# TCOM 501 Networking Theory Fundamentals by utg65734

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```									           TCOM 501:
Networking Theory & Fundamentals

Lecture 2
January 22, 2003
Prof. Yannis A. Korilis

1
2-2   Topics
 Delay in Packet Networks
 Introduction to Queueing Theory

 Review of Probability Theory

 The Poisson Process

 Little’s Theorem

 Proof and Intuitive Explanation
 Applications
2-3   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
2-4   Basic Queueing Model
Buffer     Server(s)

Arrivals                             Departures
Queued   In Service

   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
2-5   Characteristics of a Queue

b        m

 Number of servers m: one, multiple, infinite
 Buffer size b

 Service discipline (scheduling): FCFS, LCFS,

Processor Sharing (PS), etc
Arrival process
Service statistics
2-6   Arrival Process
n 1        n     n 1
n

tn              t

  n : interarrival time between customers n and n+1
  n is a random variable


 { n , n  1} is a stochastic process

Interarrival times are identically distributed and have
a common mean
E[ n ]  E[ ]  1/ l
   l is called the arrival rate
2-7   Service-Time Process
n 1            n        n 1

sn
t

 s n : service time of customer n at the server
 {sn , n  1} is a stochastic process

Service times are identically distributed with common mean
E [ sn ]  E [ s ]  m
   m is called the service rate

 For   packets, are the service times really random?
2-8   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
2-9   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
2-10   Probability Fundamentals

 Exponential Distribution
 Memoryless Property

 Poisson Distribution

 Poisson Process

 Definition and Properties
 Interarrival Time Distribution

 Modeling Arrival and Service Statistics
2-11   The Exponential Distribution
   A continuous RV X follows the exponential distribution
with parameter m, if its probability density function is:
 me m x   if x  0
f X ( x)  
 0         if x  0

Probability distribution function:
1  e m x    if x  0
FX ( x )  P{ X  x}  
 0            if x  0
2-12   Exponential Distribution (cont.)
   Mean and Variance:
1                              1
E[ X ]           , Var( X ) 
m                          m2
Proof:
                         
E[ X ]   x f X ( x ) dx   x m e  m x dx 
0                          0
                 1
  xe m x   
  e m x dx 
0
0                  m
                                                           2              2
E[ X ]   x m e
2           2     mx
dx   x e    2 mx 
 2  xe m x dx        E[ X ] 
0
0
0           m              m2
2       1            1
Var( X )  E[ X 2 ]  ( E[ X ])2                         
m2       m2           m2
2-13   Memoryless Property
   Past history has no influence on the future

P{ X  x  t | X  t}  P{ X  x}

Proof:
P{ X  x  t , X  t} P{ X  x  t}
P{ X  x  t | X  t}                         
P{ X  t}         P{ X  t}
e m ( x t )
  mt  e m x  P{ X  x}
e
   Exponential: the only continuous distribution with the
memoryless property
2-14   Poisson Distribution
   A discrete RV X follows the Poisson distribution with
parameter l if its probability mass function is:
lk
P{ X  k }  e  l        , k  0,1, 2,...
k!
   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
2-15   Poisson Distribution (cont.)
   Mean and Variance
E[ X ]  l, Var( X )  l
Proof:
                                 
lk               
lk
E[ X ]   kP{ X  k }  e             l
 k k !  e  ( k  1)!l

k 0                              k 0                        k 0

lj
 e l
l
 e  l l el  l
j 0    j!
                                     
lk                 
lk
E[ X ]   k P{ X  k }  e
2            2                      l
k         2
e    l
 k ( k  1)!
k 0                                k 0           k!              k 0

lj              
lj                  
lj
 e l  ( j  1)
l
 l  je             l
 le    l
            l2  l
j 0             j!             j 0             j!                j 0   j!
Var( X )  E[ X 2 ]  ( E[ X ])2  l 2  l  l 2  l
2-16   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  X 1  X 2  ...  X n

Sn follows Poisson distribution with parameter l
l  l1  l2  ...  ln
2-17   Sum of Poisson Random Variables (cont.)
P r o o f : For n = 2. Generalizat ion by induc-
t ion. T he pm f of S = X 1 + X 2 is
m
X
P f S = mg =              P f X 1 = k; X 2 = m ¡ k g
k= 0
m
X
=           P f X 1 = kg P f X 2 = m ¡ kg
k= 0
m
X             k
¡ ¸ 1¸ 1     ¡ ¸2     ¸ m¡ k
2
=           e            ¢e
k= 0           k!           ( m ¡ k) !
m
1 X           m!
=     e¡ ( ¸ 1 + ¸ 2 )                  ¸ k ¸ m¡ k
1 2
m ! k= 0 k!( m ¡ k) !
m
¡ ( ¸ 1+ ¸ 2) ( ¸ 1 + ¸ 2)
= e
m!
Poisson w it h param et er ¸ = ¸ 1 + ¸ 2 .
2-18   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
2-19   Sampling a Poisson Variable (cont.)
P r o o f : For n = 2. Generalize by induct ion. Joint pm f:

P f X 1 = k1 ; X 2 = k2 g =
=    P f X 1 = k1 ; X 2 = k2 j X = k1 + k2 g P f X = k1 + k2 g
³k + k ´                                ¸ k 1 + k2
1         2    k 1 k2       ¡ ¸
=                     p1 p2 ¢e
k1                             ( k1 + k2 ) !
1
=               ( ¸ p1 ) k 1 ( ¸ p2 ) k 2 ¢e¡ ¸ ( p1 + p2 )
k 1 !k 2 !
k1                        k2
¡ ¸ p1 ( ¸ p1 )           ¡ ¸ p2 ( ¸ p2 )
=    e                     ¢e
k1 !                      k2 !

² X 1 and X 2 are indep endent
k1                                             k2
² P f X 1 = k1g =     e¡ ¸ p1 ( ¸ k 11!)
p
, P f X 2 = k2 g =   e¡ ¸ p2 ( ¸ k 22!)
p

X i f ollow s Poisson dist ribut ion w it h param et er ¸ pi .
2-20       Poisson Approximation to Binomial
   Binomial distribution with               Proof:
parameters (n, p)                                      n k
P{ X  k }    p (1  p ) n  k
k 
n                                                                 nk
P{ X  k}    p k (1  p )nk          ( n  k  1)...( n  1)n  l   l 
k

                             1  
k                                     k!             n  n
   As n→∞ and p→0, with np=l                 ( n  k  1)...( n  1)n
 1 
moderate, binomial distribution                      nk               n

converges to Poisson with                  l
n

parameter l                                1    e
n
 l
 n
 l
k

 1    1
n

 n
l   lk
P{ X  k }  e
n

k!
2-21   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+] of length 
 Depends only on its length 

 Follows Poisson distribution with parameter l

 l ( l )
n
P{ A(t   )  A(t )  n}  e               , n  0,1,...
n!
Average number of arrivals l; l is the arrival rate
2-22       Interarrival-Time Statistics
   Interarrival times for a Poisson process are independent
and follow exponential distribution with parameter l
tn: time of nth arrival; n=tn+1-tn: nth interarrival time
P{ n  s}  1  e  l s , s  0

Proof:
 Probability distribution function

P{ n  s}  1  P{ n  s}  1  P{ A(tn  s )  A(tn )  0}  1  e  l s

   Independence follows from independence of number of arrivals in
disjoint intervals
2-23   Small Interval Probabilities
   Interval (t+ d, t] of length d
P{ A(t  d )  A(t )  0}  1  ld   (d )
P{ A(t  d )  A(t )  1}  ld   (d )
P{ A(t  d )  A(t )  2}   (d )

Proof:
 ld          ( ld )2
P{ A(t  d )  A(t )  0}  e  1  ld            1  ld   (d )
2
 ld                ( ld )2 
P{ A(t  d )  A(t )  1}  e ld  ld  1  ld             ld   (d )
              2 
1
P{ A(t  d )  A(t )  2}  1   P{ A(t  d )  A(t )  k}
k 0
 1  (1  ld   (d ))  ( ld   (d ))   (d )
2-24       Merging & Splitting Poisson Processes
l1                                                           lp
p
l
l1  l2
1-p
l(1-p)
l2

   A1,…, Ak independent Poisson        A: Poisson processes with rate l
processes with rates l1,…, lk       Split into processes A1 and A2
   Merged in a single process           independently, with probabilities p
A= A1+…+ Ak                          and 1-p respectively
A is Poisson process with rate       A1 is Poisson with rate l1= lp
l= l1+…+ lk                          A2 is Poisson with rate l2= l(1-p)
2-25   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
2-26   Little’s Theorem

l                       N

T
 l: customer arrival rate
 N: average number of customers in system

 T: average delay per customer in system

N  lT
2-27   Counting Processes of a Queue
(t)

N(t)

b(t)

t
 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
2-28        Time Averages
   Time average over interval [0,t]      Little’s theorem N=λT
   Steady state time averages            Applies to any queueing system
provided that:
1 t
Nt     N ( s )ds   N  lim N t       Limits T, λ, and d exist, and
t 0                t 
λ= d
a (t )
lt                  l  lim lt
t                t 
We give a simple graphical proof
1 a (t )                      under a set of more restrictive
Tt            Ti
a (t ) i 1
T  lim Tt
t          assumptions
b (t )
dt                  d  lim d t
t                 t 
2-29       Proof of Little’s Theorem for FCFS
(t)             FCFS system, N(0)=0
(t) and b(t): staircase graphs
N(t)
i                                                       N(t) = (t)- b(t)
Ti
b(t)                               t
S (t )   N ( s )ds
0
T2
T1
t
   Assumption: N(t)=0, infinitely often. For any such t
 (t ) 1 Ti
 (t )
 (t )
1 t
N ( s)ds   Ti   N ( s)ds 
t
0            i 1  t 0             t  (t )
 N t  ltTt

If limits Nt→N, Tt→T, λt→λ exist, Little’s formula follows
We will relax the last assumption
2-30       Proof of Little’s for FCFS (cont.)
(t)

N(t)
i                                         Ti
b(t)

T2
T1

   In general – even if the queue is not empty infinitely often:
b (t )  T 1 t             (t )  T
b                         
b (t )                (t )            (t )                  (t)

 Ti  0 N ( s)ds   Ti  t b (t )  t 0 N ( s)ds  t  (t )
t
1          i          1         i

i 1                 i 1

 d tTt  N t  ltTt
   Result follows assuming the limits Tt →T, λt→λ, and dt→d exist,
and λ=d
2-31   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

pn (t )  P{N (t )  n}

   Expected number of customers in system at t
                      
E [ N (t )]   n.P{N (t )  n}   npn (t )
n 0                   n 0
2-32   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
lim pn (t )  pn , n  0,1,...
t 

   Expected number of customers in steady-state [stochastic aver.]

EN   npn  lim E [ N (t )]
t 
n 0
   For an ergodic process, the time average of a sample function is
equal to the steady-state expectation, with probability 1.
N  lim Nt  lim E[ N (t )]  EN
t     t 
2-33   Probabilistic Form of Little (cont.)
   In principle, we can find the probability distribution of the delay
Ti for customer i, and from that the expected value E[Ti], which

ET  lim E[Ti ]
i 

   For an ergodic system


Ti
T  lim      1
 lim E[Ti ]  ET
i    i           i 

Probabilistic Form of Little’s Formula: EN  l.ET
Arrival rate define as
E[ (t )]
l  lim
t     t
2-34   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
2-35   Moment Generating Function
1. D e¯ nit ion: for any t 2 IR:
8 Z1
>
>        et x f X ( x ) dx ;   X cont inuous
<
tX       X¡ 1
M X ( t) = E [e ] =
>
>       et x j P f X = x j g ; X discret e
:
j

2. If t he m om ent generat ing funct ion M X ( t) of X
exist s and is ¯ nit e in som e neighb orhoo d of t = 0,
it det erm ines t he dist ribut ion of X uniquely.

3. Fundam ent al P rop ert ies: f or any n 2 IN :
dn
( i)     n
M X ( t ) = E [X n et X ]
dt
dn
( ii)       M X ( 0) = E [X n ]
dt n

4. M om ent Generat ing Funct ions and Independence:
X ; Y : independent )                 M X + Y ( t ) = M X ( t) M Y ( t )
T he opp osit e is not t rue.
2-36      Discrete Random Variables

Distribut ion       Prob. Mass Fun.               Moment Gen. Fun.        Mean     Variance
(paramet ers)          P f X = kg                     M X (t)             E [X ]   Var(X )
¡ n¢
Binomial         k pk (1 ¡ p) n¡ k              (pet + 1 ¡ p) n        np      np(1 ¡ p)
(n; p)          k = 0; 1; : : : ; n

pet            1         1¡ p
Geometric            (1 ¡ p) k ¡ 1 p                 1¡ (1¡ p)et        p          p2
p                 k = 1; 2; : : :
³          ´                         h               ir
k¡ 1        r          k¡ r              pet            r       r (1¡ p)
Negative Bin.        r¡ 1p (1 ¡ p)                        1¡ (1¡ p)et        p          p2
(r ; p)           k = r; r + 1; : : :

¡ ¸ ¸k                     ¸ (et ¡ 1)
Poisson                   e k!                       e                 ¸          ¸
¸                k = 0; 1; : : :
2-37     Continuous Random Variables

Dist ribution   Prob. Density Fun.      Moment Gen. Fun.          Mean     Variance
(paramet ers)         f X (x)               M X (t)               E [X ]   Var(X )

1                  et b ¡ et a          a+ b     (b¡ a) 2
Uniform over             b¡ a                  t(b¡ a)              2          12
(a; b)           a< x < b

Exponent ial           ¸ e¡ ¸ x                  ¸
¸¡ t
1
¸
1
¸
¸                  x¸ 0
2   2                    2
Normal        p 1 e¡ (x ¡ ¹ ) =2¾       e¹ t + (¾t)       =2
¹         ¾2
2¼¾
2        ¡ 1 < x< 1
(¹ ; ¾ )

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