GA-based optimal dimensioning of three-level traffic shaper for statistical multiplexing in ATM networks

624 pp. 624-631

Kyeong Soo KIM * Byeong Gi LEE *

for

GA-based optimal dimensioning of three-level traffic shaper

statistical multiplexing in ATM networks

Abstract

In this paper we discuss genetic algorithm (GA)-based optimal dimensioning of the three-level traffic shaper (~rs) for statistical multiplexing in ArM networks. As the objective function for the optimal dimensioning, we consider three alternatives - - the variance of the output cell service periods, the variance of the output cell rates, and the sum of squared differences of adjacent cell service periods. We perform simulations for optimal dimensioning of the TLTS for statistical multiplexing of VBR video services, and we confirm that the objective function based on the variance of the output cell rates is the most suitable among the three alternatives. The framework of the optimal dimensioning procedure introduced in this paper can be easily extended to other optimal design problems in ATM networks for which traditional gradient-based optimal dimensioning methods or random search techniques are not easily applicable due to the nonlinearities and complexities of the related functions and the discreteness of the involved parameters.

Key words : Traffic control, Statistical multiplexing, ATM, Genetic algorithm, Optimization, Variable rate, Videocommunication service, Cell loss.

R~sum~

Cet article pr(sente le dimensionnement par algo- rithmes g tn t t iques (GA) d'un r#gulateur ~ trois niveaux (TLTS) pour le multiplexage statistique clans les r(seaux ATM. Trois fonctions de dimensionnement optimal sont consid#r~es : la variance des p t r iodes de service de cellules en sortie, la variance du dtbit de cellules en sortie, et la somme des difftrences au cart6 des ptriodes de services de cellules adjacentes en sortie. En rue du dimensionnement optimal du rLTS pour le multiplexage statistique de services vidto VBR, des simulations ont 6t6 r6alis~es ; celles-ci confirment que la fonction objec- tif bas~e sur la variance du dtbit de cellules en sortie constitue la meilleure des trois solutions consid~rtes. L'approche pour le dimensionnement optimal pr~sent(e dans cet article peut facilement ~tre appliqu~e ~ d'autres probldmes d'optimisation clans les r6seaux ArM pour les- quels les m~thodes classiques fond~es sur I' optimisation du gradient ou des techniques de recherche altatoires ne sont pas applicables facilement du fait de la non- lin(arit6 et de la complexit~ des fonctions impliqu(es et du fait du caractdre discret des paramdtres.

Mots ci~s : Maltrise trafic, Multiplexage statistique, Multiplexage temporel asynchrone, Algorithme gtnttique, Optimisation, Dtbit trans- mission variable, Service vidtocommunication, Perte cellule.

D I M E N S I O N N E M E N T OPTIMAL BASI~ SUR DES A L G O R I T H M E S GI~NI~TIQUES

D ' U N RI~GULATEUR DE TRAFIC A TROIS N I V E A U X

POUR LE M U L T I P L E X A G E STATISTIQUE D A N S LES R E S E A U X ATM

Contents

I. Introduction. II. Three-level traffic shaper and performance measures.

III. cA-based optimal dimensioning procedure and its objective functions.

IV. Numerical examples. V. Conclusion.

References (16 ref.).

* Department of Electronics Engineering, Seoul National University, Seoul, 151-742, Korea.

ANN. T~L~COMMtm., 50, n ~ 7-8, 1995 1/8

K. S. K I M . - - T H R E E - L E V E L T R A F F I C S H A P E R F O R S T A T I S T I C A L M U L T I P L E X I N G 625

I. I N T R O D U C T I O N

statistical multiplexing gain. Finally we will perform a comparative performance analysis of the three alternative objective functions through numerical examples for VBR video services.

In asynchronous transfer mode (ATM) networks, traffic shaping is a fundamentally important function that controls the ATM cell traffic to achieve a desired modifica- tion of the traffic characteristics [1, 2]. There have been introduced a number of algorithms which can provide various traffic shaping functions, such as reducing the peak rate, spreading bursty traffics, and others [3, 4, 5]. In addition to these variable bit rate (VBR) data oriented functions, source clock frequency recovery function is additionally required for VBR video or other real-time services, and a new traffic shaper named three-level traffic shaper (TLTS) was recently proposed to meet this requi- rement by regulating the minimum cell rate as well as the maximum cell rate [6].

In designing a traffic shaper, the optimal dimensioning of the involved parameters is one of the most important issues, as is the case in most engineering designs. The optimal dimensioning refers to the procedure that determines the parameter values in an algorithm or a scheme such that they can produce the best possible performances without violating the given constraints. The performance objectives and the constraints of most existing traffic shaping algorithms, however, cannot be expressed in simple closed-form functions. As can be seen in [3, 4, 5, 6], they are complicated nonlinear functions in general, and some of the input parameters often take on only discrete values which makes it difficult to apply traditional gradient- based optimization methods and random search techniques [8]. This is why most existing researches focused mainly on the performance analyses without discussing the optimal dimensioning issues.

In order to overcome these difficulties, we can employ the genetic algorithm (GA) [7] as the function optimizer for optimal dimensioning of traffic shapers. The GA is an iterative search procedure that emulates biological evolutionary processes such as the survival of the fittest, and is thus known to be one of the most promising methods that can solve complex problems in practical applications, quickly and reliably [8]. In employing the GA for optimal dimensioning it is important to define a proper objective function to minimize for the given application, because a proper objective function enables to get the desired performance while an improper one could lead to a failure or a poor result.

In this paper, we are going to discuss the GA-based optimal dimensioning of the TLTS for statistical multiplexing of cell streams in ATM networks and analyze its performance for some alternative objective functions. For this, we will first describe the operation and performance measures of the TLTS for use in optimal dimensioning procedure. Then we will discuss the GA-based optimal dimensioning procedure of the TLTS and define three alternative objective functions for improving the

II. THREE-LEVEL TRAFFIC SHAPER AND PERFORMANCE MEASURES

We first consider the operation of the TLTS and the two-state Markov-modulated Poisson process (MMPP)- based source modeling of the VBR video sources. Then we carry out a queueing system analysis of the TLTS for the MMPP source model and determine the performance measures such as the cell loss probability, the underflow probability, and the maximum cell delay within the TLTS.

II.1. Operation of the TLTS.

The TLTS consists of a buffer of length K , a server whose service time is controlled by the queue length in the buffer, and a controller for this control. Figure 1

C o n t r o l l e r [

19 11 ~I

19 1 z m.I

K (= 13 ) . . I

F1G. 1. - - Three-level traffic shaper. R~gulateur de trafic ?l trois niveaux.

i~ To network

l = r = l

Cell time

depicts the operation of the TLTS, where q denotes the queue length in the buffer, T the service period, and In (n = 0, 1, 2, 3) the queue threshold values. The server provides a service at the beginning of each adaptive period T. The controller samples the queue length q immediately after the service, and controls the server for the adjustment of the corresponding cell service period T. The relation between q and T is :

(1) T = t~ if li-1 <_ q < li,

for some prespecified values ti (i = 1, 2, 3) with ti >_ tj for i < j . Note that l0 = 0, and 13 ---- K , Therefore the TLTS can limit the minimum cell rate as well as the maximum or peak cell rate of an input traffic.

2/8 ANN. TI~LI~COMMUN., 50, n ~ 7-8, 1995

626 K . S . KIM. -- THREE-LEVEL TRAFFIC SHAPER FOR STATISTICAL MULTIPLEXING

II.2. Source modeling. (3) Tn = ti if li-1 ~ Un < li,

As the source model for the VBR video services, we use the two-state MMPP which is characterized by a transition rate matrix Q = [qi,j], i, j = 1 or 2, and an arrival rate vector r = @1, r2). Note that ql,1 = -q l ,2 and q2,2 = -q2,1. There are several MMPP parameter matching techniques available for modeling of real bursty traffics [9, 10]. Among them, we employ the parameter matching technique in [10] for use in VBR video traffic modeling. This is a new method which

p =

(4) U n - + l =- min(U~ + An, K) ,

(5) Un+l = max(Un+ 1 - 1, 0).

With these random variables and a two-state MMPP input, we describe the queueing system of the TLTS as a two-dimensional Markov chain (MC). The state of the MC is defined by (U(n), J(n)), and the state space is {0, 1, . . . , K - 1} x {1, 2}. Then the transition probability matrix takes the upper block Hessenberg

A1,o+AI , I A1,2

A1 ,o A~ ,t

0 Ai,o

. . .

0

0

O

O

form as follows :

O Al,o

O ( 3

A, ,K-I ~ i : 1 r ml , i

A1,K 2 Ei~176 Al,i

AI ,K-:~ ~i~1(-2 Al,i

Z A~ ,K-l~ i=K-( l~-1) A l , i

0 A2,o

0 A2,o

A2,1r-(h+l) ~i~ A2, i

A2 ,K-12 ~ie~ A2,i

O C 2 ~

A:~.o . A:~d,-_(t~+x ) ~i=K-12 A:~,i

O 0

0 Aa,0 }-~i= 1 A:~,i

0

1

2

11 - 1

Ii ,

12 -- 1

12

K - 1

reflects the importance of the peak rate and accurately characterizes the correlated traffics such as VBR video

services. According to this method, the four parameters of the two-state MMPP are determined as follows :

( 2 ) r l = m q- V / - ~ , r2 = m -- V / ~ / P , 1 - p

q l , 2 = p/7-, q2,1 : - - ' , T

where m is the mean arrival rate, u the variance, a the peak-to-mean ratio, 7 the autocovariance time coefficient defined in [ l l ] , and p the probability of being in state 2, i.e., p -- a / (a + 1).

II.3. Queueing system analysis.

For queueing system analysis, we first define random variables Tn, Un, U~-, An, and Jn as follows : T~ is a random variable for the duration of the n-th service period in unit of cell time; U~- and Un the number of cells in the buffer immediately prior to and immediately after the beginning of the nth service period, respecti- vely; An the number of cells arrived during the duration time Tn; Jn the state of the MMPP at the beginning of the nth service period. These random variables are illus- trated in Figure 2. As can be confirmed from the figure, the random variables meet the relations :

where O denotes a 2 • 2 zero matrix, and Ai,j a 2 • 2 arrival matrix whose (/, m)th element is :

(7) Aid(l, m ) = P r { A n =j , Jn+x=mlTn=ti, Jn=l}.

The arrival matrix Ai,j can be evaluated using the efficient procedures introduced in [12].

The steady state probability vector u for the transition probability matrix P can be determined from the linear equations :

(8) / u . P = u,

t u . e2K --~ 1,

where u = (u0, ul ," ' ,UK-1) , and e2K is a 2K- dimensional unit column vector. Note that ui(i = O, 1, �9 . . , K - 1) itself is a two dimensional vector, ie ui = (ui,1, ui,z) whose individual element indicates the limiting probability distribution :

(9) ui,j : lira Pr{Un = i, Jn = j} , j = 1, 2. n--- -+ O ~

11.4. Performance measures.

In the ~TS it is undesirable to get either overflow or underflow in the buffer. As the overflow is connected to loss of cells, so is the underftow connected to loss

ANN. TI~L[~COMMUN., 50, n ~ 7-8, 1995 3/8

K. S. KIM. -- THREE-LEVEL TRAFFIC SHAPER FOR STATISTICAL MULTIPLEXING 627

U-rl-I Un-I

I L

An- 1

�9 �9 �9 [ I t I �9 " ' ] I I I �9 �9 �9 I I I

- % -7 / / Cell d e p a r t u r e Cell d e p a r t u r e Cell d e p a r t u r e

(if ex is t s ) (if ex is t s ) (if ex is t s )

Cell T ime

FIG. 2. - - Relations among random variables.

Relations entre variables aldatoires.

of control of the maximum cell interval or the corresponding minimum cell rate. Therefore it is necessary to take the cell loss probability and the underflow probability as the performance measures of the TLTS. TO be more specific, the cell loss probability, PcL, refers to the probability that an arrived cell is lost due to buffer overflow; and the underflow probability, PUF, the probability that the server cannot find cells in the buffer at the beginning of a service time. The delay characteristics of the TLTS is another important factor for consideration. In some cases it is critical to design the traffic shaper to satisfy the required delay constraint. So we also take the maximum delay, Dm~x, that a cell can experience in the TLTS as the third performance measure. Following the derivations given in [6], we can obtain the following expressions for the three performance measures :

3 lm 1 2

/lO) E E E m = l i=lm 1 j = l [ I Z ( t m A t ) ] ( J )

oc

E (i + k - K)[Am,k'e2l(J), k = K i4-1

2

(11) PUF = E [ A I , o " e2](i)uo,i, i=1

(12) 3

D m a x = (/1 -- 1)/:1 q- E ( l j -- l j 1)~j, j=2

where At is the cell time, x(j) denotes the j th element of the vector x, and p( t ) denotes the expectation vector whose j th element is the expected number of cell arrivals in interval (0, t] given J(0) -- j [13]. For the two-state MMPP, ]Z(t) is given by :

1 -- E (ql,~+q2,1)t (13) ~(t)---- rlq2,1+ r2ql,2 te 2+

ql,2 + q2,1 (ql,2 q- q2,1) 2

I r l q l , 2 - - r 2 q 1 , 2 I '

- - r lq2 ,1 q- r2q2,1

I I I . G A - B A S E D O P T I M A L

D I M E N S I O N I N G P R O C E D U R E

A N D I T S O B J E C T I V E F U N C T I O N S

We now introduce the cA-based optimal dimensioning procedure of the TLTS. First we present the framework of optimal dimensioning based on the GA, and describe the constituent objective function, constraint, and penalty function. In this framework, we then consider three alternative objective functions for statistical multiplexing in ATM networks.

I I I . I . G A - b a s e d o p t i m a l d i m e n s i o n i n g p r o c e d u r e .

Optimal dimensioning procedure of the TLTS refers to a designing procedure that finds the TLTS parameter values which yield the best desired performance without violating the given constraints. For the TLTS, there are six parameters to determine; the queue thresholds 11, 12, 13, and the cell service periods tl , t2, t3. The performance objective and the constraints vary depending on the application of the TLTS. The statistical multiplexing gain is one of the most important performance objectives for traffic shaping function, and cell delay, cell loss probability, and underflow probability are the main constraints in the optimal dimensioning procedure. For real-time services such as videoconference, bits generated by the coder in each frame must be delivered to the decoder with delays not exceeding a maximum of about four frames [3]. But the delay allowable for the traffic shaper is much smaller than this, as the major source of delay comes from the network in the form of propaga- tion and queueing delays. The cell loss probability must remain small enough to guarantee the quality of service (QOS) negotiated during the call set-up. The underflow probability has no direct relation with the Qos, but it is connected to loss of control of the maximum cell interval. So it must be kept as low as possible.

4/8 ANN. TI~LI~COMMUN., 50, n ~ 7-8, 1995

628 K. S. KIM. -- THREE-LEVEL TRAFFIC SHAPER FOR STATISTICAL MULTIPLEXING

The performance objectives and the constraints of the TLTS, however, cannot be given in simple closed- form expressions. As presented in Section II, they are complicated nonlinear functions, in general, and the parameters take on only integer values. Therefore we now employ the GA [7] as the optimization process for optimal dimensioning of the TLTS.

TO devise an optimal dimensioning procedure based on the GA, we minimize the objective function f(x) subject to the constraints ci(x) >_ 0 for the constraint functions ci(x), i = 1, 2, 3, where x = (/1, 12, 13, t l , t2, t3) . We can transform this into the unconstrained problem :

3

(14) minimize f(x) + Cp E (I)[ci(x)], i=1

where �9 is a penalty function and Cp is a penalty coefficient [14]. In this paper we define the penalty function and the constraint functions as follows :

f y2 if y < 0, (15) ~(Y)

[ O, otherwise,

and

(16) { c l ( x ) = C l -Pc~(x),

~2(x) C2 P~(x),

c3(x) C3 Dmax(X),

where C1, C2, and Ca are the maximum values respec- tively of the cell loss probability, the underflow probability, and the maximum delay of the TLTS, which are also functions of the TLTS parameter vector x. Then the remaining problem is to define the objective function f .

and

(19) 3 li --1

E{T}=Et{ E uj.e2. i=1 j : l i - i

III.2.2. Variance of the output cell rates.

We define the second objective function f2 as the variance of output cell rates instead of the variance of the output cell service periods. Let R denote the random variable for the output cell rates of the TLTS. Then in view of Figure 3, we find that for PUF ~ 0 the output cell rate distribution of the TLTS is

b t~l b tj ~l

n o r m a l i z e d b i t r a t e

t~. (3 At) _l_ t ; (3 At)

004 , , 1 - -

~ --1 ' / ~ m e

1 I

I

/'t~i m e

FIG. 3. - - Relation between the cell interdeparture time and the normalized bit rate.

Relat ion entre le temps entre les ddparts des cellules et le dgbit binaire normal i s s

(20)

{ Pr{R = r} =

~ . / t~ V ~1{-1 z / ~ Z. . .~j : l i_ l Uj " e2 ,

0,

where

if r : 1/ti (i : 1, 2, 3),

otherwise,

3 l i - 1

(21) C= E ti E uj "e2. i=1 j = l i 1

III.2. Objective functions for statistical multiplexing.

Therefore we can obtain f2 as follows :

(22) f2(x) ~ E{R 2 } - E{R} 2,

If we consider the network performances such as statistical multiplexing gain, it is desirable to shape the characteristics of VBR video sources to be similar to those of constant bit rate (CBR) traffics. In this context we can consider the following three alternative objective functions for the oA-based optimal dimensioning :.

III.2.1. Variance of the output cell service periods.

We define the first objective function f l as the variance of the output cell service periods, that is,

(17) f l (x) ~ E{T 2 } - E{T} 2.

Note that, for Puv ~ O, the two expectations in (17) are given by :

3 l i - 1

(18) E{T2}=E ti2 E uj.e2, i 1 j = l i - 1

where

(23)

and

(24)

3 ~ 1~--1

E { R 2 } = E E u j -e2 , /=1 j : l { - 1

3 1 l i - 1

E { R } = E C E uj.e2. i=1 j = l i 1

III.2.3. Sum of squared differences of adjacent cell service periods.

As the third alternative objective function f3 we take the sum of the squared differences of adjacent cell service periods. In this case we can directly minimize the sum of the squared differences given by :

(25) fa(x) = ( t l -- t2) 2 ~- (t2 -- t3) 2.

ANN. TflLs 50, n ~ 7-8, 1995 5/8

K. S. KIM. -- THREE-LEVEL TRAFFIC SHAPER FOR STATISTICAL MULTIPLEXING

Note that in the ideal CBR case in which t l = t2 -- t 3 these three objective functions have their minimum value of zero.

IV. NUMERICAL EXAMPLES

We consider some numerical examples of the GA- based optimal dimensioning for the TLTS along with the related performance evaluations of the statistical multiplexing for VBR video services. For these examples, we use two sets of video data given in [15] which are generated at the rate of 25 frames per second. It is assumed for the data sets that each frame is composed of cells with 48 octet user data; and the distributions for the number of cells per frame are

S : mean = 173.7423, variance = 9845.146,

maximum = 835, p = 0.98459,

A : mean = 439.375, variance = 22358.56, maximum = 1406, p = 0.9731,

where p is the autocorrelation coefficient. We set the output line rate of the TLTS, which is the access line rate to a statistical multiplexer, to the DS-3 rate of 45 Mbit/s, so that the cell time becomes 8.5333 x 1 0 - 6 seconds. In this environment, we can obtain, using (2), the following MMPP parameters :

S : r l = 9781.5868, r2 = 3212.0417,

ql,2 = 0.32137, q2,1 = 0.06687,

A : r l = 17671.4611, r2 = 8894.6606, ql,2 = 0.51940, q2,1 = 0.16231.

629

Next, we consider the constraints that the shaped traffic must satisfy. For the probabil i ty of cell loss, we set the limit, C1, to 10 -8 , which is strict enough to meet the requirements for videoconference services specified in [16]. For the probabili ty of underflow, we set the upper limit, C2 to 10 - s . This upper limit corresponds to one occurrence of underflow event in every 10.67 hours, in the case of a cell flow with average rate of 1 Mbit/s. We restrict the maximum delay within the TLTS, D,~ax, not to exceed more than half a frame, i.e. Ca = 2343 cell time. This condition is also strict enough to meet the QOS of most video services [3].

For the implementation of the GA, we use the GAUCSD system which was developed at the University of Cali- fornia, San Diego [7]. To efficiently encode TLTS parameters into binary strings in the 6A, we employ the differences :

(26) dli ~ li - l i -1 ,

d t i ~ t i - t i+z,

i = 1, 2, 3,

i = 1, 2, 3,

where t4 is a dummy variable whose value is zero. Then we represent all dli's by an 8 bit unsigned integer, and all dti's by a 7 bit unsigned integer. We set the penalty coefficient Cp in (14) to 105~ the population size to 48, the crossover rate to 0.6, the mutation rate to 0.0026, and take the default values provided by the GAUCSD for other parameters (*).

The TLTS parameters obtained through the experiments are as summarized in Table I. For each distribution

(*) Refer to [7] for the details o f these parameters .

TABLE I. - - A s u m m a r y o f TLTS p a r a m e t e r s ob ta ined t h r o u g h the op t ima l d i m e n s i o n i n g exper iments .

R~sumds des paramdtres du rdgulateur obtenus par des expdriences de dimensionnement optimal.

S A

f~ f2 f3 f~ f2 f3

queue th resho lds 11 10 4 22 8 122 52

12 43 26 50 149 156 67

13 119 139 78 2 3 4 182 184

cel l serv ice pe r iods q 82 167 56 28 16 16

t 2 36 22 32 13 6 12

t 3 5 10 5 3 4 6

p e r f o r m a n c e m e a s u r e s PCL 3.58 �9 10 -17 2.41 �9 10 -7 2 .59 �9 1@ 15 1.36 �9 10-17 2 . 5 0 " 10-15 3 . 7 9 . 10-12

PuF 3.43 �9 10-11 7.91 �9 10 - l ~ 6 .06 �9 10 - u 3.27 �9 10- 9 4 .12 �9 10- 23 2.93 �9 1@ u

Dma x 2 3 0 6 2115 2212 2284 2244 1698

6/8 ANN, TI~LI~COMMUN., 50, n ~ 7-8, 1995

630 K. S. K I M . - - T H R E E - L E V E L T R A F F I C S H A P E R F O R S T A T I S T I C A L M U L T I P L E X I N G

we p e r f o r m e d three runs of expe r imen t s wi th d i f ferent

r andom streams. F r o m the table, we can conf i rm that

none o f the constra ints are violated.

For the evaluat ion o f the statistical mul t ip lex ing per-

fo rmances , we assume that both the output line rate and

the access line rates o f the statistical mul t ip lexer are

the DS-3 rate o f 45 Mbit /s . Note that this a s sumpt ion is

sl ightly di f ferent f rom that o f [15], in wh ich access line

rate ran at 8.5 Mbit /s . We set the buffer size o f the sta-

t istical mul t ip lexer to 22 cells , which gives a m a x i m u m

buffer delay o f 200 #s. In this env i ronment , we carry

out several runs o f s imula t ions for each dis t r ibut ion o f

v ideo data vary ing the n u m b e r o f sources. In each run

o f s imulat ion, we take 109 cells. The cell loss probabi l i -

t ies ob ta ined through the s imulat ions are as summar i zed

in Tables II and III r espec t ive ly for each dis t r ibut ion S

and A, wh ich are ske tched in Figures 4 and 5. In the

tables the least cell loss probabi l i ty in each row is made

bold faced for clarification. Note that, in the cases fx ,

f2 , and f3, the cell loss probabi l i ty includes the cell loss

occur red in the TLTS, wh ich is in pract ice negl igible .

10-3 25 t~ e~ O E. 10-4

o

"~ 10-5

10-6

'10-7 ~i . . . . /

I0-8

10-9 14 15

i �84 ......

), �9 i �9

i

16 17

unshaped _ _

fl f2 f3

i J

18 19 20 number of sources

FIG. 4. - - Cell loss probabilities vs number of sources for the distribution S.

Probabilit6 de perte de cellules en fonction du nombre de sources pour la r~partition S.

TABLE II. - - Cell loss probabilities for the distribution S.

Probabilitds de pertes de cellules pour la distribution S

number of sources unshaped fl f2 f3

14 3.74 �9 I(P 7 9.29 �9 11~ 8 5.00 �9 10- 9 1.79 �9 10- 7

15 1.68 �9 10 -6 1.07 �9 10- 7 7.86 �9 10- 9 6.49 �9 10- 7

16 7.69 �9 10 -6 7.54 ~ 10 -6 2.84 �9 10 --6 2.98 �9 10 -6

17 3.22 ~ 10- 5 1.88 �9 10 -5 1.31 �9 10- s 2.39 �9 10- 5

18 1.19 �9 10 -4 2.83 �9 10- 5 7.22 �9 10- 5 7.92 �9 10- 5

19 3.68 �9 10- 4 2.84 ~ 10 -4 2.84 ~ 10 -4 2.66 ~ 10 -4

20 " 8.41 �9 10 -4 7.43 �9 10 -4 6.14 ~ 10 -4 6.05 �9 10 -4

T A B L E I I I . - - C e l l l o s s p r o b a b i l i t i e s f o r t h e d i s t r i b u t i o n A .

Probabilitds de pertes de celiules pour la distribution A.

number of sources unshaped fl f2 f3

6 3.91 �9 10 -7 5.36.10- 7 0 . 0 0 0 . 0 0

7 4.18 �9 10- 5 2.35 �9 10- 5 5.35 �9 10 --6 1.04 �9 10- 5

8 8.14 ~ 10 -4 5.89 �9 10 -4 5.12 ~ 10 -4 3.47 �9 10 -4

9 6.39 �9 10- 3 4.29 �9 10 -3 4.49 �9 10- 3 3.89 �9 10 -3

1 0 2 . 4 1 �9 10- 2 2.10 �9 1 0 - 2 2 . 0 5 �9 10- 2 2.12 �9 10- 2

We observe f rom Tables II and III, and Figures 4 and

5 that the objec t ive funct ion f 2 , the var iance o f the out-

put cell rates, pe r fo rms the bes t in mos t cases ; and the

ob jec t ive func t ion f3, the sum o f squared d i f fe rences o f

adjacent cell service per iods o f the TLTS, is comparab le in

pe r fo rmance to f2. As a whole , the cell loss probabi l i t ies

for the shaped cases are less than those for the unshaped

case. Thus, we can conf i rm that the p e r f o r m a n c e o f an

opt imal d imens ion ing p rocedure is s ignif icant ly influen-

ced by the se lec t ion o f the objec t ive funct ion.

ANN. TI~LI~COMMUN., 5 0 . n ~ 7 - 8 . 1 9 9 5 7 / 8

K. S. KIM. -- THREE-LEVEL TRAFFIC SHAPER FOR STATISTICAL MULTIPLEXING 631

10-1

o ~. 10 -2

o - - 1 0 . 3

10-4

, 1 0 - 5

10-6

j - 2~i;;i?:; ;72~2";~--~

. - 27: / j . ,

/ / ; /

10-3 i , 6 7 8

unshaped _ _ fl I2 fa

9 10 number of sources

FIG. 5. - - Cell loss probabilities vs number of sources for the distribution A.

P r o b a b i l i t d d e p e r t e d e c e l l u l e s en f o n c t i o n d u n o m b r e de s o u r c e s p o u r la r d p a r t i t i o n A .

tiplexing, and that f3 (the sum of squared differences of adjacent output cell service periods) is comparable in performance to f2. In addition, it was confirmed that the cell loss probabilities for the shaped traffics are less than those for the unshaped case. Therefore we find that the statistical multiplexing performances of the three objective functions differ under the finite delay constraints, and that the cell rate statistics heavily affect the performances.

The GA-based optimal dimensioning technique presented in the paper is the first trial to link the results of the performance analysis with the practical design problems of traffic shapers in ATM networks. This technique takes advantage of the t ime-domain statistics of the output traffic of the TLTS. It will be also worthwhile to consider a frequency-domain approach in which the objective function relies on the frequency-domain statistics.

M a n u s c r i t r e fu le 3 j a n v i e r 1995.

V. C O N C L U S I O N S

In this paper we have studied GA-based optimal dimensioning of the TLTS for statistical multiplexing in ATM networks. Differently from other conventional traffic shapers the TLTS which regulates the minimum as well as the maximum cell rate requires a careful study on parameter decision because its effect on the network performance significantly depends on its parameters. The

traditional gradient-based optimization methods or random search techniques, however, are difficult to use for this purpose because the involved functions take complicated nonlinear forms and some of their parameters take on only integer values. The GA-based approach enabled us to overcome this difficulty in optimization procedure and rendered a means to optimally decide the TLTS parameters in practical applications.

For the queueing analysis of the TLTS, we employed the two-state MMPP source model because it is appro-

priate for source modeling of real bursty traffics such as VBR video services. As the measure for the TLTS performance, we took the cell loss probability, the underflow probability, and the maximum cell delay within the TLTS because these parameters are essential to characterize the OOS and the controllabili ty of the TLTS.

The three alternative objective functions f l , f2, and f3 we took for the GA-based optimal dimensioning are taken based on the statistics of the output traffic, assu- ming that it is desirable to shape the traffic characteristics of a VBR source to be similar to those of a CBR one. If there is no delay constraint and no limit in buffer size, all the objective functions get the minimum value of

zero, which means that the output traffic becomes a CBR traffic. According to the simulations performed on two

sets of video data, it turned out that f2 (the variance of the output cell rates) perfonns the best among the three alternatives as the objective function for statistical mul-

REFERENCES

[1] ***. Traffic control and congestion control in B-ISDN. ITU-T Recommendation 1.371 (March 1993).

[2] ***. The ArM Forum. ATM User-Network Interface Specification, version 3.0, PTR Prentice Hall (1993).

[3] OTT (T.), LAKSHMAN (T. V.), TABATABAI (A.). A scheme for smoothing delay-sensitive traffic offered to ATM networks. Proc. of lnfocom (1992).

[4] BERNABEI (F.), GRATT (L.), LISTANTI (M.), SARGHINI (A.). Analysis of ON-OFF source shaping for ATM multiplexing. Proc. oflnfocom (1993).

[5] SIDI (M.) et al. Congestion control through input rate regulation. 1EEE Trans. COM (March 1993), 41, n ~ 3, pp. 471-477.

[6] K~M (K. S.), LEE (B. G.). Three-level traffic shaper and its application to source clock frequency recovery for VBR video services. Proc. of lC3N (1993).

[7] SCHRAUDOLPH (N. N.). A user's guide to GAUCSD 1.4. Computer Science and Eng. Dept., Univ. California, San Diego (July 1992).

[8] GOLDBERG (D. E.). Genetic and evolutionary algorithms come of age. ACM Communications (March 1994), 37, n ~ 3, pp. 113-119.

[9] HEFFES (H.), LUCANTONI (D. M.). A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. 1EEE J. SAC (Sep. 1986), 4, pp. 856-868. LEE (J. W.), LEE (B. G.). Performance analysis of ATM cell multiplex with MMPP input. 1EICE Trans. COM (Aug. 1992), E75-B, pp. 709-714.

HEFFES (H.). A class of data traffic processes. Covariance function characterization and related queueing results. Bell Syst. Tech. J. (July/Aug. 1980), 59, pp. 897-929. LUCANTONI (D. M.), RAMASWAMI (V.). Efficient algorithm for solving the non-linear matrix equations arising in phase type queues. Commun. Statist.-Stochastic Models (1985), 1, n ~ 1, pp. 29-51. RAMASWAMI (V.). The N/G/1 queue and its detailed analysis. Adv. Appl. Prob. (1980), 12, pp. 222-261. GOLDBERG (D. E.). Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, Mass (1989). COHEN (D. M.), HEYMAN (D. P.). Performance modeling of video teleconferencing in ArM networks. IEEE Trans. CSVT (Dec. 1993), 3, n ~ 6, pp. 408-420. ***. Integrated video services (IVS) in broadband ISDN. CCITT Study Group XVIII. Temporary Document 9, Melbourne (Dec. 1991).

[101

[lll

[121

[13]

[14]

[15]

[16]

8/8 ANN. TI~LECOMMUN., 50, n ~ 7-8, 1995

Documents

GA-based optimal dimensioning of three-level traffic shaper for statistical multiplexing in ATM networks