Probability Review Thinh Nguyen

Probability Theory Review Sample space Bayes’ Rule Independence Expectation Distributions

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

Set Operations Union Intersection Complement Properties  Commutation  Associativity  Distribution  De Morgan’s Rule S

Axioms and Corollaries Axioms If If A 1, A 2, … are pairwise exclusive Corollaries

Conditional Probability Conditional Probability of event A given that event B has occurred If B 1, B 2,…,B n a partition of S, then (Law of Total Probability) S B1B1 B3B3 B2B2 A

Bayes’ Rule If B 1, …, B n a partition of S then

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

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(ζ) to each outcome ζ in the sample space of a random experiment S x SxSx ζ X(ζ) = x

Cumulative Distribution Function Defined as the probability of the event {X≤x} Properties x 2 1 F x (x) ¼ ½ ¾ x

Types of Random Variables Continuous  Probability Density Function Discrete  Probability Mass Function

Probability Density Function The pdf is computed from Properties For discrete r.v. dx f X (x) x

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

Queuing Theory

Example Send a file over the internet Send a file over the internet packetlink buffer Modem card (fixed rate)

Delay Models time place A B C propagation transmission Computation (Queuing)

Queue Model

Practical Example

Multiserver queue

Multiple Single-server queues

Standard Deviation impact

Queueing Time

Queuing Theory The theoretical study of waiting lines, expressed in mathematical terms inputoutput queue server Delay= queue time +service time

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.

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

Notation for queuing systems M for Markovian (exponential) distribution D for Deterministic distribution G for General (arbitrary) distribution :Where A and B can be omitted if infinite omitted if infinite

The M/M/1 System Poisson Process output queue Exponential server

Arrivals follow a Poisson process 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! Readily amenable for analysis Readily amenable for analysis Reasonable for a wide variety of situations Reasonable for a wide variety of situations

Model for Interarrivals and Service times  Customers arrive at times t 0 < t 1 < Poisson distributed  The differences between consecutive arrivals are the interarrival times :  n = t n - 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(S n  s) = 1 - e -  s

System Features Service times are independent service times are independent of the arrivals Both inter-arrival and service times are memoryless Pr(T n > t 0 +t | T n > t 0 ) = Pr(T n  t) future events depend only on the present state  This is a Markovian System

Exponential Distribution

Markov Models Buffer Occupancy n+1 n n-1 n departure arrival

Probability of being in state n

Steady State Analysis

Markov Chains n-1 n n+1

Substituting Utilization

Substituting P 1 Higher states have decreasing probability Higher utilization causes higher probability of higher states

What about P 0 Queue determined by

E(n), Average Queue Size

Selecting Buffers For large utilization, buffers grow exponentially

Throughput Throughput=utilization/service time =  /T s For  =.5 and T s =1ms Throughput is 500 packets/sec

Intuition on Little’s 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

Applying Little’s Law

Probability of Overflow

Buffer with N Packets

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

Example

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

Network of M/M/1 Queues

M/G/1 Queue Q S 0 S Assume that every customer in the queue pays at rate R when his or her remaining service time is equal to R. Total cost paid by a customer: Expected cost paid by each customer: 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. The customers pat at rate since each customer pays on the average and customers go through the queue per unit time.