6.041 / 6.431 15. Poisson Process - II

LECTURE 15 Poisson process — II • Readings: Finish Section 6.2. • Review of Poisson process • Merging and splitting • Examples • Random incidence Review • Defining characteristics – Time homogeneity: P (k, τ) – Independence – Small interval probabilities (small δ): P (k, δ) ≈ ⎧⎪⎪⎨⎪⎪⎩ 1 − λδ, if k =0, λδ, if k =1, 0, if k> 1. • Nτ is a Poisson r.v., with parameter λτ : P (k, τ)= (λτ )ke−λτ k! , k =0, 1,... E[Nτ ]=var(Nτ )= λτ • Interarrival times (k = 1): exponential: fT1(t)= λe−λt , t ≥ 0, E[T1]=1/λ • Time Yk to kth arrival: Erlang(k): fYk(y)= λkyk−1e−λy (k − 1)! , y ≥ 0 Poisson fishing • Assume: Poisson, λ =0.6/hour. – Fish for two hours. – if no catch, continue until first catch. a) P(fish for more than two hours)= b) P(fish for more than two and less than five hours)= c) P(catch at least two fish)= d) E[number of fish]= e) E[future fishing time | fished for four hours]= f) E[total fishing time]= Merging Poisson Processes (again) • Merging of independent Poisson processes is Poisson All flashes (Poisson) Red bulb flashes (Poisson) �1 �2 Green bulb flashes (Poisson) – What is the probability that the next arrival comes from the first process? Light bulb example • Each light bulb has independent, exponential(λ) lifetime • Install three light bulbs. Find expected time until last light bulb dies out. Splitting of Poisson processes • Suppose email traffic through server is a Poisson process and destinations are independent. MIT Server Email Traffic leaving MIT Foreign USA � p (1 -p) � � • Each output stream is Poisson. (Why?) • Example: Y = X1 + ···+ XN N,X1,X2,... independent N: geometric(p); Xi: exponential(λ) – Find the distribution of Y Random incidence for Poisson • Poisson process that has been running forever • Show up at some “random time” (really means “arbitrary time”) x x Time Chosen time instant x xx • What is the distribution of the length of the chosen interarrival interval? Random incidence in “renewal processes” • Series of successive arrivals – i.i.d. interarrival times(but not necessarily exponential)• Example: Bus interarrival times are equally likely to be 5 or 10 minutes • If you arrive at a “random time”: – what is the probability that you selected a 5 minute interarrival interval? – what is the expected timeto next arrival?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.