Simulating events: the Poisson process

Transcription

1 Simulating events: te Poisson process p. 1/15 Simulating events: te Poisson process Micel Bierlaire Transport and Mobility Laboratory

2 Simulating events: te Poisson process p. 2/15 Siméon Denis Poisson Siméon-Denis Poisson ( ). Frenc matematician. Poisson random variable Poisson process Non omogeneous Poisson process

3 Simulating events: te Poisson process p. 3/15 Poisson random variable Number of successes in a large number n of trials (binomial distribution) wen te probability p of a success is small. Denote λ = np. Property: Pr(X = k) = e λλk k!. E[X] = Var(X) = λ.

4 Simulating events: te Poisson process p. 4/15 Poisson random variable Binomial distribution, p = 0.2 Poisson distribution, λ = np = n = 100

5 Simulating events: te Poisson process p. 5/15 Poisson random variable Binomial distribution, p = 0.1 Poisson distribution, λ = np = n = 100

6 Simulating events: te Poisson process p. 6/15 Poisson process Events are occurring at random time points N(t) is te number of events during [0,t] Tey constitute a Poisson process wit rate λ > 0 if 1. N(0) = 0, 2. # of events occurring in disjoint time intervals are independent, 3. distribution of N(t+s) N(t) depends on s, not on t, 4. probability of one event in a small interval is approx. λ: lim 0 Pr(N() = 1) = λ, 5. probability of two events in a small interval is approx. 0: lim 0 Pr(N() 2) = 0.

7 Simulating events: te Poisson process p. 7/15 Poisson process Property: Inter-arrival times: N(t) Poisson(λt), Pr(N(t) = k) = e λt(λt)k k! S k is te time wen te kt event occurs, X k = S k S k 1 is te time elapsed between event k 1 and event k. X 1 = S 1 Distribution of X 1 : Pr(X 1 > t) = Pr(N(t) = 0) = e λt. Distribution of X 2 : Pr(X k > t S k 1 = s) = Pr(0 events in ]s,s+t] S k 1 = s) = Pr(0 events in ]s,s+t]) = e λt.

8 Simulating events: te Poisson process p. 8/15 Poisson process X 1 is an exponential random variable wit mean 1/λ X 2 is an exponential random variable wit mean 1/λ X 2 is independent of X 1. Same arguments can be used for k = 3,4... Terefore, te CDF of X k is, for any k, Te pdf is F(t) = Pr(X k t) = 1 Pr(X k > t) = 1 e λt. f(t) = df(t) dt = λe λt. Te inter-arrival times X 1, X 2,... are independent and identically distributed exponential random variables wit parameter λ, and mean 1/λ.

9 Simulating events: te Poisson process p. 9/15 Poisson process Simulation of event times of a Poisson process wit rate λ until time T : 1. t = 0, k = Draw r U(0,1). 3. t = t ln(r)/λ. 4. If t > T, STOP. 5. k = k +1, S k = t. 6. Go to step 2.

10 Simulating events: te Poisson process p. 10/15 Non omogeneous Poisson process Assume tat te rate varies wit time, and call it λ(t). Te events constitute a non omogeneous Poisson process wit rate λ(t) if 1. N(0) = 0 2. # of events occurring in disjoint time intervals are independent, 3. probability of one event in a small interval is approx. λ(t): lim 0 Pr((N(t+) N(t)) = 1) = λ(t), 4. probability of two events in a small interval is approx. 0: lim 0 Pr((N(t+) N(t)) 2) = 0.

11 Simulating events: te Poisson process p. 11/15 Non omogeneous Poisson process Mean value function: Poisson distribution: m(t) = t 0 λ(s)ds, t 0. N(t+s) N(t) Poisson(m(t+s) m(t)) Link wit omogeneous Poisson process: Consider a Poisson process wit rate λ. If an event occurs at time t, count it wit probability p(t). Te process of counted events is a non omogeneous Poisson process wit rate λ(t) = λp(t).

12 Simulating events: te Poisson process p. 12/15 Non omogeneous Poisson process Proof: 1. N(0) = 0 [OK] 2. # of events occurring in disjoint time intervals are independent, [OK] 3. probability of one event in a small interval is approx. λ(t): [?] lim 0 Pr((N(t+) N(t)) = 1) = λ(t), 4. probability of two events in a small interval is approx. 0: [OK] lim 0 Pr((N(t+) N(t)) 2) = 0.

13 Simulating events: te Poisson process p. 13/15 Non omogeneous Poisson process N(t) number of events of te non omogeneous process N (t) number of events of te underlying omogeneous process Pr((N(t+) N(t)) = 1) = k Pr((N (t+) N (t)) = k,1 is counted) = Pr((N (t+) N (t)) = 1,1 is counted) = Pr((N (t+) N (t)) = 1)Pr(1 is counted) lim 0 Pr((N(t+) N(t))=1) = lim 0 Pr((N (t+) N (t))=1) Pr(1 is counted) = λp(t).

14 Simulating events: te Poisson process p. 14/15 Non omogeneous Poisson process Simulation of event times of a non omogeneous Poisson process wit rate λ(t) until time T : 1. Consider λ suc tat λ(t) λ, for all t T. 2. t = 0, k = Draw r U(0,1). 4. t = t ln(r)/λ. 5. If t > T, STOP. 6. Generate s U(0,1). 7. If s λ(t)/λ, ten k = k +1, S(k) = t. 8. Go to step 3.

15 Simulating events: te Poisson process p. 15/15 Summary Poisson random variable Poisson process Non omogeneous Poisson process Main assumption: events occur continuously and independently of one anoter Typical usage: arrivals of customers in a queue Easy to simulate