Properties of Poisson process 1st of all:memory-less The additional time to wait is independent on when it starts P(X>t+s X>t)=P(X>s) 0 Think about the coin-tossing process,though not Poisson,it is memory-less -X is the number of trials until the first "head" Additional trials to get a"head"is independent of previous trials Weigiang Sun Communication Networks
Weiqiang(Sun Communica/on(Networks Proper/es(of(Poisson(process • Think(about(the(coinWtossing(process,(though(not(Poisson,(it(is( memoryWless( – X(is(the(number(of(trials(un/l(the(first(“head”( – Addi/onal(trials(to(get(a(“head”(is(independent(of(previous(trials( • 1st(of(all:(memoryFless& • The(addi/onal(/me(to(wait(is(independent( on(when(it(starts( P(X>t+s|X>t)(=(P(X>s) 11
Properties of Poisson process(cont.) Merging property -Let A1,A2,...Ak be independent Poisson process of rate入1，入2，，入k,A=∑Ai is also Poisson with rate入=∑入i λ1 入1↓↓↓↓↓ λ2 x2↓↓ 入k 入1+入2出出↓↓↓↓↓↓ Weigiang Sun Communication Networks 12
Weiqiang(Sun Communica/on(Networks Proper/es(of(Poisson(process((cont.) • Merging&property& – Let(A1,(A2,(…,(Ak(be(independent(Poisson(process( of(rate(λ1,(λ2,…, λk,(A=(∑Ai(is(also(Poisson( with(rate(λ=(∑ λi( λ1 λ2 λk ∑ λi λ1 λ2 λ1+λ2 12
Properties of Poisson process(cont.) Selection property -Suppose a random selection is made from a Poisson process (A),each arrival is selected with probability p,independent of the others,the resulting process is a Poisson process with rate p 入1 ↓↓↓↓↓↓ λ2 ∑i 入k p入↓↓↓ ↓↓ Splitting property The above property also leads to random splitting property,why and how? Weigiang Sun Communication Networks 13
Weiqiang(Sun Communica/on(Networks Proper/es(of(Poisson(process((cont.) • SelecHon&property& – Suppose(a(random(selec/on(is(made(from(a(Poisson( process((λ),(each(arrival(is(selected(with(probability( p,(independent(of(the(others,(the(resul/ng(process(is(a( Poisson(process(with(rate(pλ λ1 λ2 λk ∑ λi λ pλ SpliIng&property& The(above(property(also(leads(to(random(splibng( property,(why(and(how?( 13