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CMPE 252A: Computer Networks

Review Set:Quick Review of Probability Theory and Random Processes

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Why Probability Theory? Information is exchanged in a computer network in

a random way, and events that modify the behavior of links and nodes in the network are also random

We need a way to reason in quantitative ways about the likelihood of events in a network, and to predict the behavior of network components.

Example 1: Measure the time between two packet arrivals

into the cable of a local area network. Determine how likely it is that the interarrival

time between any two packets is less than T sec.

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Probability Theory A mathematical model used to quantify

the likelihood of events taking place in an experiment in which events are random.

It consists of: A sample space: The set of all possible

outcomes of a random experiment. The set of events: Subsets of the sample

space. The probability measure: Defined according

to a probability law for all the events of the sample space.

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Random Experiment A random experiment is specified by

stating an experimental procedure and a set of measurements and observations.

Example 2: We count the number of packets received

correctly at a base station in a wireless LAN during a period of time of T sec.We want to know how likely it is that the next packet received after T sec is correct.

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Our Modeling Problem ... We want to reason in quantitative

terms about the likelihood of events that can be observed.

This requires: Describing all the events in which we are

interested for the experiment. Combining events into sets of events that

are interesting (e.g., all packets arriving with 5 sec. latency)

Assigning a number to each of those events reflecting the likelihood that they occur.

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Probability Law Probability of an event: A number

assigned to the event reflecting the likelihood with which it can occur or has been observed to occur.

Probability Law: A rule that assigns probabilities to events in a way that reflects our intuition of how likely the events are.

What we need then is a formal way of assigning these numbers to events and any combination of events that makes sense in our experiments!

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Probability Law Let E be a random experiment (r.e.) Let A be an event in E The probability of A is denoted by P(A) A probability law for E is a rule that assigns

P(A) to A in a way that the following conditions, taken from our daily experience, are satisfied: A may or may not take place; it has some

likelihood (which may be 0 if it never occurs). Something must occur in our experiment. If one event negates another, then the

likelihood that either occurs is the likelihood that one occurs plus the likelihood that the other occurs.

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Probability Law More formally, we state the same as

follows. A probability law for E is a rule that

assigns a number P(A) to each event A in E satisfying the following axioms: AI: AII: AIII:

P(A) 0 ≤1 )( =SP

P(B)P(A) BAPBA +=⇒= ) ( UI φ

Everything else is derived from these axioms!

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Important Corollaries

1.

2.

3.

4.

5.

∑=⇒≠∀Φ=k

kk

kjin APAPjiAAAAA )()( ... , , 21 UI

)(1)( APAP c −=

1)( ≤AP

0)( =ΦP

)()()()( BAPBPAPBAP IU −+=

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Probability Law

All probabilities must be in [0, 1] The sum of probabilities must be at most 1

A

S

x

)(1 SP=

)(0 Φ=PΦ

)(AP

)(xP

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Conditional ProbabilityEvents of interest occur within some context, and that context changes their likelihood.

time

packet collisions

timeinterarrival times

We are interested in events occurring given that others take place!

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Conditional Probability The likelihood of an event A occurring given

that another event B occurs is smaller than the likelihood that A occurs at all.

We define the conditional probability of A given B as follows:

0)(for )(

)()|( >

∩= BP

BPBAP

BAP

We require P(B) > 0 because we know B occurs!

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Theorem of Total Probability Purpose is to divide and conquer We can describe how likely an event is by

partitioning it into mutually exclusive pieces.

timesuccess failure

busy period idle period busy period

1B 2B 3B

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Theorem of Total Probability1B 2B

nB...

A

)()()|()( ∑∑ ∩==i

ii

ii BAPBPBAPAP

5BA∩

Intersections of A with B’s are mutually exclusive

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Independence of Events In many cases, an event does not depend on prior

events, or we want that to be the case. Example: Our probability model should not have to

account for the entire history of a LAN. Independence of an even with respect to another

means that its likelihood does not depend on that other event.

A is independent of B if P(A | B) = P(A) B is independent of A if P(B | A) = P(B) So the likelihood of A does not change

by knowing about B and viceversa! This also means: )()()( BPAPBAP =∩

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Random Variables We are not interested in describing the likelihood

of every outcome of an experiment explicitly. We are interested in quantitative properties

associated with the outcomes of the experiment. Example:

What is the probability with which each packet sent in an experiment is received correctly?We don’t really care!

What is the probability of receiving no packets correctly within a period of T sec.?We care! This may make a router delete a neighbor!

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Random Variables We implicitly use a measurement that assigns a

numerical value to each outcome of the experiment. The measurement is [in theory] deterministic; based

on deterministic rules. The randomness of the observed values of the

measurement is completely determined by the randomness of the experiment itself.

A random variable X is a rule that assigns a numerical value to each outcome of a random experimentYes, it is really a function!

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Random Variables Definition: A random variable X on a

sample space S is a function X: S ->R that assigns a real number X(s) to each sample point s in S.

∞− ∞+

S

5s3s

2s1s4s

0)( 2sX )( 1sX )()( 54 sXsX =

ℜ⊂∈= }|)({ SssXS iiX which is called the event space

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Random Variables Purpose is to simplify the description of the problem. We will not have to define the sample space!

∞+∞−

0

1S

s

xsX =)(

))(( xsXP =

Possible values of X

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Types of Random Variables Discrete and continuous: Typically

used for counting packets or measuring time intervals.

time0 t

time

Busy period next packet

time?

1 2 3 4

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Discrete Random Variables We are interested in the probability that a discrete

random variable X (our measurement!) equals a certain value or range of values.

Example:We measure the delay experienced by each packet sent from one host to another over the Internet; say we sent 1M packets (we have 1M delay measurements). We want to know the likelihood with which any one packet experiences a delay of 5 ms or less.

Probability Mass Function (pmf) of a random variable X :

The probability that X assumes a given value x

)()( xpxXP X==

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Discrete Random VariablesCumulative Distribution Function (cdf) of a random variable X: The probability that X takes on any value in the interval ],( x−∞

)()( xXPxFX ≤=The pdf and pmf of a random variable are just probabilities and obey the same axioms AI to AIII. Therefore,

)()( b)XP(a

for )()(

0)(lim ;1)(lim ;1)(0

aFbF

babFaF

xFxFxF

XX

XX

XXxX x

−=≤<<≤

==≤≤−∞→+∞→

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Continuous Random Variables The probability that a continuous r.v. X assumes a

given value is 0. Therefore, we use the probability that X assumes a

range of values and make that length of that range tend to 0.

The probability density function (pdf) of X, if it exists, is defined in terms of the cdf of X as

dx

xdFxf X

X

)()( =

∫

∫∫∞+

∞−

∞−

=+∞→

≤≤∞−===≤≤

yprobabilit a is pdf because ,1)( then if

)()()( ;(x)dxfb)XP(ab

a

X

dttfx

xXPdttfxF

X

x

XX

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What We Will Use We are interested in:

Using the definitions of well-known r.v.s to compute probabilities

Computing average values and deviations from those values for well-known r.v.s

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Mean and Variance

What is the average queue length at each router?

What is our worst case queue?

BA

D

C

3

5

4

6

1

2

7

For VC1 use 3For VC2 use 2…..For VCn use 3

VC1

VC2

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Mean and VarianceExpected value or mean of X:

us!for defined always isMean

c.r.vfor )()(

d.r.v.for )()(

∫

∑∞+

∞−

=

=

dtttfXE

kpxXE

X

Xk

k

time

queue size

mean

too much?

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Variance

Describes how much a r.v. deviates from its average value, i.e., D = X - E(X)We are only interested in the magnitude of the deviation, so we use 22 ))(( XEXD −=

The variance of a r.v. is defined as the mean squared variation )( 2DE

))](([)( 22 XEXEXVar −==σ

Important relation: )()()( 22 XEXEXVar −=

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Properties of Mean and Variance

Useful when we discuss amplifying random variables or adding constant biases.

)()(

)()(

0)(

)( ;)()(

2 XVarccXVar

XVarcXVar

cVar

bbEbXaEbaXE

=

=+=

=+=+

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Examples of Random Variables We are interested in those r.v. that permit us

to model system behavior based on the present state alone.

We need to count arrivals in a time interval count the number of times we need to repeat

something to succeed count the number of successes and failures measure the time between consecutive

arrivals The trick is to map our performance questions

into the above four types of experiments

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Bernoulli Random Variable Let A be an event related to the

outcomes of a random experiment. X = 1 if outcome occurs and 0

otherwise This is the Bernoulli r.v. and has two

possible outcomes: success (1) or failure (0)

qpXPPP

pqXPPP

X

X

−=====−=====

1)1()1(1)0()0(

1

0

We use it as a building block for other types of counting

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Geometric Random Variable Used to count the number of attempts

needed to succeed doing something. Example: How many times do we have to

transmit a packet over a broadcast radio channel before it is sent w/o interference?

time

Assume that each attempt is independent of any prior attempt!

Assume each attempt has the same probability of success (p)

failure failure failure success!

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Geometric Random Variable We want to count the number k of trials needed for

the first success in a sequence of Bernoulli trials!

,...2,1 ;)1()()(

trialain success ofy probabilit1 =−===

=− kppkXPkp

pk

X

Why this is the case is a direct consequence of assuming independent Bernoulli trials, each with the same probability of success: k-1 failures needed before the last successful trial

Memoryless property: The probability of having a success in k additional trials having experienced n failures is the same as the probability of success in k trials before the n failed trails

)()|( kXPnXknXP ==>+=

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Binomial Random Variable X denotes the number of times success

occurs in n independent Bernoulli trials.

success 1 where0); , ... 0, 0, 1, , ... 1, 1, ,1(

such that outcome specific heConsider t

==∈

sSs

knkcn

ckk ppAPAPAPAPAPsP −+ −== )1()()...()()....()()(

:tindependen are trialsBecause

121

s0' and s1' have we; 1 1 n-kk nkk +

qpAPAPpAP

AAAAAs

iA

ci

cn

ckk

i

=−=−==

∩∩∩∩∩∩=

=

+

1)(1)( ;)(

......

then, in trial success Let

121

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Binomial Random Variableknkc

nckk ppAPAPAPAPAPsP −+ −== )1()()...()()....()()( 121

!)!(

!

k

n and slots, in the s1'for

positions choose towaysdifferent k

n are There

kkn

nn

k

−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

Because each outcome is mutually exclusive of the others:

nk

ppkkn

npp

k

nkP knkknk

n

≤≤

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

=−⎟⎟⎠

⎞⎜⎜⎝

⎛= −−

0

)1(!)!(

!)1()(

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Poisson Random Variable We use this r.v. in cases where we need to count

event occurrences in a time period. An event occurrence will typically be a packet

arrival. Arrivals are assumed to occur at random over a

time interval. The time interval is (0, t] We divide the time interval into n small subintervals

of length tnt Δ=/

• The probability of a new arrival in a given subinterval is defined to be tΔλ

• is constant, independent of the subinterval.λ

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Poisson Random Variable

A sequence of n independent Bernoulli trials; with X being the number of arrivals in (0, t]

arrivals. 1or 0 having ofy probabilit the tocompared

negligible is in arrival one than more ofy probabilit The

:0 and Make

t

tn

Δ→Δ∞→

By assumption, whether or not an event occurs in a subinterval is independent of the outcomes in other subintervals.We have:

tΔ

…. time

t0

arrival

1 2 3 4 n

1 2 3 k

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Poisson Random Variablearrivals ofnumber theis ;)1()( kpp

k

nkP knk

n−−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

nttnttPp /}/in arrival{ λλ =Δ==Δ=kkn

k n

t

n

t

kkn

nPkXPtkP ⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡ −−

====− λλ

1 !)!(

!)(]}[0, in arrivals {

: then,0 and Make →Δ∞→ tn

knk

kk n

t

n

t

k

t

knn

nP

−

⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡ −⎥⎦

⎤⎢⎣

⎡−

=λλλ

11!)(

)!(!

:get toRearrange

[ ] knk

k n

t

k

t

n

knnnn −

⎥⎦

⎤⎢⎣

⎡−⎥⎦

⎤⎢⎣

⎡ +−−−11

!

)()1)..(2)(1( λλ

€

Pk =nk

n

t

k

t⎥⎦

⎤⎢⎣

⎡ −λλ

1!)(

)1(

nxx

ex

x −=⎟⎠

⎞⎜⎝

⎛ +≡ ∞→ with;1

1lim also and

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Poisson Random Variable

We see that the Poisson r.v. is the result of an approximation of i.i.d. arrivals in a time interval.

We will say “arrivals are Poisson” meaning we can use the above formulas to describe packet arrivals

The probability of 0 arrivals in [0, t] plays a key role in our treatment of interarrival times.

€

Pk = P{k arrivals in (0,t]} =λ t( )

k

k!e−λt

tetPP λ−== ]}(0, in arrivals 0{0

tePtP λ−−=−= 11]}(0, in arrivals some{ 0

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Important Properties of Poisson Arrivals

The aggregation of Poisson sources is a Poisson source.

If packets from a Poisson source are routed such that a path is chosen independently with probability p, that stream is also Poisson, with rate p times the original rate.

It turns out that the time of a given Poisson arrival is uniformly distributed in a time interval

• The parameter λ is called the arrival rate

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Exponential Random Variable Consider the Poisson r.v., then teP λ−=0• Let Y be the time interval from a time origin (chosen arbitrarily) to the first arival after that origin• Let ] (0,in arrivals ofNumber )( ττ =X then

λτττ −===> eXPYP )0)(()(• We can obtain now a c.d.f. for Y as follows:

0 ;1)(1)()( ≥−=>−=≤= − ττττ λτeYPYPFY

and the p.d.f. of Y is then

0 ;)()( ≥=≤= − τλττ λτeYPdtd

fY

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Memoryless Property of Exponential Random Variable

After waiting h sec. for the first arrival, the probability that it occurs after t sec. equals the prob. that the first arrival occurs after t sec.

Knowing that we have waited any amount of time does not improve our knowledge of how much longer we’ll have to wait! )()|( hYPtYhtYP >=>+>

time

time > t?time? time >t ?

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Mean and Variance Results

You have to memorize these!You should be able to derive any of the above

Exponential: 22 /1 ;/1)( λσλ ==YE

Poisson: tXE λσ == 2)(

Geometric:22 / ;/1)( pqpXE == σ

Binomial: npqnpXE == 2 ;)( σ

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Random Processes

Definition: A random process is a family of random variables {X(t) | t is in T } defined on a given sample space, and indexed by a parameter t, where t varies over an index set T.

t

xx

x

xx

xx

x

x

xx

1t 2t 3t 4t 5t

a state of X(t)

X(t5) is a R.V.statespace ofrandomprocess

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Poisson Process We assume that events occur at random instants of time at

an average rate of lambda events per second. Let N(t) be the number of event occurrences (e.g., packet

arrivals) in (0,t]. N(t) is a non-decreasing, integer-valued, continuous time

random process.

t1a 2a 3a

4a 5a

54321

a r.v.a r.v.

4)( 2 =tN4)( 1 =tN

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Poisson Process As we did for the case of the Poisson r.v., we divide the time axis into very

small subintervals, such that The probability of more than one arrival in a subinterval is much smaller than the

probability of 0 or 1 arrivals in the subinterval. Whether or not an arrival occurs in a subinterval is independent of what takes place in

any other subinterval. We end up with the same Binomial counting process and with the length of

the subintervals going to 0 we can approximate:

( ),...1,0 ,

!]}(0,in arrivals {])([ ==== − ke

k

ttkPktNP t

kλλ

Again, inter arrival times are exponentially distributed with parameter lambda

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Poisson Process Why do we say that arrivals occur “at random” with a Poisson process? Suppose that we know that only one arrival occurs in (0, t] and x is the

arrival time of the single arrival. Let N(x) be the number of arrivals up to time x, 0 < x ≤ t Then:

N(t) -N(x) = increment of arrivals in (x, t]

}1)({

}0)()( and 1)({

}1)({

}1)()({

}1)(|1)({][

==−=

==

===

===≤

tNPxNtNxNP

tNPtNxNPtNxNPxXP

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Poisson Process Arrivals in different intervals are independent.

}1)({

}0)()(P{ }1)({][

==−=

=≤tNP

xNtNxNPxXP

t

x

te

exxXP

t

x

==≤ −

−

λ

λλ

λλ ] ][e)[(

][)x-(t-

The above probability corresponds to the uniform distribution!

Hence, in the average case, a packet arrives in the middle of a fixed time interval with Poisson arrivals