Post on 31-Dec-2016
1
TCOM 501: Networking Theory & Fundamentals
Lecture 2January 22, 2003
Prof. Yannis A. Korilis
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 linkFocus: Queueing & Transmission Delay
2-4 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
Arrivals Departures
Buffer Server(s)
Queued In Service
2-5 Characteristics of a Queue
Number of servers m: one, multiple, infinite Buffer size b Service discipline (scheduling): FCFS, LCFS, Processor Sharing (PS), etcArrival processService statistics
mb
2-6 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
is called the arrival rate
n 1n 1n n
nt t
n
n{ , 1}n n
[ ] [ ] 1/nE E
2-7 Service-Time Process
: service time of customer n at the server is a stochastic process
Service times are identically distributed with common mean
is called the service rate
For packets, are the service times really random?
n 1n 1n
ns
t
ns{ , 1}ns n
[ ] [ ]nE s E s
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 , if its probability density function is:
Probability distribution function:1 if 0( ) { }
0 if 0
x
X
e xF x P X x
x
if 0( )
0 if 0
x
X
e xf x
x
2-12 Exponential Distribution (cont.) Mean and Variance:
Proof: 2
1 1[ ] , Var( )E X X
0 0
0 0
2 2 20 20 0
2 22 2 2
[ ] ( )
1
2 2[ ] 2 [ ]
2 1 1Var( ) [ ] ( [ ])
xX
x x
x x x
E X x f x dx x e dx
xe e dx
E X x e dx x e xe dx E X
X E X E X
2-13 Memoryless Property Past history has no influence on the future
Proof:
Exponential: the only continuous distribution with the memoryless property
{ | } { }P X x t X t P X x
( )
{ , } { }{ | }{ } { }
{ }x t
xt
P X x t X t P X x tP X x t X tP X t P X t
e e P X xe
2-14 Poisson Distribution A discrete RV X follows the Poisson distribution with
parameter 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
{ } , 0,1,2,...!
k
P X k e kk
2-15 Poisson Distribution (cont.) Mean and Variance
Proof: [ ] , Var( )E X X
0 0 0
0
2 2 2
0 0 0
2
0 0 0
2 2 2
[ ] { }! ( 1)!
!
[ ] { }! ( 1)!
( 1)! ! !
Var( ) [ ] ( [ ])
k k
k k k
j
j
k k
k k k
j j j
j j j
E X kP X k e k ek k
e e ej
E X k P X k e k e kk k
e j je ej j j
X E X E X
2
2-16
Sum of Poisson Random Variables Xi , i =1,2,…,n, are independent RVs Xi follows Poisson distribution with parameter i
Partial sum defined as:
Sn follows Poisson distribution with parameter
1 2 ...n nS X X X
1 2 ... n
2-17
Sum of Poisson Random Variables (cont.)P roof: For n = 2. Generalization by induc-tion. T he pmf of S = X 1 + X 2 is
P f S = mg =mX
k=0P f X 1 = k;X 2 = m ¡ kg
=mX
k=0P f X 1 = kgP f X 2 = m ¡ kg
=mX
k=0e¡ ¸1¸k1
k! ¢e¡ ¸2 ¸m¡ k2
(m ¡ k)!
= e¡ (¸1+¸2) 1m!
mX
k=0
m!k!(m ¡ k)!¸
k1¸m¡ k2
= e¡ (¸1+¸2) (¸1 + ¸2)m
m!Poisson with parameter ¸ = ¸1 + ¸2.
2-18 Sampling a Poisson Variable X follows Poisson distribution with parameter 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 independentXi follows Poisson distribution with parameter ipi
2-19
Sampling a Poisson Variable (cont.)P roof: For n = 2. Generalize by induction. J oint pmf:
P f X 1 = k1;X 2 = k2g == P f X 1 = k1;X 2 = k2jX = k1 + k2gP f X = k1 + k2g=
³ k1 + k2k1
´pk1
1 pk22 ¢e¡ ¸ ¸k1+k2
(k1 + k2)!= 1
k1!k2!(¸p1)k1(¸p2)k2 ¢e¡ ¸(p1+p2)
= e¡ ¸p1 (¸p1)k1
k1! ¢e¡ ¸p2 (¸p2)k2
k2!
² X 1 and X 2 are independent
² P f X 1 = k1g = e¡ ¸p1 (¸p1)k1
k1! , P f X 2 = k2g = e¡ ¸p2 (¸p2)k2
k2!
X i follows Poisson distribution with parameter ¸pi.
2-20
Poisson Approximation to Binomial
Binomial distribution with parameters (n, p)
As n→∞ and p→0, with np= moderate, binomial distribution converges to Poisson with parameter
Proof:
{ } (1 )k n knP X k p p
k
{ } (1 )
( 1)...( 1) 1
( 1)...( 1) 1
1
1 1
{ }!
!
k n k
n k
nk
n
n
k
n
k
k
n
nP X k p p
k
n k n nn n
n k n nn
en
n
Pk
k
X k e
2-21 Poisson Process with Rate {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
Average number of arrivals ; is the arrival rate
( ){ ( ) ( ) } , 0,1,...!
n
P A t A t n e nn
2-22 Interarrival-Time Statistics Interarrival times for a Poisson process are independent
and follow exponential distribution with parameter tn: time of nth arrival; n=tn+1-tn: nth interarrival time
{ } 1 , 0snP s e s
Proof: Probability distribution function
Independence follows from independence of number of arrivals in disjoint intervals
{ } 1 { } 1 { ( ) ( ) 0} 1 sn n n nP s P s P A t s A t e
2-23 Small Interval Probabilities Interval (t+ , t] of length
{ ( ) ( ) 0} 1 ( ){ ( ) ( ) 1} ( ){ ( ) ( ) 2} ( )
P A t A tP A t A tP A t A t
Proof:2
2
1
0
( ){ ( ) ( ) 0} 1 1 ( )2
( ){ ( ) ( ) 1} 1 ( )2
{ ( ) ( ) 2} 1 { ( ) ( ) }
1 (1 ( )) ( ( )) ( )k
P A t A t e
P A t A t e
P A t A t P A t A t k
2-24
Merging & Splitting Poisson Processes
A1,…, Ak independent Poisson processes with rates 1,…, k
Merged in a single processA= A1+…+ Ak A is Poisson process with rate= 1+…+ k
A: Poisson processes with rate Split into processes A1 and A2
independently, with probabilities p and 1-p respectivelyA1 is Poisson with rate 1= pA2 is Poisson with rate 2= (1-p)
p
(1-p)
p
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 nAs 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
: customer arrival rate N: average number of customers in system T: average delay per customer in system
Little’s Theorem: System in steady-state N T
N
T
2-27 Counting Processes of a Queue
N(t) : number of customers in system at time t (t) : number of customer arrivals till time t (t) : number of customer departures till time t Ti : time spent in system by the ith customer
(t)
N(t)
t
(t)
2-28 Time Averages 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 exist, and λ=
We give a simple graphical proof under a set of more restrictive assumptions
0
( )
1
1 ( ) lim
( ) lim
1 lim( )( ) lim
t
t tt
t tt
a t
t i tti
t tt
N N s ds N Nta t
t
T T T Ta t
tt
2-29
Proof of Little’s Theorem for FCFS
Assumption: N(t)=0, infinitely often. For any such t
If limits Nt→N, Tt→T, λt→λ exist, Little’s formula followsWe will relax the last assumption
FCFS system, N(0)=0(t) and (t): staircase graphs N(t) = (t)- (t)Shaded area between graphs
t
0( ) ( )
tS t N s ds
(t)
T1
N(t)
T2
Tii
(t)
( )
1
( )
0 01
1 ( )( ) ( )( )
t
itt t
i t t ti
TtN s ds T N s ds N Tt t t
2-30 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 t→ exist, and λ=
(t)
T1
N(t)
T2
Tii
(t)
( ) ( )
1 1
( ) ( )
0 01 1
( ) 1 ( )( ) ( )( ) ( )
t t
i it tt t
i ii i
t t t t t
T Tt tT N s ds T N s dst t t t t
T N T
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
Expected number of customers in system at t
( ) { ( ) }np t P N t n
0 0
[ ( )] . { ( ) } ( )nn n
E N t n P N t n np t
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
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.
lim ( ) , 0,1,...n ntp t p n
0
lim [ ( )]n tn
EN np E N t
lim lim [ ( )]tt tN N E N t EN
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 converges to steady-state
For an ergodic system
Probabilistic Form of Little’s Formula: Arrival rate define as
lim [ ]iiET E T
1lim lim [ ]i
ii i
TT E T ET
i
.EN ET
[ ( )]limt
E tt
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 Function1. De¯nition: for any t 2 IR:
M X (t) = E [etX ] =
8>><>>:
Z 1
¡ 1etxf X (x) dx; X continuous
Xj
etxj P f X = xj g; X discrete
2. If the moment generating function M X (t) of Xexists and is ¯ nite in some neighborhood of t = 0,it determines the distribution of X uniquely.
3. Fundamental P roperties: for any n 2 IN :
(i) dn
dtnM X (t) = E [X netX ]
( ii) dn
dtnM X (0) = E [X n]
4. M oment Generating Functions and Independence:X ;Y : independent ) M X +Y (t) = M X (t)M Y (t)
T he opposite is not true.
2-36 Discrete Random VariablesDistribution Prob. Mass Fun. Moment Gen. Fun. Mean Variance(parameters) P fX = kg MX (t) E [X ] Var(X )
Binomial ¡ nk¢pk(1¡ p)n¡ k (pet +1¡ p)n np np(1¡ p)
(n;p) k = 0;1;: ::;n
Geometric (1¡ p)k¡ 1p pet1¡ (1¡ p)et
1p
1¡ pp2
p k = 1;2;:::
NegativeBin.³ k¡ 1
r¡ 1´pr (1¡ p)k¡ r
h pet1¡ (1¡ p)et
i r rp
r(1¡ p)p2
(r;p) k = r;r + 1;:::
Poisson e¡ ¸ ¸ kk! e (et ¡ 1) ¸ ¸
¸ k = 0;1;:::
2-37 Continuous Random VariablesDistribution Prob. Density Fun. Moment Gen. Fun. Mean Variance(parameters) f X (x) MX (t) E [X ] Var(X )
Uniform over 1b¡ a
etb¡ etat(b¡ a)
a+b2
(b¡ a)212
(a;b) a < x < b
Exponential ¸e¡ ¸x ¸¸¡ t
1¸
1¸
¸ x ¸ 0
Normal 1p2¼¾e¡ (x¡ ¹ )2=2¾2 e¹ t+(¾t)2=2 ¹ ¾2
(¹ ;¾2) ¡ 1 < x < 1