The Poisson Process

1. The Poisson Process

2. Definition What is A Poisson Process? The Poisson Process is a counting that counts the number of occurrences of some specific event through time Examples: - Number of customers arriving to a counter - Number of calls received at a telephone exchange - Number of packets entering a queue

3. The Poisson Process 3rd Event Occurs • X1, X2, … represent a sequence of +ve independent random variables with identical distribution • Xn depicts the time elapsed between the (n-1)th event and nth event occurrences • Sn depicts a random variable for the time at which the nth event occurs • Define N(t) as the number of events that have occurred up to some arbitrary time t. 1st Event Occurs 2nd Event Occurs 4th Event Occurs X4 X3 X1 X2 time t=0 The counting process { N(t), t>0 } is called a Poisson process if the inter-occurrence times X1, X2, … follow the exponential distribution

4. N (t=10 min) = 2 4th Bus Arrival 1st Bus Arrival 2nd Bus Arrival 3rd Bus Arrival X4=2 min X1=5 min X2=4 min X3=7 min time S1 = 5 min S2 = 9 min t=0 S3 = 16 min S4 = 18 min The Poisson Process: Example For some reason, you decide everyday at 3:00 PM to go to the bus stop and count the number of buses that arrive. You record the number of buses that have passed after 10 minutes Sunday

5. N (t=10 min) =4 2nd Bus Arrival 1st Bus Arrival 3rd Bus Arrival 4th Bus Arrival 5th Bus Arrival X1=1 min X2=2 min X3=4 min X4=2 min X5=6 min time S2 = 3 min S3 = 7 min S1 = 1 min t=0 S4 = 9 min S5 = 15 min The Poisson Process: Example For some reason, you decide everyday at 3:00 PM to go to the bus stop and count the number of buses that arrive. You record the number of buses that have passed after 10 minutes Monday

6. N (t=10 min) =1 1st Bus Arrival 2nd Bus Arrival X1=10 min X2=6 min time S1 = 10 min t=0 S2 = 16 min The Poisson Process: Example For some reason, you decide everyday at 3:00 PM to go to the bus stop and count the number of buses that arrive. You record the number of buses that have passed after 10 minutes Tuesday

7. The Poisson Process: Example Given that Xi follow an exponential distribution then N(t=10) follows a Poisson Distribution

8. The Exponential Distribution The exponential distribution describes a continuous random variable Cumulative Distribution Function (CDF) 1 0 Probability Density Function (PDF) λ 0

9. Mean of the Exponential Distribution

10. Variance of the Exponential Distribution

11. The Laplace transform of any PDF for a continuous random variable may be used to deduce different parameters and characteristics of the distribution Laplace Transform of Exponential PDF What could F(S) be used for

12. Probability Density Function for Sk The probability density function for the sum of k independent random variables (X1, X2, …, Xk) could be deduced from the Laplace transform of fXn(x) as follows From Laplace Transform Tables

13. Cumulative Distribution Function for Sk From Laplace Transform Tables

14. Probability Mass Function for the Poisson Distribution The Poisson Distribution

15. where Mean of the Poisson Distribution On average the time between two consecutive events is 1/λ This means that the event occurrence rate is λ Consequently in time t, the expected number of events isλt

16. Variance of the Poisson Distribution

17. The Moment Generating Function of any PMF for a discrete random variable may be used to deduce different parameters and characteristics of the distribution Moment Generating Function of Poisson Distribution What could G(Z) be used for

18. Remaining Time of Exponential Distributions Xk+1 is the time interval between the kth and k+1th arrivals Condition: T units have passed and the k+1th event has not occurred yet Question: Given that X*k+1 is the remaining time until the k+1th event occurs What is Pr[X*k+1≤x] kth Event Occurs k+1th Event Occurs T X*k+1

19. Remaining Time of Exponential Distributions Xk+1 follows an exponential Distribution, i.e., Pr[Xk+1≤t]=1-e-λt The remaining time X*k+1 follows an exponential distribution with the same mean 1/λ as that of the inter-arrival time Xk+1

20. The Memoryless Property: The waiting time until the next arrival has the same exponential distribution as the original inter-arrival time regardless of long ago the last arrival occurred Memoryless Property of Exponential Distribution and the Poisson Process The Memoryless Property of Exponential Distributions In the Poisson process, the number of arrivals within any time interval s follows a Poisson distribution with mean λs

21. Merging of Poisson Processes • {N1(t), t ≥ 0} and {N2(t), t ≥ 0} are two independent Poisson processes with respective rates λ1 and λ2, • {Ni (t)} corresponds to type i arrivals. • The merged processN(t) = N1(t) + N2(t), t ≥ 0. Then {N(t), t ≥ 0} is a Poisson process with rate λ = λ1 + λ2. • Zk is the inter-arrival time between the (k − 1)th and kth arrival in the merged process • Ik= i if the kth arrival in the merged process is a type i arrival, • For any k = 1, 2, . . . , P{Ik = i | Zk = t} =λi/(λ1+λ2) , i= 1, 2, independently of t .

22. Splitting of Poisson Processes • {N(t), t ≥ 0} is a Poisson process with rate λ. • Each arrival of the process is classified as being a type 1 arrival or type 2 arrival with respective probabilities p1 and p2, independently of all other arrivals. • Ni (t) is the number of type i arrivals up to time t . • {N1(t)} and {N2(t)} are two independent Poisson processes having respective rates λp1 and λp2.