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Lectures6-9

# Lectures6-9 - Arrival Process or Counting Process ECE-CSE...

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1 ECE & CSE ECE-CSE 861: Introduction to Computer Communication Networks Ness B. Shroff ECE & CSE Lectures 6-9 ECE & CSE Arrival Process or Counting Process Definition: 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 Definition 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 Definition 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. 0 ) ( lim 0 = h h f h ! ) ( n} N(t) s) P{N(t n s e n s λ λ = = +

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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 ). t ECE & CSE Stationary Increments Property Strong Property (but less so than Ind. Inc): For any t, s 0 the distribution of N(t+s) – N(t) is independent of t.
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Lectures6-9 - Arrival Process or Counting Process ECE-CSE...

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