MODELING AND SIMULATION OF SELF-SIMILAR TELETRAFFIC

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.Fractal-binomial-noise-driven Poisson processes: and .~et ,Sv*) -,~;*)+Ti(fl.

.Superposition of fractal renewal processes (SFRP); Step 7: Find a new .i* such that }*= argmin;{S\J~. and.M/G/oo processes (MGIP): compute ,~I-.S'. .

.Pareto-modulated Poisson processes (PMPP): Step 8: Repeat Step 6 through .S'tep 8 to obtain .\" as in (.;).

.Spatial renewal processes and fractional Gaussian noise: .Step 9: .4dvance the .~imulation clock, i.e., S-,~*j. and

.Superposition of autoregressive processes (SAP). .~et y=-v+.\".For the standard fractal renewal process (FRP), inter-event Step 10: Repeat .S'tep 6 through step JO within time slot oftimes are independent random variables (Bobbio et al, length l1t.2003). The marginal probability density function (PDF) of Step 11: (~ompute -\"i=POIS:S'(v), set y=O, and i=i+ I.such a fractal renewal process can be defined as (I), Step 12: Repeat .S'tep 6 through step J J until i=n, where II

{0. t s .4, i.~ the number ~f sample points..f~ )= Q4" ~,"+I) 4 (1) An approximate self-similar sequence {.\~i~\"]~Y2,... J W[]S

" t , t > -obtained. It took 1 min. 38 sec. on the Centrino-based HP

where 0<6<2. Selecting O in this inter\'al proves far PC ( 1.76 GHz) to generate a traffic sample sequence of

superior to 0«')<1 for the same required values that the 1,048,576 numbers (e.g., about 524*106 inter-event times).

inter-event time PDF can be further improved (Borst & Table 1 shows the mean values of estimated HurstMitra, 1998). The improved PDF of the FRP decays as a parameters H and 95% confidence intervals for the melIDpower law, as shown in (2), in parentheses, obtained using the wavelet-based H

It)= {a4~le-ItIA. 0 < t s .4. 2 ~~ima~or ~~ the F"!:NDP method ~vith four input, val~es,

.f \i {~~"~4t~("+I', t > .4. ( ) +4-9:.9~, R-~OO, M,-4 to 14, a?d H=().6 to 0.9. T~t: re~ul(:; i

con1lrnl that the most appropriate aggregate levellsi\f=IO :1whi:h Tis. co~~inuous fo~ all t, producing smoother spectral ..Table \ J

dens It} function (Rade\ ,2005). Mean Values ot Estimated HA method based on the Fractal-binomial-noise-driven M 0.6 0.7 0.8 09

PoisStm processes (FBNDP) adds M indep,,?ndent and 0.6023 0.6864 0.7817 0.8469 -~identically distributed (iid) alternating FRPs to generate a 4 (0.575.0.630) (0.659.0.714) (0.754.0.809) (0.819.0.874)~j

fractal binomial noise process that serves as the rate 0.6085 0.6871 0.7804 0.8496function for a I:>oisS{m process. The FBNDP requires five 6 (0.581.0.636) (0.660.0.715) (0.753,0.808) (0.822.0877)I.nput parametel.' t ') aenerate ' eli' ' I.ml.lal .' e rIli ' nces. 4 " 0.6080 0.6870 0.7803 0.8489s , 1:- .s -~ s, t: c. ., u, 8 (0.581,0.636) (0.660.0.715) (0.753.0.808) (0.821.0876)

R, l1t, and M. The resultIllg Hurst parameter H assumes the 0.6089 0.6875 0.7827 0.8502value (a+l)/2. The suggested algorithm is advancing with 10 0.581,0.636 0.660.0.715 0.755.0.810 0.823.0878

the intervals l1t. If S is a simulation clock, which advances 12 -0.6053 0.6903 0.7832 0.8501. t . d rlil ., th I ed t . f. th . th FRP .~ (0.578.0.633) (0.663.0.718) (0.756.0.811) (0.823.0878)III Ime, an ,,)- :s le e ap~ Imelo. e .1- 0.6049 0.6895 0.7842 0.8497sequence, then S"=TO\l + TI \1 +...+Tk\l for some k and 14 (0.577.0.632) (0.662.0.717) (0.757.0.812) (0.822.0877)

.i=I,2,... M, where Tk\ll is the inter-arrival time. The

sequence of seU'-similar pseudo-random numbers .\~i..\],... A method based on a supcl-position of the fractal renew[]!is generated through the f()llowing steps of Algorithm 1. processes (SFRP) uses a group of independent []ndAI~orithm I: identical fractal renewal processes (FRP). This method

,,\'tep 1: For each.i=1 ,2,. ..,Af, generate To(fl.fi.om (3), requires three parameters, i.e., (l and .4 fl-om the individual

,j)- { -0-1Alog[U(O".-I)(0".'-tr,)-I, 1"~1. FRPs, and.M, which is the number ofFRPs SUpClpOSedTo- ~41cl/'I-"'. r"<I. (3) Th~result~lg~urstpara~eterH,and. the m~an~and the

variance a of the margIllal output dIstributIon 111 rel[]tedwhere ( counting process during the unit time interval, can be

r- = 1+(0 -1)e" U (4) defined according to (7):0 H = (a + 1)/2

and U i.~ an iid untfi)/7II~v distributed random variable over 11 = E[ .\" ] = }.. (7)theuniti1ltelval[0,1):set,S\II=l;i\ll. 2- c " " -a}..

Step 2: Find.i* and S"' such that.i*=argmin;r;s'J)}. {1 -r ar[-\ ,,] -(1 + (1/ to»

Step 3: Calculate (5). .where

{0. ~r S'j"<.4. -}..=MO[I+(o-I)~le~"]-1.4~1 (8)x = I if S'j') ~ 4 (:> ) is an aggregate arrival rate of events in the unit time

..." interval. The sequence of seU.-similar pseudo-r[]ndom,\'tep 4: .!f .\"~ 1, .thell .\0 should be drawn .fi.om. a Poisson numhers .\-;i~\"] , ...is generated by the following steps of

probablhf}' dlstl1butl011 wlth }..= 1. rr\"=o, then.\o=O. AI~orithm 2:

Step 5: .S'~: i=l, and y=O- .4dvance the simulati011 cloch-. ,,\'tep 1: For each }=1,2,... M, alld i=O, gellerate Tu(l'fi-om

i.e...S-,S'~. ..I equation.~(3) and(4);.~et,,\)V'=l;i(fl-Step 6: L011Stl71Ct a new mter-event tIme Ti\l .fi-om (6), Step 2: Find .i* .~uch that }*=argminj{S"11 and set 10=

, .{ " ~1 41 [ 1 !] 1 T ~" (\7 I

) -U.OgL, L)~e. u.

T;J = e~I.4U~II", U < e~", (6) Step 3: .4clvance the .~imulation clock, i.e- S-,S';*I.

356

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Step .1: .Set ;=i+ I. ('mlstrnct a l1ew illter-evellt til11e 1:;(j) linearly dependent on the shape parameter a:: of a beta-

~*) n*) (j) distribution, while (~I can be selected arbitrary; forfi.OIII eq//atiol1 (6) alld set .~ I -0)"' + T:. .

I 1 examp e, (~I=I In all cases that are InvestIgated.

~tep 5: , F~~ld a lIew .i* s//ch that .i*= 3I.gmi~i{$J~, alld The PDF ./(.\") of the beta-distribution is defined as ( 10),

COlIIP//tt .S -S. { "1-1 (1 "2-1 *1 x -x)

Step 6: .4dvallce the .~il11//latioll clock, ;:e. S-.~l .( -° < x < I

Step 7: Repeat .SteJ~ 4 thro//gh Step 6//llt;1 i=11 is reached, .r x) -{3(al,a2) ..(10)

»'here II i.~ the 11//l11ber l?r.~al11ple poillt,~. 0. othern lse

wheref3«(~J,(~.')isgivenas(II).The obtaIned results ~~om the sl~~latlon wIth M= 10, -r a r a

A=3.8 and other condItIons that sImIlar to the FBNDP P((~I.(~2) = 1x"1-1(I-x)"2-1dx= ( I) ( 2) (11)

method, and with the mean values of estimated H and 95% I r«(~1 +a2)

confidence inten'als t()r the mean in parentheses, are This method, also based on a superposition of the

presented in Table 2. autoregressive processes, consists of the following steps.

Table 2 At .th J, gorl m :

Mean Values of Estimated H

M 0.6 0.7 0.8 0.9 ,\'tepJ: .Set;=i+J.DeteI711;lIezl;alldz::i//sillg(9).

0.6076 0.6986 0.7929 0.8603 ,\'tep 2: ('alc//late the .~//111.\';= ZJ i + Z::i, i= 1,2, 4 (0.580.0.635) (0.671.0.726) (0.765.0.820) (0.833.0.888) Step 3: Repeat Step J alld,S'tep 2 ulltil ;=II, where II i.~ the

0.6076 0.6986 0.7929 0.8655 11//l11ber l?r.~al11ple poillt.~.6 (0.580.0.635) (0.671.0.726) (0.765.0.820) (0.833.0.888) The asymptotically self-similar traffic sequence :.\] ~\'::, ...}

8 (0.5~;!Z.~47) (0.6~1~~~;36) (0.7~;.9~~27) (0.8~.:.6g. 0!' l,l~48,57~ numbers. was ge~erated in 3.5 seconds on the

0.6166 0.7091 0.7986 0.8686 Centnno-ba~edHPPC(I.76CJHz).

10 (0.589.0.644) (0.682.0.737) 0.771,0.826) (0.841.0.896) Unlike other sequential generators, the SAP generator does

0.6134 0.7085 0.7978 0.8718 not require an aggregation level to be assumed as an input12 (0.586.0.641 ) (0.681.0.736) (0.770.0.825) (0.844.0.899) parameter. But it requires the shape parameter aJ as input

0.6143 0.7063 0.7983 0.8708 t M I f t . t d 11 bt . d . th14 (0.587.0.642) (0.679.0.734) (0.771.0.826) (0.843.0.898) parame er. ean va ues 0 es Ima e 0 alne WI

algorithms for sequential generators are shown in Table 3

" The mean values for 0.6 and 0.7 of estimated H with 95% and Table 4, respectively, for Hurst parameter, equal to

( confidence intervals for the means in parentheses tor 0.6,0.7 and 0.8,0.9.

FBNDP algorithm are 0.6086 (0.581, 0.636) and 0.6875 , .Table 3

(0.66, 0.715), respectively. Their Llli(% ) is + 1.44 and -Mean Values of EstImated H and AH

! 1.789, respectively. For H=O.6, 0.7 and 0.8 the overall Methods 0.6 0.7

relative en.or is less then 5%, but t()r H=0.9 it is greater if AH if AH

then 5%. The same characteristics, obtained with SFRP FBNDP 0.6086 +1.440 0.6875 -1.789

algolithm are 0.6166 (0.589, 0.644) and 0.7091 (0.682, (0.581,0.636) (0.660,0.715)

0.737),. and their Llli(%) are +2.759 and -1.296, SFRP (0.5~:!g.~44) ,2.759 (0.6~.;.0~~37) +1.296

respectIvely. SAP 0.5989 -0.182 0.6852 -2.112

The SFRP approach shows more accurate results and the (0.595,0.603) (0.680.0.690)

relative error for H=O.6, 0.7, 0.8 and 0.9 is +2.76%,

+1.29%, -0.17% and -3.49%, respectively. Similar to Table 4.

FBNDP algorithm, H values, estimated with SFRP Mean Values of Estimated H and AH

algorithm, ranged from positi,'ely biased to negatively Methods 0.8 0.9

biased, as the H ,'alue increased. if AH if AH

A method based on the superposItIon. of autore~r~ss~ve FBNDP 0.7827 -2.157 0.8502 -5.538

processes (SAP) generates asymptotIcally selt-slmllar (0.755.0.810) (0.823.0.878)

sequences when aggregating several independent SFRP 0.7986 -0.174 0.8686 -3.491

autoregressive processes. In the simplest case this can be (0.771.0.826) (0.841.0.896)

presented as the sum of two autoregressive processes of SAP 0.7845 -1.937 0.8971 -0.325

th t .. t d h . (9) 10.781.0.788\ 10.894.0.900\elrsorer,assownln : , ~'., ,-.' ~,

~ - 4 .. + v~ -.~ Th I . I . f. h FBNDP h du u 1..1 .u (9) e cumu atlve re atlve en:or or t e met o was

z2i = .42iZ2J-I + .v2i. I = 1.2 less then 5011), except tor H=O.9. For the SEFR method the

where AJi and A::; are randomly chosen from a beta- cumulative relative error was +6.21%, +1.78%, -1.430;1,

distribution B«(~J,(~::)on [0,1] with the shape parameters al and -5.37%, tor H=().6, 0.7,0.8 and 0.9, respectively.

and (~::, where (~J>O, (~::>O; -~'1i and -~'::; are a pair of Similar to the FBNDP method, estimated H values ranged

independent and identically distributed sequences of from positively biased to negatively biased, as ~he H value

random variables ~.ith mean of zero and variance d=l. Increased. For (~::=2.9 and 8.1 all values of the Hurst

Using the least-square fitting we can find that parameter tro~ the traffic sampl~ sequence of the S~P

a~=7 .7929*log(H)+4. 9513. Then, the Hurst parameter H is method were hIgher than the requIred value~. The ~elatlve

error was +24.14% and +10.45%, respectIvely: In that

357

case, the results were overestimated. For a1=21.3 and 71.5, sample sequence with 1,048,576 numbers. The algorithm,the relative error was -0.010/1. and -8.45%. based on the F -ARIMA method, is too rigorous to be used

to generate long sample sequences.FIXED-LENGTH SEQUENCE GENERA TORS OF In this work we propose a new generator of pseudo-SELF-SIMlLAR TELETRAFFIC random self-similar sequence based on fl-actional Gaussian

noise (FGN) and Daubcchies wavelet (DW), called the

The mostfi-equently studied discrete-time models of self- FGN-DW method. The use of Daubechies wavelet makes

similar trafTic belong to fractional autoregressive po~sible to produce more accurate self-similar sequences,integrated moving-average (F-ARIMA) and fractional where wavelets are closer to the tI-ue values. Wavelets can

Gaussian noise (FGN) processes, and possible fixed-length provide compact representation for a class of FaN

sequence generators are based on. processes, because the structure of wavelets natw.ally.Fractional autoregressive integrated moving-average matches the self-similar structure of long-range dependent

processes (F -ARIMA): processes (Daubechies, 1992)..Fast Fourier transfolm (FFT); The wavelet analysis transforms a traffic sequence onto a.Fractional Gaussian noise and Daubechies wavelets: timc-scale grid, where the term scale is used instead of

.Random midpoint displacement (RMD); frequency, because the mapping is not directly related to

.Successive random additions. frequency as in the F ourier transformation. The waveletLet consider F -ARIMA(O,d,O) method for generating self- transfolmation delivers good resolution in both time and

similar sequences, where d is the fractional differencing scale, as compared to the Fourier transformation, which

paranreter, 0<d<1/2, and let generate the process X=:-\i: provides only good frequency resolution. The developed

i=O, 1,2,. ..,n} with a normal marginal distribution, the algorithm consists of the fl)llowing steps.

mean of zero and the variance a~ ' and the autocorrelation AI~orithm 5:

f' t . (ACF) f }(k--1) +1 )th td t'. d ' (12) .\'tepl: Given: H. .Start.fi)ri=1 and continue untili=n.

unc Ion, lP"J c-.. ,- ,... a e me a~ , I I f I iI: I: f ."r(l-d r(k+d cacuateasequencel~ vaue.\' Ul,,2, ;n: usmg(IJ),

P = }' / y = ) ) ( 12 ) f( A H) =c IA 11-2H +0 (1 A Im.."-2H21 ) ( 13

)k k 0 r(d)r(k+l-d)' ..f .

where u,here c f = a2(27r)-1 sin(Jr:H)r(2H + I). 0( .) represents the

}' =02 (=1)~(1-2d) residual elTm.and .I; =.l(JI:i/n:H),k 0

r(k -d + l)r(l- k -d) the value l?f the .fi.equencie.\' .{; con.espond.\' to the spectral

The fixed-length sequence of self-similar pseudo-random den,\'il}' l~f an FGJ\! process fi)r.{; ranging betu'een Jtij, + JC

numbers is generated through the following steps of Step 2: Multipl}' V;} b}' realization,\' l?f the independenlAlgorithm 4. e.\1Jonential random variahle with the mean l~f one 10

Algorithm 4: -obtain: .~ ) , becau.\,e the ,\,pectral den.\'il}' e,\'timated fiir a

Step 0: .Set N,,=O and Do=l. .\~o. the .fir.\' pseudo random given .fi.equency i.\, distributed a.\,}'mptotical~v a.\' the

element in the output se([-similar sequence, i.\' independent e.\"pmlential randmn variahle with the ll/eal/

generated .ji.om the nmmal disl1"ibution N(O.a~). .f(}.,H).u,here a~ is the required variance ~f the -\~;. Step 3: Generate a sequence U~I. [2. 1'n} of coIl/pIe\'

Step i: (i=I n-l). Compute mean; and var; l?f .\-; number.\' such that 1}~I=..[Ji and the phase ofJi is

recul:\'ive~v. u.\'ing the .following equations: unifonnlv di.\'l1.ibuted between O and 2Jr. This rando/1lI. -

j\' = P -~ ,l. P -pha.\,e technique preselve.\, the specl1.al densiry1 , .fI"1"-IJ 1-} .

}- .con'esponding to: .I; : .It al.\,o make.\' the mal'ginal

Di = Di-1 -Ni~1 / D1-J ' di.\'l17hution ~f the .final sequence no111lal and produces the

,I. =N/D- re q uirement.\' forFGj\!'1'11 1 I .1' .

l/Ji;=I/Ji-Ii-I/JiiI/Ji-Li-;. j=1 ,i-1. Step-l: (~alculate two .\,ynthetic co~fficients of

m1hono111lal Daubechie.\' wavelets. The output ,\,equencewhere l/Jij .1 = 0, J = 0 11 -1. IS gIven by :-\1~\~1, In} repre.\,entin,~ appro.\"imate~v selfsil1/ilar

(i ) (j -d -1)!(i -d -j)! FGN process i.\, obtained hy app~ving the invel:1'e

l/Jii = -.( - d -

1)'( .- d) 1 ' Daltbechies u'avelet.\' l1.an~fo111lation l~f the .\,equenceJ .1 .1 1' 1' 1-'l 1. 2. ..., n j .

mean; = \' l/Jii-\"i-;' It took only 2 seconds on Ce~trino-based HP PC (176~ GHz), to generate a sequence of 1,048,576 numbers usIng

var = (1- ,1.2)vat this algorithm., '1'11 ,-I.

?enerate .\~;.fi.mn N(mean;. var;). Increase i by 1. ~f ANALYSIS OF NUMERICAL RESULTS

1=11. then stop.

Aself-similartrafficsequence:-\I~\~2,...~\"n}wasobtained. Th H ' t . t ,. ,. t . t' th , 1 l ' dHIt t k4h '

1 . d24 "' th C t . b ' d e ur~ palamel:I e~tImae~ or e \\aveet-)a~e00 our~, mill. an ~el:. on e en nno- a~e . h . 1. bl 5 (H -n 6 d I) 7 d l '

11HP PC (1.76GHz) to eneI-ate the F-ARIMA traffic estlmatorares own.m .~,e =v. an ,..).an aJeg 6 (H=O.8 and 0.9) wIth dIfferent wavelet coefficIents.

358

Table 5 considered. (a) how accurately self-similar process can be.

Mean Values of Estimated H and AH generated; (b) how quickly the methods generate long self-

Wave- 0.6 0.7 similar sequences, and (c) how appropriately self-similarlet if AH if AH processes can be used in sequential simulation. Thecoeft- algorithms developed on the base of difteren1 methods for

2 0.6019 +0.323 0.6984 -0.229 the sequentional generators and the fixed-Iength sequence(0574630) {0671 0726) - h d I . d . 1 ., If. ,-.- , '~ ~ '

o " generators perfolm t e mo elng an slmua1lon of se -4 0.60~6 +0.433 0.7039 + .--4 .. t h . f 1 I f.f . d N .

I(0.575:0.630) (0.676.0.731) simIlar )e avl.or O. rea te etra IC .ata. umenca

8 0.6026 +0.430 0.7031 +0.445 examples and simulatIon results are provided.(0.575.0.630) (0.676.0.731 )

16 0.6013 +0.21" 0.6987 -0.185 REFERENCES(0.574.0.629) (0.671, 0.726)

Bobbio. A.. Horv!\th. A.. Scarpa. M.. Telek. M. (2003). AcyclicTable 6 discrete phase type distributions: Properties and a parameter

Mean Values of Estimated H and AH estimation algorithm. -PertormalJce Evaluation. 54(1 ). 1-32.Wave- 0.8 0.9 Borsl S.C., Mitra. D. (1998). Virtual Partitioning tor Robustlet ..AH Resource Sharing: Computational Techniques torcoeff. H AH H Heterogeneous Traffic. -IEEE Jolmlal 011 Selected .4reas ;II

2 0.7943 -0.709 0.8898 -1.137 COI11I11UII;Ca,;oll.5.16. (5), 668-678.(0.767.0.822) (0.862.0.917) Boxma, 0., Cohen. J. (2000). Selt:Similar Network Traffic and

4 0.8055 +0.684 0.9074 +0.821 Pertomlance Evaluation. .John Wiley & Sons. New York.(0.778.0.833) (0.880.0.935) Daubechies, I. (1992) Tell LeclIlre.5 "vll rt'm'elets. 1'o/.61 t<f

8 0.8039 +0.486 0.9049 +0.545 CBMS-NFS Regiollal Coliferellc,~ Series ill .4ppl;ed

(0.776.0.831) (0.877, 0.932) Malllel11at;cs. SlAM Press, Philadelphia, Pennsylvania.]6 0.7962 -0.474 0.8938 -0.694 F . R " () M d I. d I ... 1., It -.. 1 t t-t-, . (0.769.0.824) (0.866.0.921) ara.l. .(-()() .o e mg an. ana ysls 0 se -slml ar ra IC m

A TM networks. PhD thesIs..Giambene.G.(2005). Queueing Theory and Telecommunications.

The obtaIned results are averaged over 30 sequences, and Networks and Applications. Springer. NY. 585 p.they show that for all input H values the F-ARlMA and the Hayes. J.. Ganesh Babu. T. (2004), Model;lIg alld .4I1a(v.5;s ofFGN-DW methods .produce sequences with less biased H .TelecOl11l11UlI;Ca';olls Nen,'orks, NY. John Wiley and Sons.

values than other methods. The means of estimated H Jeong. H.-D. (2002). Modelling of Sel1:Similar Teletraffic tor

values, obtained with the suggested wavelet-based Simulation. Plill Thesis. University ofCanterbury. USA.algorithm, are shown in Table 7. Kushner. H.. (2001 ). Heavy Traffic Analysis of Controlled

Table 7 Queueing and Communications Networks. Springer. NY.I t M V I t ' E t . t d H d A u Mehdi. J. (2003). Stochastic Models ill Oueueillg Theonl,

wave e ean a ues o s Ima e an l'1Jl A d . P ~ .

-.ca emlc ress.coefuclcnt 06 07 08 09 ..~ S I - S.. 1 N k T t]- dPark. K.. Wllllngcr. W. (.:.000). .e t-, Iml ar etwor. ra IC an2 1.94ge-04 2.154e-04 2.393e-04 2.670e-04 Pertormance Evaluation. .John Wiley & Sons. New York.4 2.508e-04 2.540e-04 2.586e-04 2.645e-04 Radev. 0. (2005). Fluid Flow Analysis tor Long-Range8 2.440e-04 2.43;e-04 2.438e-04 2.456e-04 Dependent Traffic. Proceedillgs t<f the ('ol11put,'r16 2.116e-04 2.05)e-04 2.007e-04 1.973e-04 Sc;ellce '2005. Chalkidiki. Greece. vol. 2, 19-24.

Rubinstein. R., Melamed. B.. (1998). Modem Sin111latioll mIdAnalysis of the mean times required to generate sequence Modelillg. New York, John Wiley & Sons.of a given length demof\!!trates that the sequential Stathis. C.. Maglaris, B. (2000). Modeling the Sel1:Similar

generators are more attractive for the practical simulation Behaviour 01- Network Traffic. Jolmlal of Col11puterstudies of computer networks than the F-ARlMA-based Network.5, 34 (1).37 47.generator, since they are much faster. However, these Trivedi. K.. (2001). Probab;lily, ~lId Sta,;.5';C.5 .willl Rel;ab;lil)'.generators require more input parameters, and selecting Queueulg. alld Col11puter L5clellce .4pphcalloll.5. New York.

.. t 1 h . 1 dd ..John Willey & Sons.appropnate values IS a pro) em t at remaIns. n a ltIon, .

the problem of how to define the relationship between the BIOGRAPHYHurs1 parame1C1. and two shape parameters of a beta-

distribution in the case of SAP also remains. DIMITAR RADEV, D.Sc. is Associate Professor at

Depal"1men1 of Communication Technique andCONCLUSIONS Technologies at University of Rousse, Bulgaria. His main

.research interests are Teletraffic Theory .Simulation andTeletraffic .exhibits self-similar properti~:.,; over- a wIde Modeling of Communication Networks: ~nd Perf.ol1l1ance

range ~)f tl~e sc.a~es that are V~I)' .d!ft~rent from the Analysis of Queuing Systems; dradev(~).abv.bgproperties of ti.adItIonal models. Self -simIlar models arc ,

appropriate f()r 1eletraffic .as they pro~ide capacity to IZABELLA LOKSHINA, PhD is Associate Professor ofestImate the network peliolmance/qualltv , allocate thc MIS' d h . f' M t M k t . d I f. t .

, .., an c air O anagemen , ar .e Ing an n Oll11a Ionresources, and ensure the QoS. One of the problems that S' ' t ' D "1 . t 1 S' UNY () t US' A H ..

, y~ em~ epal m\;n a, neon a, , .el mamcom

l)uter network researchers face dunng simulatIOn IS . h . t t Art .f... 1 1 t II .d C 1researc In eres ~ arc I Icla n e Igence an omp ex

how to generate long synthetIc sequential self-slmllar S 1 M d I . d S'. 1 t . 1 k h .. ( ;: , ' 1 t ed' , YS em o e Ing an, Imu a Ion; o .s I IV "oneon a. u

sequences. To solve the problem three aspects must be' -

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