6.262.Lec5

6.262.Lec5 - DISCRETE STOCHASTIC PROCESSES Lecture 5...

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6.262 Lect. 5 2/17/2010 1 DISCRETE STOCHASTIC PROCESSES Lecture 5 Review: Bernoulli & Poisson Processes – Similarities and Differences Stationary and Independent Increments PMF’s and Densities Splitting and Merging Review: Non-Homogeneous Poisson Processes Time-Dependent Splitting Produces Non-Homogeneous Poisson Processes Example: M/G/ Queue A bit more detail on order statistics Convergence of Sequences of Random Variables Deterministic convergence of a sequence of numbers. Pointwise deterministic convergence of a sequence of functions. Sequences of random variables: three examples Two new, quite strong types of convergence Sure convergence of a sequence of random variables. Almost sure convergence and the strong law of large numbers
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6.262 Lect. 5 2/17/2010 2 Bernoulli Process (p) Poisson Process ( ) (Discrete-time) stationary & (Continuous-time) stationary & independent increments independent increments Increment Distribution (X) Geometric Exponential Number of Arrivals N(m) or N(t) to Date Binomial Poisson Time S m = X 1 + --- + X m of m-th Arrival Pascal Erlang Density λ (() ) ( 1 ) , 0 km k m PNm k p p m k k ⎛⎞ == ⎜⎟ ⎝⎠ () (( ) ) , 0 , 0 ! k t t PNt k e k t k = =≥ 1 ( 1 ) , 1 1 mk m m k PS k p p k m m 1 , 1 0 (1 ) ! m mm t S te f tm t m −− = ≥≥ 1 ( 1 ) , 1 n PX n p p n 1, t 0 t PX t e =−
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6.262 Lect. 5 2/17/2010 3 Merging Two Processes If N 1 (n) is a Bernoulli process with probability p 1 and N 2 (n) is an independent Bernoulli process with probability p 2 Then the process N 3 (n) = max {N 1 (n), N 2 (n)}, n > 0, is a Bernoulli process with probability p 1 + p 2 -p 1 p 2. If N 1 (t) is a Poisson process with rate λ 1 and N 2 (t) is an independent Poisson process with rate λ 2, Then N 3 (t) = N 1 (t) + N 2 (t) is a Poisson process with rate λ 1 + λ 2.
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6.262 Lect. 5 2/17/2010 4 Splitting a Process Let N 1 (n) be a Bernoulli process with probability p 1 and assume that each arrival of N 1 (n) is assigned to a process N 2 (n) with probability p and each arrival of N 1 (n) is assigned to a process N 3 (n) with probability (1-p), and all assignments are independent. Then N 2 (n) and N 3 (n) are Bernoulli processes with probabilities p 1 p and p 1 (1-p), respectively. Let N 1 (t) be a Poisson process with rate λ 1 and assume that each arrival of N 1 (t) is assigned to a process N 2 (t) with probability p and each arrival of N 1 (t) is assigned to a process N 3 (t) with probability (1-p), and all assignments are independent. Then N 2 (t) and N 3 (t) are Poisson processes with rates λ 1 p and λ 1 (1-p), respectively. Furthermore, N 2 (t) and N 3 (t) are independent random processes.
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6.262 Lect. 5 2/17/2010 5 1 () 0 , le t () tN t λ > t Non-Homogeneous (i.e., Time-Varying) Poisson Processes: To create an inhomogeneous (i.e., time-varying) Poisson process with a time- dependent rate be a homogeneous Poisson process with 1, = and let Nt be the inhomogeneous Poisson process with rate ( ) , t defined by () ( ) ( ) 1 0 .
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This note was uploaded on 10/21/2010 for the course EE 5581 taught by Professor Moon,j during the Spring '08 term at Minnesota.

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6.262.Lec5 - DISCRETE STOCHASTIC PROCESSES Lecture 5...

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