1 Exponential Distribution & Poisson Process Memorylessness & other exponential distribution properties; Poisson process and compound P.P.’s

2 Exponential Distribution: Basic Facts Pr{ T  t ) =  0 t e – u d u = 1 – e – t ( t  0) f ( t ) = { e – t, t  0 0, t < 0 PDF CDF Mean Variance E [ T ] = 1 Var [ T ] = 1 2

3 Key Property: Memorylessness 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 Pr{T  a + b | T  b} = Pr{T  a} a, b  0 Memoryless Property

4 Properties of Exponential Distribution If X 1 and X 2 are independent exponential r.v.’s with parameters (rate) l 1 and l 2 respectively, then P(x1< x2) = l 1/( l 1+ l 2) That is, the probability X1 occurs before X2 is l 1/( l 1+ l 2) Minimum of Two Exponentials: If X 1, X 2, …, X n are independent exponential r.v.’s where X n has parameter (rate) l i, then min( X 1, X 2, …, X n ) is exponential with parameter (rate) l 1 + l 2 + … + l n Competing Exponentials:

5 Properties of Exponential RV. The probability of 1 event happening in the next  t is Pr{ T  t ) =  e –  t =  t When  t is small, (–  t ) n  0 = 1 – { 1 + (–  t )+  (–  t ) n /n! ) } Exponential is the only r.v. that has this property.

6 Counting Process 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 ]. 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 ]

7 Stationary & Independent Increments 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. 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

8 Poisson Process Definition 1 A counting process { N ( t ), t  0} is a Poisson process with rate l, l > 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 t Pr{ N(t+s) – N(s) = n } = ( t) n e – t /n!, n = 0, 1,... Where is arrival rate and t is length of the interval Notice, it has stationary increments

9 Poisson Process Definition 2 A function f is said to be o ( h ) (“Little oh of h”) if A counting process { N ( t ), t  0} is a Poisson process with rate l, l > 0, if N(0) = 0 and the process has stationary and independent increments and Pr { N(h) = 1} = h + o ( h ) Pr { N(h)  1} = o ( h ) Definitions 1 and 2 are equivalent because exponential distribution is the only variable has this property

10 Interarrival and Waiting Times t N(t)N(t) T2T2 T1T1 T3T3 T4T4 S1S1 S2S2 S3S3 S4S4 The (total) waiting time until the nth event has a gamma distribution The times between arrivals T 1, T 2, … are independent exponential r.v.’s with mean 1/ : P(T1>t) = P(N(t) =0) = e - t

11 Poisson Process – Pure Birth Process Pure Birth Process – Poisson Process States0123…… …00000 … Poisson Process Rate Matrix

12 An Example Suppose that you arrive at a single teller bank to find five other customers in the bank. One being served and the other four waiting in line. You join the end of the line. If the service time are all exponential with rate 5 minutes. What is the prob. that you will be served in 10 minutes ? What is the prob. that you will be served in 20 minutes ? What is the expected waiting time before you are served?

13 Other Poisson Process Properties Then { N 1( t ), t  0} and { N 2( t ), t  0} are independent Poisson processes with respective rates p and (1- p ). Let N 1( t ) and N 2( t ), respectively, be the number of type I and type II events up to time t. Poisson Splitting : Suppose { N ( t ), t  0} is a P.P. with rate, and suppose that each time an event occurs, it is classified as type I with probability p and type II with probability 1- p, independently of all other events.

14 Other Poisson Process Properties Competing Poisson Processes : Suppose { N 1 ( t ), t  0} and { N 2 ( t ), t  0} are independent P.P. with respective rates 1 and 2. Let S n i be the time of the n th event of process i, i = 1,2.

15 Other Poisson Process Properties Conditional Distribution of Arrival Times : Suppose { N ( t ), t  0} is a Poisson process with rate l and for some time t we know that N ( t ) = n. Then the arrival times S 1, S 2, …, Sn have the same conditional distribution as the order statistics of n independent uniform random variables on (0, t ). If Y 1, Y 2, …, Yn are random variables, then Y (1), Y (2), …, Y ( n ) are their order statistics if Y ( k ) is the k th smallest value among Y 1, Y 2, …, Yn, k = 1, …, n.

16 Compound Poisson Process A counting process { X ( t ), t  0} is a compound Poisson process if: where { N ( t ), t  0} is a Poisson process and { Y i, i = 1, 2, …} are independent, identically distributed r.v.’s that are independent of { N ( t ), t  0}. By conditioning on N ( t ), we can obtain