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6.262.Lec4

# 6.262.Lec4 - DISCRETE STOCHASTIC PROCESSES Lecture 4...

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Lecture 4 - 2/12/2010 Discrete Stochastic Processes 1 DISCRETE STOCHASTIC PROCESSES Lecture 4 – Poisson Processes Review: Bernoulli Process Discrete-Time Memoryless Property & Stationary and Independent Increments Poisson Processes: First Definition (Independent Exponential Interarrivals) Poisson Process as a Limit of the Bernoulli Process Continuous-Time Memoryless Property Continuous-Time Stationary and Independent Increments Second Definition of Poisson Process (Stationary Independent Increments and Poisson PMF) Third Definition of Poisson Process (Baby Bernoulli View of Poisson process.) Merging Poisson Processes Splitting Poisson Processes Time-Varying (i.e., Inhomgeneous) Poisson Processes Introduction to Order Statistics and Conditional Arrival Epochs

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Lecture 4 - 2/12/2010 Discrete Stochastic Processes 2 Bernoulli Process – A Simple Discrete-Time Renewal Process The Poisson process can be derived by a limiting argument from the Bernoulli process. More importantly, much intuition about the Poisson process comes from the Bernoulli process. Definition: (Gallager, pp 15-16) A Bernoulli Process is a sequence Y 1 , Y 2 ,--- of iid binary random variables with P(Y k = 1) = p, P(Y k = 0) =1–p. New Notation: We say an arrival occurs at each time m that Y m =1, and we assume there is no arrival at time zero. We consider the times T k, of the k-th arrival and the interarrival times , X k = T k –T k-1, , k 1, between the (k-1)st and kth arrivals. And we consider the partial sums S n , n = 1, 2, --- given by which give the number of arrivals to date at time n (i.e., the number of arrivals at times n.) 1 n nk k SY = = 1 k ≥≥
Lecture 4 - 2/12/2010 Discrete Stochastic Processes 3 1 The interarrival times X have a geometric distribution: ( ) (1 ) , m 1. The number S of arrivals up though time n has a binomial distribution: P(S ) ) , 0 k n. The time T o k m k n kn k n k PX m p p n kp p k == ⎛⎞ ⎜⎟ ⎝⎠ f the k-th arrival has a Pascal distribution: 1 P(T ) ) , n k 1. 1 k k n np p k

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Lecture 4 - 2/12/2010 Discrete Stochastic Processes 4 Discrete-Time Memoryless Property of the Geometric Distribution The Geometric interarrival time is special (very special) because of the Memoryless property : For each n>0, m>0, ( >n+m| >n) = ( >m) PX X Since the number of tosses until the next "heads" in a sequence of independent coin tosses with probability p of giving heads is geometrically distributed, if you have already waited for 100 tosses, the entire distribution of the remaining time until the next head has the same distribution as when you began observing 100 tosses ago! ( > n+m| > n) = ( > n+m)/P(X > n) X so memoryless iff ( > n+m) = ( > n)P(X > m) The geometric distribution is memoryless, since (to be more rigorous) 1 1 1 0 () 1 1 ( 1 ) 1( 1 ) 1 ( 1 ) , s o P(X > n+m) = p ( ) . r k k r kr r k nm PX r p p pp p p p pP X n P X m = = + >= ≤= = −− = = => > =
Lecture 4 - 2/12/2010 Discrete Stochastic Processes 5 Theorem 1’—The Next Event in a Bernoulli Process At each time n the wait until the next arrival is geometric with probability p and is independent of () for all k .

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6.262.Lec4 - DISCRETE STOCHASTIC PROCESSES Lecture 4...

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