Section 8 Information Streams and Traffic Processes

Information Streams

and Traffic Processes

Professor Izhak Rubin

Electrical Engineering Department

UCLA

© 2014-2015 by Izhak Rubin

(c) Prof. Izhak Rubin 2

Arrival Point Processes (1) Messages arrive at the system in accordance with a stochastic point

process:

where An denotes the arrival time of nth message, n ≥ 1. We generally set A0=0.

Associated with the arrival process A is the arrival counting process.

where Nt ≡ N(t) denotes the arrival count at t, expressing the number of messages arriving at the system during (0, t].

(1) ,1.. ,1, 1 pwAAnAA nnn

(2) ,0 ,0, 0 NtNN t

N(t)

t A0 A1 A2 A3

1

2

0

N(t)

T2 T3 T1


Inter-Arrival Times and

Renewal Point Processes The inter-arrival times are denoted as

The arrival process A is typically assumed to be a Renewal point

process, under which {Tn, n ≥ 1} is a sequence of independent identically

distributed (i.i.d.) random variables. Such a process is statistically

characterized by the inter-arrival time distribution

We set A(t) = 0, for t ≤ 0.

(3) .1 ,1 nAAT nnn

(4) .tTPtA

t A0 A1 A2 A3

1

2

0

N(t)

T2 T3 T1

A4


Arrival Processes (2) The mean inter-arrival time is denoted as

The message arrival intensity (or rate) is given by

The arrival process is said to be a continuous-time arrival point process if A(t)

is a continuous distribution having the probability density function

a(t) = dA(t)/dt, t ≥ 0.

The arrival process is said to be a discrete-time arrival point process if A(t) is a

discrete distribution, so that arrivals can occur only at times

{tk = kσ+t0, k=0,1,2,…, }.

Slot duration = σ. We normalize the slot time, setting σ = 1. For a discrete-time

arrival process, inter-arrival times are measured in [slots], and the inter-arrival

time distribution is denoted as

)5( .TE

)6( .1

(7) .1 , jjtjTP n


Arrival Processes (3)

A0

A0

A1

A1

A2

A2

A3

A3

A4

A4

(a)

(b)

T1 T3

T4

t

t

Fig. 1. Realizations of (a) continuous-time and (b) discrete-time stochastic

point processes


Arrival Processes (4) For a continuous-time arrival point process, the Laplace

transform for the inter-arrival time p.d.f. a(t) is give by

For a discrete-time point process, the Z-transform of the inter-

arrival time probabilities {t(j), j ≥ 1} is denoted as

)8( 0)Re( ,0

*

sdttaesA st

)9( 1. ,1

*

zzjtzTj

j


Arrival Processes (5) For a discrete-time point process, we set Mk to denote the

number of messages arriving during the kth slot.

The associated counting process is N = {Nk, k = 0,1,2,…,} where N0=0, and

so that Nk expresses the total number of message arrivals during the first k slots.

)10( 1, ,1

kMNk

i

ik


Arrival Processes (6)

0

1

2

3

4

5

6

7

N(t)

t

Fig. 2. A realization of a counting process N = [N(t), t≥0]


Arrival Processes (7) For a discrete-time renewal point process, the

probability distribution of An is given by the

discrete-convolution

1*1 1

1

1

, 1, (15)

, and ; (16)

, 1. (17)

n

n

jn n

i

nj

n

j

P A j t j n

t j t j t j t j i t i

z P A j T z n


Memoryless Stochastic Point

Process

An inter-arrival time variable T is said to be memoryless, if it

satisfies

so that the residual time to next arrival is statistically independent

of the time since the last arrival.

For a memoryless distribution A(t), we obtain from the defining

identity that

, , 0 (18)P T t s T t P T s t s

(19) .0st, ,111 sAtAstA


Memoryless Arrival Processes

The only distributions that solve

the latter identity are given by:

1. The continuous-time exponential

distribution

2. and the discrete-time geometric

distribution.


Counting and Arrival Time

Distributions

where

)10( ,

1

1

1

tFtF

tAPtAP

nNPnNPnNP

nn

nn

ttt

, 1. (11)n nP A t F t n


Arrival Time Distribution for

Renewal Processes For a renewal point process, since

we have

where A(t)*n denotes the nth order convolution of A(t) with itself. For a continuous-time process,

and the p.d.f. is: fn(t)=a(t)*n, where a(t)*1=a(t) and Fn*(s)=[A*(s)]n.

(13) .1 ,

ntAtFn

n

(14) .1 ,)(0

1

ndssastAtA

t nn

(12) .1 ,1

nTAn

j

jn


Poisson Process: Definition and

Properties A Poisson process A={An, n≥0} is a Renewal point process governed by an

exponential inter-arrival time distribution:

where

is the unit step function. For the process we have

Recall that it is a unique (the only one) continuous-time renewal point process which is memoryless.

)15( 1 tUetTPtA t

)16( 0 ,0

0 ,1

t

ttU

)20( .Re ,

(19) ,

)18( ,

)17( ,

2

1

ss

sA

tUeta

TVar

TEα

t


Poisson Process: Arrival

Times The nth arrival occurs at time An and is governed by Erlang distribution

En,λ(t) of order n, n ≥ 1, and parameter λ > 0, and corresponding density

en,λ(t), t ≥ 0:

where the Erlang p.d.f. is given by

We have

)21(

1

0,,

n

setE

tUetAPtF

t

nn

t

nn

)22( .!1

1

, tUn

ette

tn

n

)23( . ; ;2

, n

AVarn

AEs

sE nn

n

n


Poisson Counting Process The counting process N={Nt, t≥0} associated with a Poisson point process is

called: Poisson counting process. The distribution of Nt can be computed by

which yields the result (showing it to be governed by a Poisson distribution):

)24( ,,1,1 tEtEtFtFnNP nnnnt

, 0,1,2,...; 0; (25)!

; t 0; (26)

; t 0. (27)

n

t

t

t

t

tP N n e n t

n

m t E N t

Var N t


Poisson Process as a Process with Stationary

Independent Increments The Poisson counting process N can be shown to

have independent increments, so that the number of arrivals (events) occurring over disjoint intervals are statistically independent.

Furthermore, N has stationary independent increments. An increment is said to be stationary if its distribution depends only on the length of the underlying interval and not on its actual position. Thus, we have:

,

, 0. (28)!

t s h t s

n

h

P N t s t s h n P N N n

hP N h n e n

n


Poisson Process as a Process with Stationary

Independent Increments (cont)

In fact, a counting process N={Nt,t≥0} can be axiomatically defined as a Poisson counting process by imposing the following requirements:

1. N0=0;

2. N has stationary independent increments;

3. Nt is governed by the Poisson distribution

, 0, 0. (29)!

n

t

t

tP N n e n t

n


Geometric Point Process as a

Bernoulli Point Process

The Geometric point process has the

following presentation:

Mn = number of message arrivals during the n-th

slot

{Mn,n≥1} = sequence of i.i.d. binary random

variables, so that

P(Mn = 1)=p, P(Mn = 0)=1- p, each n.


Realization of Geometric Point

Process

t (k) A0 A1 A2 A3

1

2

0

N(t) Nk

T2 T3

M1=1

T1

M2=0 M3=1 M4=1

3

N1=1 N2=1

N3=2

N4=3

Nk = Number of events occurring in k slots (i.e., at time t = k).


Geometric (Bernoulli, Binominal) Process A Geometric (Bernoulli) point process A = {An, n≥1} is a discrete-time renewal

point process for which the inter-arrival time is governed by a Geometric

distribution ( inter-arrival time T is measured in slots):

where 0 < p ≤1. We have

The arrival rate is thus equal to

1

( ) 1 , j=1,2,..., (30)j

t j P T j p p

1

1

2

( ) , 11 1

1, . (31)

PzT z z p

z p

pE T p Var T

p

1

[arrivals/slot] (32)

:

E(M) = 1 * P(M=1) + 0 * P(M=0) = P(M=1) = p [mess/slot].

pE T

Also


Geometric (Bernoulli, Binominal) Process:

Arrival and Counting Variables The time An of the nth arrival is governed by the z-transform [T*(z)]n, which yields

the negative-binomial distribution:

1( ) 1 , , 1, (33)

1

k nn

n

kP A k p p k n n

n

The counting process N = {Nk, k ≥ 0} is also known as a Binomial counting

process, noting that Nk = N(t) at t=k (the k-th time mark) is governed by a

Binomial distribution:

( ) 1 , 0,1,..., (33 )k nn

k

kP N n p p n k b

n

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Section 8 Information Streams and Traffic Processes