The exponential distribution and the Poisson process 1-1
Exponential Distribution: Basic Facts PDF f(t) = { λe λt, t 0 0, t < 0 CDF Pr{T t) = 0 t λe λu du = 1 e λt (t 0) Mean E[T] = 1 λ Variance Var[T] = 1 λ 2 1-2
Key Property: Memorylessness Memoryless Property Pr{T > a + b T > b} = Pr{T > a} a, b 0 Reliability: Amount of time a component has been in service has no effect on the amount of time until it fails Inter-event times: Amount of time since the last event contains no information about the amount of time until the next event Service times: Amount of remaining service time is independent of the amount of service time elapsed so far 1-3
Example Suppose that the amount of time one spends in a bank is exponentially distributed with mean 10 minutes, λ = 1/10. a. What is the probability that a customer will spend more than 15 minutes in the bank? b. What is the probability that a customer will spend more than 15 minutes in the bank given that he is still in the bank after 10 minutes? Solution P(X >15) = exp( 15 λ ) = exp( 3/2) = 0.22 P(X >15 X >10) = P(X >5) = exp( 1/2) = 0.604 14-4
Properties of Exponential Distribution Minimum of Two Exponentials: 1-5 If X 1, X 2,, X n are independent exponential r.v. s where X i has parameter (rate), then λ i is exponential with parameter (rate) λ + λ + + Competing Exponentials: 1 min(x 1, X 2,, X n ) λ n If X1 and X2 are independent exponential r.v. s with parameters (rate) λ and respectively, then P(x1< x2) = That is, the probability X1 occurs before X2 is 2 λ1 λ1 + λ2 λ1 λ + λ 1 2 1 λ 2
Properties of Exponential RV. The probability of 1 event happening in the next t is Pr{T t ) = 1 e λ t = 1 { 1 + ( λ t )+ ( λ t ) n /n! ) } = λ t When t is small, ( λ t ) n 0 Exponential is the only r.v. that has this property. 1-6
Counting Process A stochastic process {N(t), t 0} is a counting process if N(t) represents the total number of events that have occurred in [0, t] Then { N(t), t 0 } must satisfy: a) N (t) 0 b) N(t) is an integer for all t c) If s < t, then N (s) N(t) and d) For s < t, N (t ) - N (s) is the number of events that occur in the interval (s, t ]. 1-7
Stationary & Independent Increments independent increments A counting process has independent increments if for any 0 s t u v, N(t) N(s) is independent of N(v) N(u) i.e., the numbers of events that occur in non-overlapping intervals are independent r.v.s stationary increments A counting process has stationary increments if the distribution if, for any s < t, the distribution of N(t) N(s) depends only on the length of the time interval, t s. 14-8 1-8
Poisson Process Definition 1 1-9 A counting process {N(t), t 0} is a Poisson process with rate λ, λ > 0, if N(0) = 0 The process has independent increments The number of events in any interval of length t follows a Poisson distribution with mean λt Pr{ N(t+s) N(s) = n } = (λt) n e λt /n!, n = 0, 1,... Where λ is arrival rate and t is length of the interval Notice, it has stationary increments 14-9
Poisson Process Definition 2 A function f is said to be o(h) ( Little oh of h ) if ( ) f h lim = 0 h 0 h A counting process {N(t), t 0} is a Poisson process with rate λ, λ > 0, if N(0) = 0 and the process has stationary and independent increments and Pr { N(h) = 1} = λh + o(h) Pr { N(h) 1} = o(h) Definitions 1 and 2 are equivalent because exponential distribution is the only distribution with this property 14-10 1-10
Interarrival and WaitingTimes N(t) The times between arrivals T1, T2, are independent exponential r.v. s with mean 1/λ: P(T1>t) = P(N(t) =0) = e -λt The (total) waiting time until the nth event has a gamma distribution S1 S 2 S 3 S 4 t n = T 1 T 2 T 3 T 4 i= 1 1-11 S n T i
Interarrival andwaitingtime Define T as the elapsed time between (n 1) st and the n th n event. { T n, n = 1,2,...} is a sequence of interarrival times. Proposition : T n, n = 1,2,... are independent identically 1 distributed exponential random variables with mean. Define as the waiting time for the n th λ S n event, i.e., the arrival time of the n th event. n S = T n i i= 1 1-12
i.e. distribution of S n is: ( ) n 1 t λt f ( t) = λe λ sn ( n 1)! which is the gamma distribution with parameters n and λ. E ( S ) n = E( Ti ) n = n i= 1 λ 1-13
Example Suppose that people immigrate into a territory at a Poisson rate = 1 per day. (a) What is the expected time until the tenth immigrant arrives? (b) What is the probability that the elapsed time between the tenth and the eleventh arrival exceeds 2 days? Solution: Time until the 10th immigrant arrives is S10. 10 E( S ) = = 10 10 λ. 2λ P( > 2) = e = 0. 133 1-14 14-14 T 11