Tuyatsetseg Badarch, Otgonbayar Bataa Mongolian University of Science and Technology On Network Traffic Modeling Framework: A Case Study with Public Safety

Tuyatsetseg Badarch, Otgonbayar Bataa

Mongolian University of Science and Technology

On Network Traffic Modeling Framework:

A Case Study with Public Safety Network

Contents “Framework on network traffic probability

models”

Recent open issues for future research: “Framework on network traffic probability models”

1. Part I. Probability model: Simple standard or mixed advanced? How to overcome a spike and tail

problem? Robust method? How we develop the good methods?Results of Public Safety Network

2. Part II. Probability model of blocking/delay: PSTN-Erlang formula or IP Erlang formula for loss and delay? An

appropriate algorithm for IP Erlang formula? Good methods! Results of Public Safety Network

3. Part III. Traffic and Probability model connected to capacity and network planning

GoS and capacity of a system? An appropriate algorithm for GoS, system capacity planning? Results of Public Safety Network

4. Results and the model validations in the case of Public Safety Network

PART I. Research Trends on Network Traffic Fitting Models

Mixed models as traffic advanced models

(mixed lognormals, mixed pareto, mixed erlangs, pareto plus

lognormals….)

Standard models as traffic candidate models

(Lognormal model, Shifted Lognormal, Pareto model, Erlang model, Weibull model)

On the world, there are 60 more standard models

But for network traffic fitting model, these above named are exact candidates.

20 30 40 50 60 70 80 90 100 110 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Average Holding Time(secs)

Pro

babi

lity

Den

sity

Weibull function approximation

Lognormal function approximationShifted Lognormal func. approx

… … …Big issues on a spike, a tail

… … …Studies are going on

Holding time (seconds)

Pro

ba

bility d

en

sity fu

nctio

n

37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

0.0

00

.02

0.0

40

.06

0.0

8

Histogram

PART I. My Research work on this Network Traffic Fitting Models

Mixed Advanced models(Mixed lognormals, mixed pareto, mixed erlangs, pareto

plus lognormals….)

Standard models as candidates (Lognormal model, Shifted Lognormal, Pareto model,

Erlang model, Weibull model)

20 30 40 50 60 70 80 90 100 110 120

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


Pro

babi

lity

Den

sity




Pro

ba

bility d

en

sity fu

nctio

n

37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

0.0

00

.02

0.0

40

.06

0.0

8

… … …Studies are going on… … … ….

Big issues on a spike, a tail

1. B. Tuyatsetseg, “Parametric-Expectation Maximization Approach for Call Holding Times of IP enabled Public Safety Networks.” WASET, Zurich, Switzerland, [online]. issue 73, pp. 568-575, Jan 2013.

1. B. Tuyatsetseg, B. Otgonbayar, “Modeling Call Holding Times of Public Safety Network”, International Journal on Computational Science & Applications (IJCSA), Australia, vol. 3, no. 3, pp. 1-19, June 2013.

2. B.Tuyatsetseg, B.Otgonbayar, “Traffic Modeling of Public Safety Network”, in Proc IEEE-APNOMS2013, Hiroshima , Japan. 25-27, Sep 2013

The algorithm of my proposed mixed lognormal model, its

performance on the Emergency Traffic

Initialize parameters2. First step:evaluate3. Second step : maximize

where4. Evaluate log likelihood, once likelihoods or

parameters converge, the algorithm is done: Else

old | , oldp z x

arg max ( , )new oldQ

( , ) ( | , ) ( , | )old old

complete

Q p z x Lnp x z

old new

Hidden traffic variables plus observed variables

Likelihood of lognormals(ex.values)

(3)

ALGORITHM OUTPUT: weights, scale,

location parameters of all components

Traffic in Erlangs

Pro

bab

ility

De

nsity

0 1 2 3 4 5 6 7 8 9 10 11 12 130

.00

.10

.20

.30

.40

.5

Histogram

Mixed LogNormal-5 modelMixed LogNormal-4 modelMixed LogNormal-3 model

ALGORITHM INPUT: the traffic values based on experiment or simulation

new

Traffic in Erlangs

Pro

ba

bility D

en

sity

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.0

0.1

0.2

0.3

0.4

0.5

( , , )old

( , , )newk k k k

Evaluate ex.values (3) using (1), (2)

Traffic random

variables for

Iteratively maximize (3)

Challenging steps to develop a mixture lognormal model

1. MLE (3 parameters: weight, location and scale) through maximum log likelihood function.

2. The marginal distribution is described: through Bayesian theorem to compute the expression of unobserved traffic data that is iterative process untill likelihood or parameters converge

3. The expected value of complete traffic data set log likelihood through both the function of log likelihood and the marginal distribution.

Log likelihood computation of a mixture of lognormal parametric

model

Logn fn:

Likelihood function:

These formulas were proved mathematically in my previous journal publications.

11 1

2

211

( ) ( , ) ( )

( )exp( )

2 2

ij j

ij

n n k

i i j y iji i

yn k

j i j

i j jji

L z L x y I x

Lnx

x y

22

1 1

1 1[ ( )] [ 2 ( ) ]

2 2ij

k n

y j j i i jjj i

Ln L z I nLn nLn Ln Lnx Lnx

Loglikelihood function (to estimate the set of parameters for the mixed lognormal distributions that maximize the likelihood function.

…comparison on log likelihoods

MLE for simple lognormal distribution

MLE for a mixture of lognormal distributions:

where indicator function for individual i that comes from the

lognormal component with

2

21

22

1 1

( )1( ( )) exp

2

1 12 ( )

2 2

ni

xii

n n

i ii i

LnxLnL x Ln

x

nLn Ln Lnx Lnx

22

1 1

( )

1 1( ) 2 ( )

2 2ij

n k

y j j i i jji j

Ln L z x y

I Ln Ln Lnx Ln Lnx

j

(1a)

The formula was proved mathematically in my previous journal publications.

MLE parameters computation of a mixture of lognormals

The weights of a lognormal components (number of density in j divided by total number of density)

The means of a lognormal components (for each density i, its assignment mean to each j lognormal components)

The variance of lognormal components

(for each density I, its assignment variance to each j

lognormal components)

(1)


Marginal distribution computation

(2)

Bayesian analytical expression (2) is used to compute the marginal distribution, (posterior

probability) of on and y

x t

evaluated at ix( )ix


Expected value computation

(3)

is the current value at iteration , for

The expected value (3) of the log likelihood is iterative, as a function of Bayesian and log

likelihood function for z

z

t t tj

1 1

1 1 1 1

22

( , ) ( ) ( )

( ) ( ) [

1 12 ( ) ] ( )

2 2

j

k nt t

j i j ij i

k n k nt

j j i j i jj i j i

ti j j i

j

q Ln x x

Ln x Ln Lnx Ln

Ln Lnx x

Marginal distribution

(2)

Log likelihoods

of lognormals

(1a)


The implementation algorithm performance , explanation by

formulas

Compute (3) ex.value using (1) weights or mixing. coeff (2) posterior prob.

Iteratively maximize (3) independently

This implementation algorithm is used in emergency incoming traffic simulation , with results in next pages.

The mixed model of traffic, its lognormal components of the proposed algorithm

1 1

( ) ( ) ( | , )j

k k

x j j k kj j

x x N x

MLE conditions:

Location (mean) shape (var)k R 0k

CHALLENGE: First of all, we should prove the exact formulas for MLE (weight, location, scale/shape)

of the mixed log function

( , , )k k k k


Pro

ba

bility d

en

sity fu

nctio

n

37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

0.0

00

.02

0.0

40

.06

0.0

8

( , , )k k k k

1

1 (0.07284255 +0.52619736 +0.34874273 +0.04059312+ 0.01162424)k

kj

(55.12360 , 61.10566 , 60.94680 , 65.46325 , 93.07292)k R

0 (0.6478691, 8.4770634, 8.4545028 ,1.1604390 ,1.6498112)k

Traffic in Erlangs

Pro

ba

bility D

en

sity

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.0

0.1

0.2

0.3

0.4

0.5

Histogram


…. PART I. My research results:

Mixed Advanced models(Mixed lognormals, mixed pareto, mixed erlangs, pareto

plus lognormals….)

Standard models as candidates (Lognormal model, Shifted Lognormal, Pareto model,

Erlang model, Weibull model)

20 30 40 50 60 70 80 90 100 110 1200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


Pro

babi

lity

Den

sity




Pro

ba

bility d

en

sity fu

nctio

n

37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

0.0

00

.02

0.0

40

.06

0.0

8


Big issues on a spike, a tail

1. B. Tuyatsetseg, “Parametric-Expectation Maximization Approach for Call Holding Times of IP enabled Public Safety Networks.” WASET, Zurich, Switzerland, [online]. issue 73, pp. 568-575, Jan 2013.

1. B. Tuyatsetseg, B. Otgonbayar, “Modeling Call Holding Times of Public Safety Network”, International Journal on Computational Science & Applications (IJCSA), Australia, vol. 3, no. 3, pp. 1-19, June 2013.

2. B.Tuyatsetseg, B.Otgonbayar, “Traffic Modeling of Public Safety Network”, in Proc IEEE-APNOMS2013, Hiroshima , Japan. 25-27, Sep 2013

The scenario of the existing Public Safety Network in Mongolia (EIN), its traffic mixed

model results

Mass traffic influences to emergency incoming traffic :

Ambulance(103) Police (102) Fire (101) Hazard (105)

The comparison on TRAFFIC pattern of Mongolian EIN, some simple statistics

Total 60 channels equally distributed but traffic has a huge difference.

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

Period in hours

Nor

mal

ized

traf

fic

in E

rlan

gs

Comparison on Emergency incoming traffic patterns

"103":Ambulance traffic"102":Police traffic"101":Fire traffic"105":Hazard traffic

Emergency mass traffic:

Ambulance call traffic (103): 57%

10275.45 seconds seamless

Traffic value : 1.3 Erlangs

Police call traffic (102):

38%


Traffic value: 0.7 Erlangs

Fire call traffic (101):

3.4 %



Hazard call traffic (105) :

2.6 %



for Mongolian biggest holiday and the continuous peak period traffic.

Part I. Numerical results of traffic mixed model:

Results on Emergency Total traffic, Ambulance, Police, Fire, and Hazard incoming traffic

Traffic in Erlangs

Pro

ba

bili

ty D

en

sity

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.0

0.1

0.2

0.3

0.4

0.5

Histogram


Police call traffic in Erlangs

Pro

ba

bili

ty D

en

sity

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

HistogramMixed LogNormal-5 modelMixed LogNormal-4 modelMixed LogNormal-3 model

Fire call traffic in Erlangs

Pro

ba

bili

ty D

en

sity

0 1 2 3 4 5 6 7 8 9 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

HistogramMixed LogNormal-5 modelMixed LogNormal-4 modelMixed LogNormal-3 model

Traffic in erlangs

Pro

ba

bili

ty D

en

sity

0 1 2 3 4 5 6 7 8 9 11 13 15 17 190

.00

0.0

50

.10

0.1

50

.20

0.2

50

.30

Histogram of Emergency total traffic

Traffic ranges, weight values

I. (1.030081, 0.1708236) II. (2.036909, 0.2698571) III. (4.885861, 0.4124513)

IV. (12.609390, 0.1468680)

Peak Traffic pattern of Mongolian EIN

I

II

III

IV

IV. (variances: 0.2436641,0.6226421,1.6315808,4.6597802)

0 20 40 60 80 100 120 140 160 1800

0.2

0.4

0.6

0.8

1

1.2

1.4

Period in hours

Nor

mal

ized

traf

fic

in E

rlan

gs

Comparison on Emergency incoming traffic patterns

"103":Ambulance traffic"102":Police traffic"101":Fire traffic"105":Hazard traffic

Time series model

Probability model

The model validation via Quantile-Quantile (Q-Q):

Results on Emergency incoming traffic

On Body/Head fitting validation

On Tail validation,

0 0.5 1-6

-4

-2

0

Normalized traffic in Erlangs

CC

DF

"103": LogNormal-5 P-P plot

General Pareto modelEmpirical dataLogNormal-5 model

0 0.5 10

0.2

0.4

0.6

0.8

1


CD

F



0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

Normalized traffic in erlang

CD

F



0 0.2 0.4 0.6 0.8-6

-4

-2

0


CC

DF



0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

Normalized traffic in Erlangs)

CD

F



0 0.2 0.4 0.6 0.8-6

-4

-2

0


CC

DF



PART II. Research Trends on Network probability of loss/delay

for channels, links

Erlang Traffic analysis for Video over IP

(Robust probability model of loss/delay , its development??)

Standard Erlang Traffic analysis : (Probability of Loss

Probability of Delay) for wired and wireless conventional voice

communications

… … …for b=0:(z-1)factorial = prod(1:b);t = ((T)^b)/factorial;s=s + t; end … … … ….


PART II. Research on Network probability of blocked/delay for

channels, links

Erlang Traffic analysis for Video over IP

(Robust probability model of loss/delay , its development??)

This is one result of my research on the Video over IP transfer

Standard Erlang Traffic analysis: (Probability of Loss

Probability of Delay) for wired and wireless conventional voice communications

… … …for b=0:(z-1)factorial = prod(1:b);t = ((T)^b)/factorial;s=s + t; end … … … ….

Studies are going on

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Link Load [%]

Pro

b of

Del

ay o

f C

CT

V-I

P v

ideo

flo

ws

Link speed : 8 Mbps, CBR:by IP formula and simulation

Link speed : 2, 4, 8 Mbps,by IP VBR method

Link speed : 2 Mbps,by IP formula and simulation


Link speed : 4 Mbps,CBR:by IP formula and simulation

Why the Video over IP model? it’s application of Public Safety Network

Video surveillance based on real time monitoring using video camera for crime prevention.

The real time video delivering to the emergency center through IP network.

This Video over IP transfer is the exact one MASS Traffic producer in the case of PSN. For this reason, we have to model the exact computation method, algorithm to evaluate the performance of this Video over IP traffic.

Hence the modeling of video over IP is the one main component of the framework.

Part II. Blocking/Delay probability analysis on Voice traffic to IP video

traffic

Voice Traffic: Probability of Loss and Probability of

Delay The average holding time The call arrival rate The total number of

available channel The total number of calls

Blocking and delay probability depends on

erlang load

IP Video Traffic : Probability of Loss and

Probability of Delay Average transfer time per frame

in unit Number of frames per unit, ( for

ex, 25-30 frames in one sec for MPEG-4)

Packet frame arrival rate per unit Packet frame transfer (Service)

rate per unit All resource utilization or traffic

intensity for resources

Blocking and delay probability depends on erlang load

Experiment

Simulation

Computation

Simulation

Computation

Experiment

?

Video over IP Traffic Results

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Link Traffic [%]

Prob

of

Los

s of

CC

TV

-IP

vide

o fl

ows

1 Mbps2 Mbps4 Mpbs8 Mbps16 Mbps32 Mbps64 Mbps70 Mbps90 Mbps100 Mbps

0 500 1000 15000

0.2

0.4

0.6

0.8

1

CCTV video packet frame transfer rate [min]

Prob

of

loss

of

CC

TV

vid

eo p

acke

ts

1 Mbps2 Mbps4 Mbps 8 Mbps 16 Mbps32 Mbps64 Mbps70 Mbps90 Mbps100 Mbps

0 500 1000 15000

0.2

0.4

0.6

0.8

1

Packet frame transfer rate [f/min]

Prob

of

Del

ay o

f C

CT

V-I

P vi

deo

flow

s

1 Mpbs2 Mbps4 Mbps8 Mbps16 Mbps32 Mbps64 Mbps70 Mbps90 Mbps100 Mbps

0

[( / !) / ( )] /

/ ! ( / !) / ( )

Ndelay c c

Nk Nc c c

k

P E N N N E

E k E N N N E

0

[ / !] / / !N

N kloss c c

k

P E N E k

Loss of the video over IP depends on traffic

Delay of the video over IP depends on traffic

Results: VBR method is most appropriate method than CBR for CCTV-Video over IP traffic loss/delay

using

Appropriate

formulas

for the propose

d analysis

Video over IP Traffic simulation verification , Video over IP-Traffic loss

analysis

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Link Load [%]P

rob

of D

elay

of

CC

TV

-IP

vid

eo f

low

s


Link speed : 2, 4, 8 Mbps,by IP VBR method

Link speed : 2 Mbps,by IP formula and simulation


Link speed : 4 Mbps,CBR:by IP formula and simulation

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Link Traffic [%]

Prob

of

Del

ay o

f C

CT

V-I

P vi

deo

flow

s

1 Mbps2 Mbps4 Mbps8 Mbps16 Mbps32 Mbps64 Mbps70 Mbps90 Mbps100 Mbps

Results: VBR method is the most appropriate method to transfer video over IP than CBR for

CCTV-Video over IP traffic loss/delay

Part III. On Traffic Capacity Region and network planning

Fundamental theories for capacity region and network planning:

Queue theory , Markob theory , Erlang-C/Erlang-B theoryFundamental parameters for capacity region and network

planning: Call Conversation Time, Arrival rate , Service rate, Probability models (optimum traffic model parameters,

blocking and delay model parameters) Described parameters for Network capacity and planning: Resource utilization, Utilization efficiency, Traffic load, Number of links of queuing system, Bandwidth, Grade of Service ,

The algorithm steps for a network traffic - capacity region and network

planning

10-1

100

101

102

10-3

10-2

10-1

100

Ø óóðõàé äóóäëàãóóäûí à÷ààëàë (Ýðëàí ã)

äóóä

ëàãà

ñàà

òàõ

ìàã

àäëà

ë áó

þó

á¿õ

øóã

àì çà

âã¿é

áàé

õ ì

àãàä

ëàë

Õî ëáî õ ø óãàì ûí òî î N=1 2 3 5 10 15 20 50 100

103-ûí GoS äóí äàæ ò¿âø èí

I ì óæ

II ì óæ

III ì óæ

E=7.968Ýðëàí ãN=15 õî ëáî õø óãàìGoS=0.0187ì àãàäëàëòàé.

Ì àõà÷ààëàëE=39.336N=40GoS=0.8786

10-1

100

101

102

10-3

10-2

10-1

100

À÷ààëàë (Ýðëàí ã)

Õàà

ëòòà

é ñó

âãèé

í ì

àãàä

ëàë

Ñóâãèéí òî î N=1 2 3 5 10 15 17 50 100

Î ðãèëà÷ààëàëE=11N=17GoS=0.0665

Äóí äàæ/à÷:E=7.968 N=15 GoS=0.0187

Ì àõà÷àààëàë:E=39.336N=40GoS=0.8786

III á¿ñ

Minà÷ààëàëE=0.447N=3GoS=0.012

II á¿ñ

I á¿ñ

"103": GoS äóí äàæò¿âø èí

The base chart for the traffic capacity region.

(Call Holding Time per call, Number of calls per unit time, …Arrival rate , Service rate, Resource utilization, utilization efficiency,

Traffic model parameters, number of links of queue system …) Probability of waiting , Grade of Service

(GoS) should less than 1.)

Second , third stages …….. based on Erlang load and probability of delay … other parameters

Final stage

First stage

( )( ) 1 c c thN E tth cGoS t p e

tht

RESULT ON EMERGENCY NETWORK CAPACITY REGION

Peak/Congested period analysis:

As a result of robust algorithm, we can get a chance to see Erlang capacity region for the system capacity and the net planning:

Pc - Prob of delay E - Average of Traffic N - Number of

emergency links/agent GoS - Grade of Service (GoS) , prob of

waiting less than threshold level, secs

10-1

100

101

102

10-3

10-2

10-1

100

Traffic intensity (Erlang)

GoS

Link numbers N=1 2 3 5 10 15 20 50 100

IIIregion

IIregion

Iregion

"103": GoS average level

AverageTraffic:E=7.968 N=15

GoS=0.0887

MinTraffic:E=0.447N=3GoS=0.012

ÌàõTraffic:E=39.336N=40GoS=0.8786

Model validation by four parameters (K-S test, pdf, cdf,

ccdf)

1. Probability pdf results, parameters, weight coeff,

2. Head/body behavior results (cdf),3. Tail behavior results (ccdf),2. Kolmogorov Smirnov test (K-S), 3. Miminum D.max values (distance between

proposed method and real data),4. Minimum Error (the error of the model),5. Convergence of the algorithm.

Results on more validation parameters

(Emergency Ambulance case)

Models/parameters

Dmax Error P value (K-S)

P value (Chi-Sq)

Mixed Lognormal

0.00768755

0.46*(10-4)

0.98 0.87

Gen pareto

0.048 0.37*(10-3)

0.884 0.71

Shifted /Ln

0.059 0.18*(10-3)

0.832 0.64

Lognormal

0.061 0.07*(10-3)

0.73 0.61

Weibull 0.07 0.19*(10-3)

0.39 0.27

On contribution of my research

This research’s contribution is complicated fundamental computational complicated tasks , as well as the implementation algorithm development, the study of Public Safety Network in Mongolia

For me, 1. Mathematical formulas were proved and verified and then published in

Switzerland and Australian computational journals in 2013 [1], [2], [3], [5]2. In this conference, the traffic model framework. case study with Public Safety

Network .The base algorithms were done for this process. Results were verified, 3. The contribution may be for fundamental area as well for network performance

analysis and special application area. Also the algorithm of the approaches may be used in networks.

4. It may be one base method for dynamic bandwidth method/bandwidth provisioning because the model described traffic in detailed manner with main functions.

References

B. Tuyatsetseg, “Parametric Modeling Approach for Call Holding Times of IP based Public Safety Networks.” WASET. Switzerland, [online journal]. issue 73, pp. 568-575, Jan 2013.

B. Tuyatsetseg, B. Otgonbayar, “Modeling Call Holding Times of Public Safety Network”, International Journal on Computational Science & Applications (IJCSA), Australia, vol. 3, no. 3, pp. 1-19, June 2013.

B. Tuyatsetseg, B. Otgonbayar, “An Adaptive Scheduling Scheme to Efficient Emergency Call Holding Times in Public Safety Network,” in Proc. IFOST2013, Ulaanbaatar, Mongolia, June 28-July 3, 2013.

J. Wang, H. Zhou, L. Li, and F. Xu, “Accurate long-tailed network traffic approximation and its queueing analysis by hyper-Erlang distributions”, in Proc. IEEE conf. Local Computer Networks, pp. 148 - 155, 2005.

B.Tuyatsetseg, B.Otgonbayar, “Traffic Modeling of Public Safety Network”, in Proc IEEE-APNOMS2013, Hiroshima , Japan. 25-27, Sep . 2013.

Thank you for your attention Questions?

Documents

Tuyatsetseg Badarch, Otgonbayar Bataa Mongolian University of Science and Technology On Network Traffic Modeling Framework: A Case Study with Public Safety