial Introduction to Levy Distributions & Fat Tails Renewal - Reward Processes: Long-Term Time Avarages of: Residual Life of Interarrival Interval Age of Interarrival Interval Duration of Interarrival Interval

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Lecture 9 - 3/3/2010 Discrete Stochastic Processes 2 R ENEWAL P ROCESSES N(t) t SS S X X 1 1 2 2 3 3 X R ENEWAL P ROCESS : A counting process Nt t a f lq ; 0 with the property that the positive ( PX i == 00 af ) interrenewal times X 1 , X 2 ,... are IID . (Ross allows P (X i = 0) < 1.) The arrival epochs S 1 , S 2 are sums of IID rv's SX in n i = = 1 . Crucial relationship: for every integer 1 n , all t > 0, Nt () n { } = S n t { } . The homogeneous Poisson process is a (very special) renewal process.
Lecture 9 - 3/3/2010 Discrete Stochastic Processes 3 Deeper reasons to study renewal processes 1) Many complex stochastic processes have "renewal points" where process "starts over" and is independent of past. Renewal theory studies long term behavior - averaging over successive renewal periods. Short term behavior, within renewal period can be studied separately. 2) Many of the "paradoxes" of probability are related to renewal issues. Treating these carefully once avoids trouble later. E XAMPLES OF S UCH R ENEWAL P ROCESSES : 1) G / G / m queue: Arrivals form a renewal process (but not the renewal process of interest here). Whenever an arrival comes into an empty system (all servers idle), this forms a renewal for the entire queueing system.

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Lecture 9 - 3/3/2010 Discrete Stochastic Processes 4 2) Markov chain starting in state s 0 . Whenever chain enters state s 0 , a renewal occurs. 3) Random access LAN: A bunch of nodes each receive packets to send according to independent renewal processes (not the process of interest here). If multiple nodes try to transmit at the same time, a "collision" occurs, and they each retransmit at a later time according to some protocol. A renewal for the system occurs when a node receives a packet and all other nodes are idle. Be careful about which renewal processes you are talking about and what constitutes a renewal. (There may be a number of renewal processes going on simultaneously and there may be events that appear to be renewals but are not.)
Lecture 9 - 3/3/2010 Discrete Stochastic Processes 5 Simplest Example A component is replaced whenever it fails. The lifetime of the original component is 1 X and the lifetime of the (i-1) st replacement part is i X . The component lifetimes are iid.

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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.Lec9 - DISCRETE STOCHASTIC PROCESSES Lecture 9...

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