6.041 / 6.431 14. Poisson Process - I

PMF of Number of Arrivals N xx x x x xx xx xx0 Timettt1 2 3 � • Finely discretize [0,t]: approximately Bernoulli • Nt (of discrete approximation): binomial • Taking δ →0(orn →∞) gives: (λτ)ke−λτ P (k, τ)= , k =0, 1,... k! • E[Nt]= λt, var(Nt)=λt LECTURE 14 The Poisson process • Readings: Start Section 6.2. Lecture outline • Review of Bernoulli process • Definition of Poisson process • Distribution of number of arrivals • Distribution of interarrival times • Other properties of the Poisson process Bernoulli review • Discrete time; success probability p • Number of arrivals in n time slots:binomial pmf • Interarrival times: geometric pmf • Time to k arrivals: Pascal pmf • Memorylessness Definition of the Poisson process xx x x x xx xx xx0 Timettt1 2 3 � • Time homogeneity: P(k, τ) = Prob. of k arrivals in interval of duration τ • Numbers of arrivals in disjoint time intervals are independent • Small interval probabilities: For VERY small δ: ⎧1 −λδ, if k =0; P (k, δ) ⎪⎪≈ ⎨λδ, if k =1; ⎪ ⎪0⎩, if k>1 . – λ: “arrival rate” Interarrival Times • Yk time of kth arrival • Erlang distribution: λk1 yk−e−λyfY(y)= , y k( 1)! ≥ 0 Example • You get email according to a Poisson process at a rate of λ = 5 messages per hour. You check your email every thirty minutes. • Prob(no new messages) = • Prob(one new message) = k − • Time of first arrival (k = 1): exponential: fY(y)= λe−λy, y ≥ 0 1– Memoryless property: The time to the next arrival is independent of the past Graph of interarrival times for r = 1, 2, 3.Image by MIT OpenCourseWare. Bernoulli/Poisson Relation �������� n = t /� x x x np =�t0 Time p =��Arrivals POISSON BERNOULLI Times of Arrival Continuous Discrete Arrival Rate λ/unit time p/per trial PMF of # of Arrivals Poisson Binomial Interarrival Time Distr. Exponential Geometric Time to k-th arrival Erlang Pascal Adding Poisson Processes • Sum of independent Poisson random variables is Poisson • Sum of independent Poisson processes is Poisson Red bulb flashes (Poisson) All flashes�1 (Poisson) �2 Green bulb flashes (Poisson) – What is the probability that the next arrival comes from the first process? MIT OpenCourseWare http://ocw.mit.edu 6.041 /6.431 Probabilistic Systems Analysis and Applied Probability Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.