Probability Review

1
Probability Review

• Thinh Nguyen

2
Probability Theory Review

• Sample space
• Bayes Rule
• Independence
• Expectation
• Distributions

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Sample Space - Events

• Sample Point
• The outcome of a random experiment
• Sample Space S
• The set of all possible outcomes
• Discrete and Continuous
• Events
• A set of outcomes, thus a subset of S
• Certain, Impossible and Elementary

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Set Operations
S

• Union
• Intersection
• Complement
• Properties
• Commutation
• Associativity
• Distribution
• De Morgans Rule

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Axioms and Corollaries

• Axioms
• If
• If A1, A2, are pairwise exclusive

• Corollaries

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Conditional Probability

• Conditional Probability of event A given that
  event B has occurred
• If B1, B2,,Bn a partition of S, then
• (Law of Total Probability)

S
B1
B2
A
B3
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Bayes Rule

• If B1, , Bn a partition of S then

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Event Independence

• Events A and B are independent if
• If two events have non-zero probability and are
  mutually exclusive, then they cannot be
  independent

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Random Variables
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Random Variables

• The Notion of a Random Variable
• The outcome is not always a number
• Assign a numerical value to the outcome of the
  experiment
• Definition
• A function X which assigns a real number X(?) to
  each outcome ? in the sample space of a random
  experiment

S
?
X(?) x
x
Sx
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Cumulative Distribution Function

• Defined as the probability of the event Xx
• Properties

Fx(x)
1
x
Fx(x)
1
¾
½
¼
2
1
0
3
x
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Types of Random Variables

• Continuous
• Probability Density Function

• Discrete
• Probability Mass Function

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Probability Density Function

• The pdf is computed from
• Properties
• For discrete r.v.

fX(x)
fX(x)
dx
x
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Expected Value and Variance

• The expected value or mean of X is
• Properties

• The variance of X is
• The standard deviation of X is
• Properties

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Queuing Theory
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Example

• Send a file over the internet

packet
link
(fixed rate)
Modem card
buffer
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Delay Models
place
Computation (Queuing)
transmission
propagation
C
B
A
time
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Queue Model
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Practical Example
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Multiserver queue
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Multiple Single-server queues
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Standard Deviation impact
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Queueing Time
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Queuing Theory

• The theoretical study of waiting lines, expressed
  in mathematical terms

input
output
server
queue
Delay queue time service time
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The Problem

• Given
• One or more servers that render the service
• A (possibly infinite) pool of customers
• Some description of the arrival and service
  processes.
• Describe the dynamics of the system Evaluate its
  Performance
• If there is more than one queue for the
  server(s), there may also be some policy
  regarding queue changes for the customers.

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Common Assumptions

• The queue is FCFS (FIFO).
• We look at steady state after the system has
  started up and things have settled down.
• Statea vector indicating the total of
  customers in each queue at a particular time
  instant
• (all the information necessary to completely
  describe the system)

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Notation for queuing systems

• omitted if infinite

Where A and B can be
D for Deterministic distribution
M for Markovian (exponential) distribution
G for General (arbitrary) distribution
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The M/M/1 System
Poisson Process
output
Exponential server
queue
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Arrivals follow a Poisson process

• Readily amenable for analysis
• Reasonable for a wide variety of situations

• a(t) of arrivals in time interval 0,t
• ? mean arrival rate
• t k? k 0,1,. ??0
• Pr(exactly 1 arrival in t,t?) ??
• Pr(no arrivals in t,t?) 1-??
• Pr(more than 1 arrival in t,t?) 0
• Pr(a(t) n) e-? t (? t)n/n!

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Model for Interarrivals and Service times

• Customers arrive at times t0 lt t1 lt .... -
  Poisson distributed
• The differences between consecutive arrivals are
  the interarrival times ?n tn - t n-1
• ?n in Poisson process with mean arrival rate ?,
  are exponentially distributed,
• Pr(?n ? t) 1 - e-? t
• Service times are exponentially distributed, with
  mean service rate ?
• Pr(Sn ? s) 1 - e-?s

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System Features

• Service times are independent
• service times are independent of the arrivals
• Both inter-arrival and service times are
  memoryless
• Pr(Tn gt t0t Tngt t0) Pr(Tn ? t)
• future events depend only on the present state
• ? This is a Markovian System

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Exponential Distribution
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Markov Models

• n1
• n
• n-1

departure
Buffer Occupancy

• n

arrival
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Probability of being in state n
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Steady State Analysis
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Markov Chains
0 1 ... n-1
n n1
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Substituting Utilization
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Substituting P1

• Higher states have decreasing probability
• Higher utilization causes higher probability
• of higher states

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What about P0
Queue determined by
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E(n), Average Queue Size
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Selecting Buffers
For large utilization, buffers grow exponentially
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Throughput

• Throughpututilization/service time ?/Ts
• For ?.5 and Ts1ms
• Throughput is 500 packets/sec

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Intuition on Littles Law

• If a typical customer spends T time units, on the
  overage, in the system, then the number of
  customers left behind by that typical customer is
  equal to

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Applying Littles Law
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Probability of Overflow
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Buffer with N Packets
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Example

• Given
• Arrival rate of 1000 packets/sec
• Service rate of 1100 packets/sec
• Find
• Utilization
• Probability of having 4 packets in the queue

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Example
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Application to Statistcal Multiplexing

• Consider one transmission line with rate R.
• Time-division Multiplexing
• Divide the capacity of the transmitter into N
  channels, each with rate R/N.
• Statistical Multiplexing
• Buffering the packets coming from N streams into
  a single buffer and transmitting them one at a
  time.

R/N

R/N

R/N

R

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Network of M/M/1 Queues
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M/G/1 Queue
Assume that every customer in the queue pays at
rate R when his or her remaining service time is
equal to R.
At a given time t, the customers pay at a rate
equal to the sum of the remaining service times
of all the customer in the queue. The queue
begin first come-first served, this sum is equal
to the queueing time of a customer who would
enter the queue at time t.
Total cost paid by a customer
Expected cost paid by each customer
S
0
Q
S