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TCOM 501: Networking Theory & Fundamentals. Lecture 2 January 22, 2003 Prof. Yannis A. Korilis. Topics. Delay in Packet Networks Introduction to Queueing Theory Review of Probability Theory The Poisson Process Little’s Theorem Proof and Intuitive Explanation Applications.

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Tcom 501 networking theory fundamentals l.jpg

TCOM 501: Networking Theory & Fundamentals

Lecture 2

January 22, 2003

Prof. Yannis A. Korilis

Topics l.jpg

  • Delay in Packet Networks

  • Introduction to Queueing Theory

  • Review of Probability Theory

  • The Poisson Process

  • Little’s Theorem

    • Proof and Intuitive Explanation

    • Applications

Sources of network delay l.jpg
Sources of Network Delay

  • Processing Delay

    • Assume processing power is not a constraint

  • Queueing Delay

    • Time buffered waiting for transmission

  • Transmission Delay

  • Propagation Delay

    • Time spend on the link – transmission of electrical signal

    • Independent of traffic carried by the link

  • Focus: Queueing & Transmission Delay

Basic queueing model l.jpg






In Service

Basic Queueing Model

  • A queue models any service station with:

    • One or multiple servers

    • A waiting area or buffer

  • Customers arrive to receive service

  • A customer that upon arrival does not find a free server is waits in the buffer

Characteristics of a queue l.jpg
Characteristics of a Queue

  • Number of servers m: one, multiple, infinite

  • Buffer size b

  • Service discipline (scheduling): FCFS, LCFS, Processor Sharing (PS), etc

  • Arrival process

  • Service statistics



Arrival process l.jpg
Arrival Process

  • : interarrival time between customers n and n+1

  • is a random variable

  • is a stochastic process

  • Interarrival times are identically distributed and have a common mean

  • l is called the arrival rate

Service time process l.jpg
Service-Time Process

  • : service time of customer n at the server

  • is a stochastic process

  • Service times are identically distributed with common mean

  • m is called the service rate

  • For packets, are the service times really random?

Queue descriptors l.jpg
Queue Descriptors

  • Generic descriptor: A/S/m/k

  • A denotes the arrival process

    • For Poisson arrivals we use M (for Markovian)

  • B denotes the service-time distribution

    • M: exponential distribution

    • D: deterministic service times

    • G: general distribution

  • m is the number of servers

  • k is the max number of customers allowed in the system – either in the buffer or in service

    • k is omitted when the buffer size is infinite

Queue descriptors examples l.jpg
Queue Descriptors: Examples

  • M/M/1: Poisson arrivals, exponentially distributed service times, one server, infinite buffer

  • M/M/m: same as previous with m servers

  • M/M/m/m: Poisson arrivals, exponentially distributed service times, m server, no buffering

  • M/G/1: Poisson arrivals, identically distributed service times follows a general distribution, one server, infinite buffer

  • */D/∞ : A constant delay system

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Probability Fundamentals

  • Exponential Distribution

  • Memoryless Property

  • Poisson Distribution

  • Poisson Process

    • Definition and Properties

    • Interarrival Time Distribution

    • Modeling Arrival and Service Statistics

The exponential distribution l.jpg
The Exponential Distribution

  • A continuous RV X follows the exponential distribution with parameter m, if its probability density function is:

  • Probability distribution function:

Exponential distribution cont l.jpg
Exponential Distribution (cont.)

  • Mean and Variance:

  • Proof:

Memoryless property l.jpg
Memoryless Property

  • Past history has no influence on the future

  • Proof:

  • Exponential: the only continuous distribution with the memoryless property

Poisson distribution l.jpg
Poisson Distribution

  • A discrete RV Xfollows the Poisson distribution with parameter l if its probability mass function is:

  • Wide applicability in modeling the number of random events that occur during a given time interval – The Poisson Process:

    • Customers that arrive at a post office during a day

    • Wrong phone calls received during a week

    • Students that go to the instructor’s office during office hours

    • … and packets that arrive at a network switch

Poisson distribution cont l.jpg
Poisson Distribution (cont.)

  • Mean and Variance

  • Proof:

Sum of poisson random variables l.jpg
Sum of Poisson Random Variables

  • Xi , i =1,2,…,n, are independent RVs

  • Xi follows Poisson distribution with parameter li

  • Partial sum defined as:

  • Sn follows Poisson distribution with parameter l

Sampling a poisson variable l.jpg
Sampling a Poisson Variable

  • X follows Poisson distribution with parameter l

  • Each of the X arrivals is of type i with probability pi, i =1,2,…,n, independently of other arrivals;p1+ p2 +…+ pn = 1

  • Xi denotes the number of type i arrivals

  • X1, X2,…Xn are independent

  • Xi follows Poisson distribution with parameter li= lpi

Poisson approximation to binomial l.jpg

Binomial distribution with parameters (n, p)

As n→∞ and p→0, with np=l moderate, binomial distribution converges to Poisson with parameter l


Poisson Approximation to Binomial

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Poisson Process with Rate l

  • {A(t): t≥0} counting process

    • A(t) is the number of events (arrivals) that have occurred from time 0 – when A(0)=0 – to time t

    • A(t)-A(s) number of arrivals in interval (s, t]

  • Number of arrivals in disjoint intervals independent

  • Number of arrivals in any interval (t, t+t] of length t

    • Depends only on its length t

    • Follows Poisson distribution with parameter lt

    • Average number of arrivals lt; l is the arrival rate

Interarrival time statistics l.jpg
Interarrival-Time Statistics

  • Interarrival times for a Poisson process are independent and follow exponential distribution with parameter l

  • tn: time of nth arrival; tn=tn+1-tn: nth interarrival time


  • Probability distribution function

  • Independence follows from independence of number of arrivals in disjoint intervals

Small interval probabilities l.jpg
Small Interval Probabilities

  • Interval (t+ d, t] of length d


Merging splitting poisson processes l.jpg
Merging & Splitting Poisson Processes

  • A1,…, Ak independent Poisson processes with rates l1,…, lk

  • Merged in a single processA= A1+…+ Ak

  • A is Poisson process with ratel= l1+…+ lk





l1+ l2




  • A: Poisson processes with rate l

  • Split into processes A1 and A2 independently, with probabilities p and 1-p respectively

  • A1 is Poisson with rate l1= lpA2 is Poisson with rate l2= l(1-p)

Modeling arrival statistics l.jpg
Modeling Arrival Statistics

  • Poisson process widely used to model packet arrivals in numerous networking problems

  • Justification: provides a good model for aggregate traffic of a large number of “independent” users

    • n traffic streams, with independent identically distributed (iid) interarrival times with PDF F(s) – not necessarily exponential

    • Arrival rate of each stream l/n

    • As n→∞, combined stream can be approximated by Poisson under mild conditions on F(s) – e.g., F(0)=0, F’(0)>0

  • Most important reason for Poisson assumption:Analytic tractability of queueing models

Little s theorem l.jpg
Little’s Theorem

  • l: customer arrival rate

  • N: average number of customers in system

  • T: average delay per customer in system

  • Little’s Theorem: System in steady-state




Counting processes of a queue l.jpg





Counting Processes of a Queue

  • N(t) : number of customers in system at time t

  • (t) : number of customer arrivals till time t

  • b(t) : number of customer departures till time t

  • Ti: time spent in system by the ith customer

Time averages l.jpg

Time average over interval [0,t]

Steady state time averages

Little’s theorem N=λT

Applies to any queueing system provided that:

Limits T, λ, and d exist, and

λ= d

We give a simple graphical proof under a set of more restrictive assumptions

Time Averages

Proof of little s theorem for fcfs l.jpg

Assumption: N(t)=0, infinitely often. For any such t

If limits Nt→N, Tt→T,λt→λexist, Little’s formula follows

We will relax the last assumption

FCFS system, N(0)=0

(t) and b(t): staircase graphs

N(t) = (t)- b(t)

Shaded area between graphs








Proof of Little’s Theorem for FCFS


Proof of little s for fcfs cont l.jpg








Proof of Little’s for FCFS (cont.)

  • In general – even if the queue is not empty infinitely often:

  • Result follows assuming the limits Tt →T, λt→λ, and dt→d exist, and λ=d

Probabilistic form of little s theorem l.jpg
Probabilistic Form of Little’s Theorem

  • Have considered a single sample function for a stochastic process

  • Now will focus on the probabilities of the various sample functions of a stochastic process

  • Probability of n customers in system at time t

  • Expected number of customers in system at t

Probabilistic form of little cont l.jpg
Probabilistic Form of Little (cont.)

  • pn(t), E[N(t)]depend on t and initial distribution at t=0

  • We will consider systems that converge to steady-state

  • there exist pn independent of initial distribution

  • Expected number of customers in steady-state [stochastic aver.]

  • For an ergodic process, the time average of a sample function is equal to the steady-state expectation, with probability 1.

Probabilistic form of little cont33 l.jpg
Probabilistic Form of Little (cont.)

  • In principle, we can find the probability distribution of the delay Tifor customer i, and from that the expected value E[Ti], which converges to steady-state

  • For an ergodic system

  • Probabilistic Form of Little’s Formula:

  • Arrival rate define as

Time vs stochastic averages l.jpg
Time vs. Stochastic Averages

  • “Time averages = Stochastic averages,” for all systems of interest in this course

  • It holds if a single sample function of the stochastic process contains all possible realizations of the process at t→∞

  • Can be justified on the basis of general properties of Markov chains