Optimization and Simulation

Transcription

1 Optimization and Simulation Simulating events: the Michel Bierlaire Transport and Mobility Laboratory School of Architecture, Civil and Environmental Engineering Ecole Polytechnique Fédérale de Lausanne M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 1 / 19

2 Siméon Denis Poisson Siméon-Denis Poisson French mathematician ( ). M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 2 / 19

3 Poisson random variable Outline 1 Poisson random variable 2 3 Non homogeneous M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 3 / 19

4 Poisson random variable Binomial random variable Context n: number of independent trials p: probability of a success X: number of successes Probability of k successes Pr(X = k) = ( n k ) p k (1 p) n k. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 4 / 19

5 Poisson random variable Poisson random variable Context: continuous trials n + : large number of trials p 0: low probability of a success np λ: success rate X: number of successes Probability of k successes Property Pr(X = k) = e λλk k!. E[X] = Var(X) = λ. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 5 / 19

6 Poisson random variable Poisson random variable Binomial distribution, p = 0.2 Poisson distribution, λ = np = n = 100 M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 6 / 19

7 Poisson random variable Poisson random variable Binomial distribution, p = 0.1 Poisson distribution, λ = np = n = 100 M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 7 / 19

8 Outline 1 Poisson random variable 2 3 Non homogeneous M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 8 / 19

9 Events are occurring at random time points N(t) is the number of events during [0,t] with 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. λh: Pr(N(h) = 1) lim = λ, h 0 h 5 probability of two events in a small interval is approx. 0: lim Pr(N(h) 2) = 0. h h 0 M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 9 / 19

10 Property N(t) Poisson(λt), Pr(N(t) = k) = e λt(λt)k k! Inter-arrival times S k is the time when the kth event occurs, X k = S k S k 1 is the 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 k : 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. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 10 / 19

11 Inter-arrival times (ctd.) X 1 is an exponential random variable with mean 1/λ X 2 is an exponential random variable with mean 1/λ X 2 is independent of X 1. Same arguments can be used for k = 3,4... Therefore, the CDF of X k is, for any k, F(t) = Pr(X k t) = 1 Pr(X k > t) = 1 e λt. The pdf is f(t) = df(t) dt = λe λt. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 11 / 19

12 Conclusion The inter-arrival times X 1, X 2,...are independent and identically distributed exponential random variables with parameter λ, and mean 1/λ. Simulation Simulation of event times of a with rate λ until time T: 1 t = 0, k = 0. 2 Draw r U(0,1). 3 t = t ln(r)/λ. 4 If t > T, STOP. 5 k = k +1, X k = t. 6 Go to step 2. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 12 / 19

13 Non homogeneous Outline 1 Poisson random variable 2 3 Non homogeneous M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 13 / 19

14 Non homogeneous Non homogeneous Rate varies with time λ(t). Non homogeneous with 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)h: Pr((N(t +h) N(t)) = 1) lim = λ(t), h 0 h 4 probability of two events in a small interval is approx. 0: Pr((N(t +h) N(t)) 2) lim = 0. h 0 h M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 14 / 19

15 Non homogeneous Non homogeneous Mean value function m(t) = t 0 λ(s)ds, t 0. Poisson distribution N(t +s) N(t) Poisson(m(t +s) m(t)) Link with homogeneous Consider a with rate λ. If an event occurs at time t, count it with probability p(t). The process of counted events is a non homogeneous with rate λ(t) = λp(t). M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 15 / 19

16 Non homogeneous Non homogeneous 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)h: [?] Pr((N(t +h) N(t)) = 1) lim = λ(t), h 0 h 4 probability of two events in a small interval is approx. 0: [OK] Pr((N(t +h) N(t)) 2) lim = 0. h 0 h M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 16 / 19

17 Non homogeneous Non homogeneous Proof (ctd.) N(t) number of events of the non homogeneous process N (t) number of events of the underlying homogeneous process Pr((N(t +h) N(t)) = 1) = k Pr((N (t +h) N (t)) = k,1 is counted) = Pr((N (t +h) N (t)) = 1,1 is counted) = Pr((N (t +h) N (t)) = 1)Pr(1 is counted) = Pr(N (h) = 1)Pr(1 is counted) lim h 0 Pr((N(t+h) N(t))=1) h = lim h 0 Pr(N (h)=1) h Pr(1 is counted) = λp(t). M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 17 / 19

18 Non homogeneous Non homogeneous Simulation of event times of a non homogeneous with rate λ(t) until time T 1 Consider λ such that λ(t) λ, for all t T. 2 t = 0, k = 0. 3 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, then k = k +1, X(k) = t. 8 Go to step 3. M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 18 / 19

19 Non homogeneous Summary Outline Poisson random variable Non homogeneous Comments Main assumption: events occur continuously and independently of one another Typical usage: arrivals of customers in a queue Easy to simulate M. Bierlaire (TRANSP-OR ENAC EPFL) Optimization and Simulation 19 / 19