Lectures6-7

# Lectures6-7 - 1 ECE& CSE ECE-CSE 861 Introduction to...

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Unformatted text preview: 1 ECE & CSE ECE-CSE 861: Introduction to Computer Communication Networks Ness B. Shroff ECE & CSE Lectures 6-7 ECE & CSE Arrival Process or Counting Process Defnition: An arrival process or counting process is that stochastic process {N(t); t ≥ 0} such that: 1) N(0) = 0 2) N(t) is integer valued 3) N(t) is non-decreasing; i.e., if s ≤ t, then N(s) ≤ N(t). 4) For s<t, N(t +s) – N(t) equals the number of events (or arrivals) in (s,t]. (right continuous property) Time N(t) t 1 t 2 t 3 ECE & CSE Poisson Process Defnition I: The arrival process {N(t), t ≥ 0} is said to be a Poisson Process with rate λ >0 , if 1) N(0) = 0 2) P{N(t) = 1 } = λ t + o(t) 3) P{N(t) ≥ 2} = o(t) 4) For any t ≥ 0, s ≥ 0, 1) N(t +s) – N(t) is independent of {N(u): u ≤ t}. 2) This is known as the Independent Increments property. 5) For any t , s ≥ 0, the distribution of N(t+s) – N(t) is independent of t. 1) This is known as the Stationary Increments property. ECE & CSE Poisson Process (Cont’d) Remember: A function f is said to be o(h) if Equivalent De¡nition II: The arrival process {N(t), t ≥ 0 } is said to be Poisson Process with rate λ , if 1) N(0) = 0 2) Independent Increments 3) The number of arrivals in any interval of length s is Poisson distributed with mean λ s, for all s, t ≥ 0. ) ( lim = → h h f h ! ) ( n} N(t) – s) P{N(t n s e n s λ λ − = = + 2 ECE & CSE Independent Increments Property N(t + s) – N(t) is independent of {N(u): u ≤ t}. Arrivals in the future (beyond time t) are independent of the entire past history up to time t ( very strong property )....
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## This note was uploaded on 03/05/2011 for the course ECE 861 taught by Professor Shroff during the Winter '11 term at Ohio State.

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Lectures6-7 - 1 ECE& CSE ECE-CSE 861 Introduction to...

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