Markov Chain of DCF Speaker : 林益宏 Date : 10/26/’05 COMM, CCU

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

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

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

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

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

Example 天氣預測 假設昨天的天氣只跟今天有關 … State=(sunny, cloudy, rainy) sunny cloud y rain y

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

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

Example

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

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

Binary Exponential Backoff t Contention Window Size CW max CW min

Backoff Contention Window Backoff time random chosen from (0,W-1) After fail transmission w is doubled, up to 2 m W W is CW min +1 2 m W is CW max +1 CW

Markov chain model 0,0 0,20,W 0 -10, ,01,11,21,W 1 -1 i,0i,1i,2 111 i,W i -1 m,0m,1m,2m,W m p 1-p p

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 Throughput Payload 平均長度 IdleSuccesscollision

F & Q