Birth and death process arrivals births l departures

Birth and death process λ arrivals (births) l μ departures (deaths) N(t) l l N(t) = # of customers at time t. Depends on how fast arrivals or departures occur Objective 1

Behavior of the system l λ>μ l λ<μ l Possible evolution of N(t) busy idle 3 2 1 1 2 3 4 5 6 7 8 9 10 11 Time 2

General arrival and departure rates l l λn l Depends on the number of customers (n) in the system l Example μn l Depends on the number of customers in the system l Example 3

Changing the scale of a unit time Time l Number of arrivals/unit time l l Follows the Poisson distribution with rate λn Inter-arrival time of successive arrivals l is exponentially distributed l l Average inter-arrival time = 1/ λn What is the avg. # of customers arriving in dt? 4

Probability of one arrival in dt l dt so small l Number of arrivals in dt, X is a r. v. l X=1 with probability p l X=0 with probability 1 -p l Average number of arrivals in dt l Prob (having one arrival in dt) = λn dt dt 5

Probability of having 2 events in dt l Departure rate in dt l l Arrival rate in dt l l μn dt λn dt What is the probability l Of having an (arrival+departure), (2 arrivals or departures) 6

Probability distribution of N(t) l Pn (t) l The probability of getting n customers by time t l The distribution of the # of customers in system t ? t+dt n n-1: arrival n+1: departure n: none of the above 7

Differential equation monitoring evolution of # customers l These are solved l Numerically using MATLAB l We will explore the cases l Of pure death l And pure birth 8

Pure birth process l In this case l μn =0, n >= 0 l λn = λ, n >= 0 Hence, 9

First order differential equation l 10

Pure death process l In this case l λn =0, n >= 0 l μn = μ 11

Queuing system μ λ l Transient phase Pn (t) l Steady state l l transient Steady state Behavior is independent of t Pn (t) t 12

Differential equation: steady state analysis l Limiting case 13

Solving the equations l n=1 (1) l n=2 (1) => 14

Pn l What about P 0 15

Normalization equation 16

Conditional probability and conditional expectation: d. r. v. l X and Y are discrete r. v. l Conditional probability mass function l l Of X given that Y=y Conditional expectation of X given that Y=y 17

Conditional probability and expectation: continuous r. v. l If X and Y have a joint pdf f. X, Y(x, y) l Then, the conditional probability density function l l Of X given that Y=y The conditional expectation l Of X given that Y=y 18

Computing expectations by conditioning l Denote l E[X|Y]: function of the r. v. Y l l Whose value at Y=y is E[X|Y=y] E[X|Y]: is itself a random variable l Property of conditional expectation (1) l if Y is a discrete r. v. (2) l if Y is continuous with density f. Y (y) => (3) 19

Proof of equation when X and Y are discrete 20

Problem 1 l Sam will read l Either one chapter of his probability book or l l One chapter of history book If the number of misprints in a chapter l Of his probability book l l Of history book l l is Poisson distributed with mean 2 is Poisson distributed with mean 5 Assuming Sam equally likely to choose either book 21 l What is the expected number of misprints he comes across?

Solution 22

Problem 2 l A miner is trapped in a mine containing three doors l First door l l Second door l l leads to a tunnel that returns him to the mine § After 3 hours of travel Third door l l leads to a tunnel that takes him to safety § After 2 hours of travel Leads to a tunnel that returns him to the mine § After 5 hours Assuming he is equally likely to choose any door l What is the expected length of time until he reaches safety? 23

Solution 24

Computing probabilities by conditioning l Let E denote an arbitrary event l X is a random variable defined by l It follows from the definition of X 25

Problem 3 l Suppose that the number of people l Who visit a yoga studio each day l l l is a Poisson random variable with mean λ Suppose further that each person who visit l is, independently, female with probability p l Or male with probability 1 -p Find the joint probability l That n women and m men visit the academy today 26

Solution l Let l N 1 denote the number of women, N 2 the number of men l l Who visit the academy today N= N 1 +N 2 : total number of people who visit l Conditioning on N gives l Because P(N 1=n, N 2=m|N=i)=0 when i != n+m 27

Solution (cont’d) l Each of the n+m visit l is independently a woman with probability p l The conditional probability l That n of them are women is § The binomial probability of n successes in n+m trials 28

Solution: analysis l When each of a Poisson number of events l l is independently classified l As either being type 1 with probability p l Or type 2 with probability (1 -p) => the numbers of type 1 and 2 events l Are independent Poisson random variables 29

Problem 4 l At a party l N men take off their hats l The hats are then mixed up and l l l Each man randomly selects one A match occurs if a man selects his own hat What is the probability of no matches? 30

Solution l E = event that no matches occur l l P(E) = Pn : explicit dependence on n Start by conditioning l Whether or not the first man selects his own hat l M: if he did, Mc : if he didn’t l P(E|Mc) § Probability no matches when n-1 men select of n-1 § That does not contain the hat of one of these men 31

Solution (cont’d) l P(E|Mc) l l Eithere are no matches and l Extra man does not select the extra hat l => Pn-1 (as if the extra hat belongs to this man) Or there are no matches l Extra man does select the extra hat l => (1/n-1)x. Pn-2 32

Solution (cont’d) l Pn is the probability of no matches l When n men select among their own hats l l => P 1 =0 and P 2 = ½ => 33

Problem 5: continuous random variables l The probability density function of a non-negative random variable X is given by l Compute the constant λ? 34

Problem 6: continuous random variables l Buses arrives at a specified stop at 15 min intervals l Starting at 7: 00 AM l l If the passenger arrives at the stop at a time l l l They arrive at 7: 00, 7: 15, 7: 30, 7: 45 Uniformly distributed between 7: 00 and 7: 30 Find the probability that he waits less than 5 min? Solution l Let X denote the number of minutes past 7 l l That the passenger arrives at the stop =>X is uniformly distributed over (0, 30) 35

Problem 7: conditional probability l Suppose that p(x, y) the joint probability mass function of X and Y is given by l l P(0, 0) =. 4, P(0, 1) =. 2, P(1, 0) =. 1, P(1, 1) =. 3 Calculate the conditional probability mass function of X given Y = 1 36

counting process l A stochastic process {N(t), t>=0} l l is said to be a counting process if l N(t) represents the total number of events that occur by time t l N(t) must satisfy § N(t) >= 0 § N(t) is integer valued § If s < t, then N(s) <= N(t) § For s < t, N(s) – N(t) = # events in the interval (s, t] Independent increments l # of events in disjoint time intervals are independent 37

Poisson process l The counting process {N(t), t>=0} is l Said to be a Poisson process having rate λ, if l N(0) = 0 l The process has independent increments l The # of events in any interval of length t is § Poisson distributed with mean λt, that is 38

Properties of the Poisson process l Superposition property l If k independent Poisson processes l A 1, A 2, …, An l Are combined into a single process A l => A is still Poisson with rate l Equal to the sum of individual λi of Ai 39

Properties of the Poisson process (cont’d) l Decomposition property l Just the reverse process l “A” is a Poisson process split into n processes l l Using probability Pi The other processes are Poisson l With rate Pi. λ 40