LECTURE 16

• Readings: Section 5.1

Lecture outline

• Random processes• Definition of the Bernoulli process• Basic properties of the Bernoulli process

– Number of successes– Distribution of interarrival times– The time of the success

Random Processes: Motivation

• Sequence of random variables:

Examples:

• Arrival example: Arrival of people to a bank.

• Queuing example: Length of a line at a bank.

• Gambler’s ruin: The probability of an outcome is a function of the probability of other outcomes (Markov Chains).

• Engineering example: Signal corrupted with noise.

The Bernoulli Process

• A sequence of independent Bernoulli trials.

• At each trial:

–

–

T T T H T T T H H T T T T T H T H T T H

$ $ $ $ $ $

• Examples:

– Sequence of ups and downs of the Dow Jones.

– Sequence of lottery wins/losses.

– Arrivals (each second) to a bank.

Number of successes in time slots

(Binomial)•

• Mean:

• Variance:

Interarrival Times

• : number of trials until first success (inclusive).

• (Geometric)

• Memoryless property.

• Mean:

• Variance:

Fresh Start and MemorylessProperties

Fresh Start

Given n, the future sequence is a also a Bernoulliprocess and is independent of the past.

MemorylessnessSuppose we observe the process for n times and no success occurred. Then the pmf of the remaining time for arrival isgeometric.

Time of the Arrival

• : number of trials until success (inclusive).

• : kth interarrival time

• It follows that:

Time of the Arrival

• : number of trials until success (inclusive).

• Mean:

• Variance:

(Pascal)•

LECTURE 17

• Readings: Start Section 5.2

Lecture outline

• Review of the Bernoulli process• Definition of the Poisson process• Basic properties of the Poisson process

– Distribution of the number of arrivals– Distribution of the interarrival time– Distribution of the arrival time

The Bernoulli Process: Review

• Discrete time; success probability in each slot = .

• PMF of number of arrivals in time slots: Binomial

• PMF of interarrival time: Geometric

• PMF of time to arrival: Pascal

• Memorylessness

• What about continuous arrival times?Example: arrival to a bank.

The Poisson Process: Definition

• Let = Probability of arrivals in an interval of duration .

• Assumptions:– Number of arrivals in disjoint time

intervals are independent.

– For VERY small , we have:

– = “arrival rate” of the process.

From Bernoulli to Poisson (1)

• Bernoulli: Arrival prob. in each time slot =

• Poisson: Arrival probability in each -interval =

• Let and :

Number of arrivalsin a -interval

Number of successesin time slots

=

(Binomial)

From Bernoulli to Poisson (2)

• Number of arrivals in a -interval as =

(Binomial)

(reorder terms)

(Poisson)

PMF of Number of Arrivals

• : number of arrivals in a -interval, thus:

• (Poisson)

• Mean:

• Variance:

• Transform:

Email Example

• You get email according to a Poisson process, at a rate of = 0.4 messages per hour. You check your email every thirty minutes.

– Prob. of no new messages =

– Prob. of one new message =

Interarrival Time

• : time of the arrival.

• “First order” interarrival time:(Exponential)

• Why:

Interarrival Time

• Fresh Start Property: The time of the next arrival is independent from the past.

• Memoryless property: Suppose we observe the process for T seconds and no success occurred. Then the density of the remaining time for arrival is exponential.

• Email Example: You start checking your email. How long will you wait, in average, until you receive your next email?

Time of Arrival

• : time of the arrival.

• : kth interarrival time

• It follows that:

Time of Arrival

• : time of the arrival.

• (Erlang)“of order ”

Bernoulli vs. Poisson

PoissonBernoulliContinuousDiscreteTimes of Arrival

/unit time/per trialArrival Rate

PoissonBinomialPMF of Number of Arrivals

ExponentialGeometricPMF of Interarrival Time

ErlangPascalPMF of Arrival Time

LECTURE 18

• Readings: Finish Section 5.2

Lecture outline

• Review of the Poisson process• Properties

–Adding Poisson Processes–Splitting Poisson Processes

• Examples

The Poisson Process: Review

• Number of arrivals in disjoint time intervals are independent, = “arrival rate”

(for very small )

(Poisson)

• Interarrival times (k =1):

(Exponential)

• Time to the arrival:

(Erlang)

Example: Poisson Catches

• Catching fish according to Poisson .

• Fish for two hours, but if there’s no catch, continue until the first one.

Example: Poisson Catches

• Catching fish according to Poisson .

• Fish for two hours, but if there’s no catch, continue until the first one.

Adding (Merging) Poisson Processes

• Sum of independent Poisson random variables is Poisson.

• Sum of independent Poisson processes is Poisson.

Red light flashes

Some light flashes

Green light flashes

• What is the probability that the next arrival comes from the first process?

Splitting of Poisson Processes

• Each message is routed along the first stream with probability , and along the second stream with probability .

– Routing of different messages are independent.

Server USA

Email traffic leaving MIT

Foreign

– Each output stream is Poisson.

Example: Email Filter (1)

• You have incoming email from two sources: valid email, and spam. We assume both to be Poisson.

• Your receive, on average, 2 valid emails per hour, and 1 spam email every 5 hours.

Valid

Incoming Email

Spam

• Total incoming email rate =

• Probability that areceived email is spam =

Example: Email Filter (2)

• You install a spam filter, that filters out spam email correctly 80% of the time, but also identifies a valid email as spam 5% of the time.

Spam Folder

Valid

Inbox

Spam

•

• Inbox email rate =

• Spam folder email rate =

Example: Email Filter (3)

Spam Folder

• Probability that an emailin the inbox is spam =

• Probability that an email in the spam folderis valid =

Valid

Inbox

Spam

• Every how often should you check your spam folder, to find one valid email, on average?