Lecture9

# Lecture9 - Continuous Time Markov Chains ECE-CSE 861...

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1 ECE & CSE ECE-CSE 861: Introduction to Computer Communication Networks Ness B. Shroff ECE & CSE Lecture 9 ECE & CSE Continuous Time Markov Chains X(t) = Number of packets in the system at time t Is X(t) a counting process? No X(t) is a continuous time stochastic process Definition: A stochastic process {X(t),t R} is said to be a continuous- time Markov chain (CTMC) if P{X(t+s}=j| X(s)=i, X(u)=k; 0 u<s} = P{X(t+s)=j|X(s)=i} In other words, a continuous-time Markov chain is a stochastic process having the Markovian property that the conditional distribution of the future state at time t+s, given the present state at s and all past state depends only on the present state, and is independent of the past. A CTMC is conditionally independent of the past (very different from a Poisson Process)! ECE & CSE Continuous Time Markov Chains If in addition P{X(t+s)=j| X(s)=i} is independent of s, then the CTMC is said to be a stationary or time- homogeneous Markov chain.

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