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Understanding Markov Chains and DCF: Applications in Stochastic Processes

This presentation outlines the fundamentals of stochastic processes, focusing on Markov processes and discrete-time Markov chains (DTMC). We define stochastic processes as families of random variables and explore different types, including discrete and continuous states. The Markov property is discussed, emphasizing that future states depend solely on the current state. Additionally, the presentation addresses the Distributed Coordination Function (DCF) used in CSMA/CA networks, including the concepts of binary exponential backoff and throughput analysis.

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Understanding Markov Chains and DCF: Applications in Stochastic Processes

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  1. Markov Chain of DCF Speaker : 林益宏 Date : 10/26/’05 COMM, CCU E-mail : g92430006@comm.ccu.edu.tw

  2. Outline • Stochastic process • Markov process • Discrete time MC (DTMC) • DCF • Summary

  3. Stochastic process • Define : A stochastic process is a family of random variables X(t) • X() : state space • t : time index X: {X(t), tT} is called a stochastic process

  4. Types of stochastic process • Discrete state, discrete time • e.g : 第t天收到的mail數 • Discrete state, continuous time • e.g : (0,t)時間內瀏覽網頁的次數 • Continuous state, discrete time • e.g : 第t天使用MSN的時間 • Continuous state, continuous time • e.g : (o,t)時間內伺服器忙碌的時間

  5. ? Markov Process • Markov Process • future evolution of stochastic process depends only on current state • Markov Chain • A discrete stateMarkov Process forms a Markov Chain (MC) if the probability of the next state depends only on current state t

  6. Discrete Time MC (DTMC) • discrete state, discrete time random process • possible set of countable states • All past history summarized in current state • Transitions between states take place only at discrete time

  7. Example • 天氣預測 • 假設昨天的天氣只跟今天有關… State=(sunny, cloudy, rainy) 0.9 sunny 0.95 0.01 0.5 0.09 0.01 cloudy rainy 0.1 0.04 0.4

  8. m-step Transition Probability Chapman-Kolmogorov equation m–step transition probability

  9. 0.3 0 1 0.6 0.4 Steady State Probability • 系統穩定性(stationary) • 無論初始值是什麼,最後系統都能趨於穩定 0.7

  10. Example 0.3 0 1 0.7 0.6 0.4

  11. DCF( Distributed Coordination Function) • CSMA/CA - Carrier Sense Multiple Access with Collision Avoidance • Sense before transmission • If idle transmit • Else backoff

  12. Binary Exponential Backoff • Backoff_Counter= INT (CW * Rnd( )) * slot time • INT (x) : maximal int ≤ x • CW : integer between CWmin and CWmax • Rnd( ) : real number between 0 and 1

  13. Binary Exponential Backoff 1023 Contention Window Size CWmax 511 255 127 63 31 31 CWmin t

  14. Backoff Contention Window • Backoff time random chosen from (0,W-1) • After fail transmission w is doubled, up to 2mW • W is CWmin+1 • 2mW is CWmax+1 CW

  15. Markov chain model 1 1 1 0,0 0,1 0,2 0,W0-1 1-p p 1 1 1 1,0 1,1 1,2 1,W1-1 1-p 1 1 1 i,0 i,1 i,2 i,Wi-1 1-p p 1 1 1 m,0 m,1 m,2 m,Wm-1 1-p

  16. Throughput Analysis 某一個station想傳送的機率 至少有一個station傳送的機率 傳送成功的機率 Payload平均長度 Throughput Idle Success collision

  17. F & Q

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