# M M Guziy - Mathematical modelling of efficiency criteria for the traffic service - страница 1

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UDC 004.738.2

M.M. Guziy, G.V. Danilina, V.O. Ignatov, M.V. Bogoljubov

National aviation university, Kiev, Ukraine

MATHEMATICAL MODELLING OF EFFICIENCY CRITERIA FOR THE TRAFFIC SERVICE

© GuziyM.M., Danilina G.V., Ignatov V.O., BogoljubovM.V., 2007

Розроблені математичні моделі критеріїв ефективності і алгоритми оцінювання ефективності обслуговування трафіку гетерогенних комп'ютерних мереж, що побудо­вані на порівнянні фактичного режиму обслуговування трафіку з оптимальним режи­мом і оцінюванні збільшення значення критерію середнього ризику у разі відхилення режиму від оптимального режиму.

Mathematical models of criteria, algorithms for efficiency evaluating of the traffic service of heterogeneous computer networks which are constructed on comparison of an actual mode of the traffic service with an optimum mode and which permit to evaluate increase of values of criteria of average risk in case of a deviation of a mode from an optimum mode are developed.

I. Introduction. The big attention [1-3] is paid to methods and algorithms of modeling of the traffic of computer networks. But there are not enough works where problems of a choice and mathematical modeling of optimization criteria of the traffic service [4, 5] are discussed. In this actual scientific direction there are no almost work which would be devoted evaluating to efficiency of the traffic service in open information networks. Considered work is focused on elimination of this blank.

II. Purpose of work. The purpose of work is the choice of metric space, a substantiation of the principle of efficiency evaluating, development of mathematical models of criteria and algorithms for evaluating efficiency of the traffic service in heterogeneous computer networks. Main principle is comparison of an actual mode of the traffic service and an optimum mode for evaluating relative increase of criterion values of average risk in case of a deviation of a mode from an optimum mode.

III. Statement of a problem. It is supposed, that are known: function of losses because of refusals in service of packages of the data at a transport level of reference model of open information systems, function of expenses for maintenance of the necessary throughput of a transport level of system of service of the traffic, coefficient of using of throughput of this system. On these entrance data the choice of optimum throughput of system of service of the traffic which provides a minimum of criterion of average risk by the method, which was worked out by us in work [5], is carried out. Results of the problem decision of optimization serve as a basis for development of the analysis method of efficiency of the traffic service in heterogeneous computer networks on the criterion of a minimum of average risk.

IV. Decision of the problem. In a role of criterion of inefficient use of system throughput of the traffic service at a transport level it is convenient to choose a relative parameter of a deviation of an actual mode from an optimum mode in kind

where D(Q)) and Dmin(Copt), accordingly, values of criterion of average risk in not optimum and

optimum modes of the traffic service in a heterogeneous computer network. Necessary conditions of use of criterion (1) look like

^(C0 , Copt )

D(C0 ) - Dmin (Copt) Dmin (Copt)

(1)

(2)

The theorem 1. Normalized values Z of criterion of average risk depend only on factors popt and po uses of system throughput of the traffic service at a transport level in optimum and not optimum modes, as

Z(popt, Copt, Co) :

■\popt

1 + po

opt

p0 + popt popt p0 (3)

The proof of the theorem 1. For an optimum mode the equation of balance of expenses and losses is fair

(1 - P)bCopt

Dmin (Copt )

C2

opt

where P is probability of that the line will be occupied in an optimum mode. From this equation it is easy to define values of parameters a and b through Dmin (Copt) / 2

Dmin (Copt) Copt

a = ■

2 P Dmin (Copt) 1

(4)

2        (1 -P)Copt

The system of parameters (4) allows finding dependence of average expenses on throughput of line

C0 in a not optimum mode as

D

[C C D (C )]= Dmin(Copt) [уо?^ opt '-^rninv- opt/J 2

Copt + Co

C) Copt

Let's take into account that the maximal value of the minimal average expenses

D = fOb = Dmin(Copt)

1

\P(1 - P)

and optimum value

C = її і—= i±-zc і—

Copt    лі ь VPopt     V   p   Copt " ^popt

(5)

(6)

(7)

Let's execute normalization (5) on (6) and we shall take into account thus optimum value (7), we shall receive

D[C0, Copt, Dmin (Copt )\

Z (popt, C0, Copt) :

Copt Co

Dmax min

(8)

1 + popt

Let's take into account in (8) that

|

C0 Copt

popt = RCoptpo = R/C0, (9)

(1))

Finally we shall receive

Z (popt, Copt, C0) :

opt

p0    , popt

1 + popt (popt p0

as was to be shown.

Consequence 1.1. From the Eg.3 follows, that at p0 = popt

Zmin (popt) :

2y[pPo

opt

(11)

1 + popt

It means that the minimal normalized value of criterion of average risk depends only on optimum value of system throughput of the traffic service at a transport level.

Consequence 1.2. From a limiting parity

lim    Z(p0,popt) = Zmm(popt) = 2 ^P°Pt (12)

p0 ~^popt 1 + popt

follows, that at all po * popt the normalized average expenses

Z(Po *Popt) > , (13)

1 popt

Therefore this law is offered to be taken as a principle efficiency evaluating of the traffic service at a transport level in heterogeneous computer networks.

The theorem 2. The normalized relative value of criterion of average risk £(po, popt), which

characterizes a deviation of an actual mode of the traffic service on transport level from an optimum mode, is completely determined by factors po And popt systems of service and does not depend on other

parameters of a mode because

S(po, popt) = 2

( p0    + popt ^

The proof the theorem 2. As and the minimal value

popt po Z(po * popt) - Zmin (popt) > 0,

Z (  )=2Vpopt

Zmin (popt)

-1 (14)

1 + popt

that a measure of an inefficiency of a mode of throughput use is expedient to choose increase of the normalized relative expenses in an actual mode in comparison with an optimum mode which plays a role of the standard

.     Z (p0 * popt) - Z min(popt)

°(po, popt) =-z(-)- ()

Zmin (popt )

Substituting in Eq.15 value Z(po * popt) from the Eq.8, and value Zmin(popt) from the Eq. 11,

we shall receive Eq.14, as was to be shown.

Consequence 2.1. For efficiency evaluating of the traffic service at a transport level it is necessary to know productivity R of the traffic source, throughput optimum by criterion of a minimum of average risk

of service system Copt and actual average throughput C0 of this system.

Consequence 2.2. Having executed necessary transformations of an analytical parity (7), we shall receive the following evident form of inefficiency criterion an of the traffic service at a transport level

Я(п    n   )    (po -popt)2 (16) ^-HoHopt

From the Eq. 16 follows, that the inefficiencies criterion (15) of the traffic service is constructed on the metrics of normalized Euclid's space. For increase of method sensitivity at small deviations it is possible to use root-mean-square value of criterion (16)

сКАь popt) = Jdi^oT^opt) = p0   p°Pt =•, (17)

■sj2p0popt

where a sign of a difference po - popt shows, in what side the value deviation of system throughput of the traffic service C0 from optimum value takes place Copt.

Consequence 2.3. Function (17) is asymmetrical concerning valuepopt, therefore a deviation of a mode aside values with smaller than popt results in the greater damage, i.e.

°1 (p0 - Popt < 0) > °2 (p0 - Popt > 0) Therefore, if on any circumstances it is impossible to use an optimum mode of the traffic service then it is better to use the traffic service on modes at P0 > p0pt.

V. A technique for efficiency evaluating of the traffic service. Let's consider a technique and algorithm of efficiency evaluating the traffic service of on a concrete example. We shall assume, that intensity of a stream of messages      = 1 messages/sec, average length of one message 10 = 60 bit. /messages, the allowed

normalized average expenses Z * in an optimum mode should not be more than 1/2. It is necessary to construct algorithm of efficiency evaluating of the traffic service to calculate optimum throughput of the communication line and to estimate efficiency of its use at two relative values of a line throughput:

£°i_=0,6; c°l=1,4.

C C ^opt ^opt

The algorithm of the problem decision contains such steps.

1. We shall calculate optimum operating ratio of throughput of the communication line from the equation which we shall receive from the Eq.11 at Zmin = 1/2:

= 2yj popt = Zmin (popt) =~ = 1/2 .

1 + popt

From this equation we shall find, that popt * 0,072.

2. Optimum throughput of the communication line we shall find on speed of the information transfer and optimum operating ratio

Copt =-=-*-* 833,33 bit/sec.

popt      popt 0,072

3. Value of the normalized expenses for not optimum modes we shall find under the Eq.8

C01              л/0,072 ( 1 ^ Z1 (pont, —— = 0,6) = -1-+ 0,6 I * 0,5667,

1 op   Copt 1 + 0,072 ^ 0,6 J

C02             л/0,072 (        1 ^ z 2(popt —— = 1,4) =-1 1,4 + I* 0,5300.

2 Hop   Copt 1 + 0,072 {      1,4 J

4. We shall define throughputs of not optimum modes

C01 = 0,6 Copt * 0,6 • 833,33 * 500, bit/sec.

C02 = 1,4 Copt * 1,4 • 833,33 * 1166,7 , bit/sec.

5. We shall define increase of the normalized expenses for not optimum modes under the Eq.15

,1 (0,072; 0,6) = Z1(0,072; 0,6) - Zmin(0,072; 1,0) =

1 Zmin(0,072; 1,0)

= 0,5667 - 0,5000 * 0,1354, 0,5

,2(0,072; 1,4) * Z2(0,072; 1,4) - Zmin(0,072; 1,0) *

2 Zmin(0,072; 1,0)

0,5300 - 0,5000

=-* 0,06.

0,5000

Thus, a deviation of a line throughput on 40 % from optimum value Copt = 833,33 bit/sec. aside the

greater value results to approximately twice to smaller relative increase of the normalized expenses in comparison with a deviation of throughput on the same size in the smaller side.

6. We shall define operating ratios of the communication line throughput for not optimum modes

pql =j- * _60 * 0,1200, 01    C01 500

P02 *-*-* 0,0514

2    C02 1166,7

Thus, increase of operating ratio of throughput in P01 * 0,1200 * 1,66 time results in increase of

Popt 0,072

relative expenses (3) approximately on 13,54 %. Reduction of operating ratio of throughput in

Popt 0,072

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M M Guziy - Mathematical modelling of efficiency criteria for the traffic service