ON SOME STATISTICAL INDICATORS OF THE TRAFFIC IN WIRELESS LOCAL AREA NETWORKS

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1 U.P.B. Sci. Bull., Series C, Vol. 70, No. 3, 008 ISSN x ON SOME STATISTICAL INDICATORS OF THE TRAFFIC IN WIRELESS LOCAL AREA NETWORKS Şerban Alexanru STĂNĂŞILĂ 1 În ultimul timp, capacitatea e protocol ale reţelelor fără fir, e exemplu pragul permis într-un canal e comunicaţie, joacă un rol important în aplicaţiile Internet. Există mai multe moele teoretice şi variante pentru a estima întârzierile în transmisie atorate coliziunilor sau congestiei. În acest articol, se stuiază un moel bazat pe schema programării carelor, poziţionat între stratul legăturii logice şi cel al controlului meiului e acces. Se au câteva estimări privin meiile şi ispersiile unor variabile aleatoare, care escriu utilizarea curentă a reţelelor fără fir. In the last time, the protocol capacity of the wireless networks, for instance the throughput as allowe in a given wireless channel, plays an important role in the Internet applications. There are many theoretical moels an variants to take into account the transmission elay ue to collisions or to subsequent transmission. In this paper, one stuies a moel base frame scheuling scheme, positione between the layers of the logical link an meium access control (MAC) an give some statistical estimations regaring the expecte values an the variations of some ranom variables, which escribe the collisions Keywors: transmission control protocol (TCP), meium access control, statistical inicators, performance evaluation. 1. The escription of the Frame Service Time By a flui unity we mean any sequence of collision perios followe by a successfully transmission frame, where a collision perio is compose of ile slots an a collie frame. In the Fig. 1, one inicates such a unit. In practice, one meets sequences of such flui unities (SIFS = short interframes space ; ACK = acknowlegment ; DIFS = istribute interframe space ; EIFS = extene interface space ). Each flui unity consists of zero or more collision perios, followe by a successful frame transmission. One can a some mechanisms like request to sen or clear to sen, but here we avoi them. 1 Eng., Amaeus, Munchen, Germany

2 14 Şeban Alexanru Stănăşilă FRAME ACK SIFS DIFS IDLE SLOTS COLLIDED FRAME EIFS SUCCESSFUL TRANSMISSION IDLE SLOTS Fig.1 Let us fix a flui unity. A funamental inicator is the frame service time enote, in what follows, by T, as being the time necessary for a successful transmission of a frame. In a sense, T can be ientifie with the length of the consiere flui unity. T is a ranom variable an can be expresse by some simpler components. Define the following ranom variables (relatively to the same probability fiel (Ω, K, P), which is unerstoo ) : N = the number of collision perios; C k = the k-th collision perio, 1 k N ; C = the total length of collision perios; S = the time taken by a successful transmission of a frame; F = the size of a frame; CF = the size of a collie frame; S k = the number of iles slots before the k-th collision or the successful transmission, 1 k N. Directly, with these, one can formulate: PROPOSITION 1. The following relations take place: T = C + S ; (1) C = C 1 + C + + C N ; () C k = S k + CF + EIFS, for any 1 k N ; (3) S = S N F + SIFS + DIFS + ACK. (4) One can mention that the ranom variables F an are ientically istribute; EIFS, SIFS, DIFS an ACK are system parameters, with their values known (e.g. in IEEE operate WLAN, SIFS = 10 μsec, ACK = 11 bits/ 1 (Mb/s); EIFS = SIFS + DIFS + ACK);[1], [3]. On the other han, the following hypothesis are legitimate: C an S are inepenent ranom variables; C 1, C,, C N are inepenent of each other, with the same istribution; S k, CF an F are

3 On some statistical inicators of the traffic in wireless local area networks 15 inepeneat of each other. One also remark that the number of ile perios in a frame service time is N + 1 an put CS = S N + 1. We will below give some statistical estimations regaring the means (expecte values) an variations of the above ranom variables.. Some statistical inicators for a flui unity Recall that for any ranom variable X (relatively to the consiere probability fiel), one can efine its cumulative istribution function F X : R [0, 1], F X (t) = P(X t). Suppose that this function is erivable an its erivative p X : R R, calle the probability ensity function, belongs to the class L 1 of the absolutely integrable functions. Denote by X(ω ) = F {p X (t)}, the Fourier jω t transform of p X, that is X(ω ) = p X (t).e t = E(e j ω t ), calle also the characteristic function of the ranom variable X. Obviously : X(0) = 1, X(ω ) 1, X (k) k (0) = ( jt). px (t) t = ( j) k.ν k, where ν k = E(X k ) is the moment of orer k of X. Particularly, for the mean an the variance of X, it will follow that : EX = ν 1 = j. X(ω ) ; Var X = ν (ν 1 ) = X(ω ) (EX). (5) ω ω We will also apply the following fact: if X an Y are inepenent ranom variables, then p X + Y = p X p Y (convolution) an thus: (X + Y)(ω ) = X(ω ). Y(ω ) an Var(X + Y) = Var(X) + Var(Y). (6) hol PROPOSITION. With transparent notations, the following relations a) T(ω ) = C(ω ). S(ω ), for any real ω ; (7) b) S(ω ) = CS(ω ). F(ω ).e j α ω, (8) where α = SIFS + DIFS + ACK ;

4 16 Şeban Alexanru Stănăşilă c) C(ω ) = = n 0 n Γ(ω ).P(N = n), (9) where Γ(ω ) is the characteristic function of the ranom variables C k s ; ) Γ(ω ) = Γ 1 (ω ). CF(ω ). e j β ω, (10) where β = EIFS an Γ 1 (ω ) is the characteristic function of the ranom variables S k s ; Proof. a) By (1) an (6), T(ω ) = (C + S) (ω ) = C(s).S(ω ), since C an S are inepenent. b) By (4), one has S = Y + α, where Y = CS + F. But Y(ω ) = CS(ω ).F(ω), accoring to (6). On the other han, for the cumulative istribution functions: F S (t) = F Y (t α ), hence p s (t) = p Y (t α ), whence S (ω ) = F {p S (t)} = F {p Y (t α )} = e j α ω.y(ω ) an one gets (8). j ω c) By (), C(ω ) = E(e C jω C1 jω CN ) = E( e...e ) = E j C1 ( e ω j CN ) E( e ω ). Since C 1,, C N are ientically istribute, C 1 (ω ) = = C N (ω ) not = Γ(ω ), hence C(ω ) = n Γ(ω ).P(N = n). n = 0 ) From (3), it follows that C k (ω ) = S k (ω ).(CF) (ω ). ).e j β ω ; but all ranom variables C k (respectively S k ) have the same probability ensity function, namely the same Γ(ω ) an Γ 1 (ω ), whence (10). Recall that for any ranom variable X, one put X instea of EX = mean. Directly from the propositions 1 an, one then gets:. COROLLARY. T = C + S ; C = N.Γ (0).j an S = N.Γ 1 (0).j + F + α In the following, we euce some explicit formulas for means an for variances. For means, the results are similar to some of those given in [1], in terms of Laplace transformations, but for variances the results are new. By a local convention, for the characteristic function U(ω ), of a ranom variable U, put U = j. U(ω ) an U = ω ω U(ω ). Thus, ν 1 = U = U, ν = U an Var U = U U,

5 On some statistical inicators of the traffic in wireless local area networks 17 PROPOSITION 3. For any 1 k N, the following relations hol: a) C k = S k + CF + β ; (11) b) Var C k = Var S k + Var CF 4β (S k + CF ). (1) Proof. a) From (11), Γ(ω ) = Γ1 (ω ). CF(ω ).e j β ω + ω ω + Γ 1 (ω ). CF(ω ).e j β ω jβ Γ 1 (ω ).CF (ω ).e j β ω ; multiplying by j an ω making, one gets : Γ = Γ CF.1+ β hence (11). b) On the other han, ω Γ(ω ) = ω Γ 1(ω ). CF(ω ).e j β ω + + Γ 1 (ω ). CF(ω ).e j β ω jβ Γ1 (ω ). CF(ω ).e j β ω + ω ω + Γ1 (ω ). CF(ω ).e j β ω + Γ 1 (ω ). ω ω ω CF(ω ).e j β ω j β Γ 1 (ω ). CF(ω ). e j β ω j β Γ1 (ω ). CF(ω ). e j β ω ω ω j β Γ 1 (ω ). CF(ω ). e j β ω β. e j β ω an making, Γ = Γ 1 + ω Γ 1. CF β Γ 1 + CF β CF + β. Therefore, by (11) : Var Γ = Γ Γ = Var Γ 1 + Var CF 4β Γ 1 4β CF, whence (1). The istribution of the ranom variable CF is practically known (being the same with that of the size of a frame). It remains the problem to estimate the repartition of S k. An amissible hypothesis is the following: any S k is a ranom variable exponentially istribute, with a parameter λ > 0 which can be establishe by experiments. In this case, if the maximum contention winow size M of S k s is M, then by putting J = 0 an VarS k = 1 M (t Sk ) pλ (t) t I. 0 1 I p λ (t) t, it follows that k = M S λ 0 t p (t) t

6 18 Şeban Alexanru Stănăşilă Recall that the probability ensity function of an exponentially istribute ranom variable is p λ (x) = λ.e λ x for x 0 an p λ (x) = 0 for x < 0. After a simple computation, it follows that, for any 1 k N, S k = λ M e λ M 1 λ M λe λ M λm 1, (13) if the prouct λ M is small ; a similar result gives Var S k. In this way, the formulas (11), (1) allow to estimate the main statistical inicators of C k, S k, whence one can estimate the behaviour of the mean T of the frame service time. 3. Statistical inicators for a full MAC unity As we have sai, a flui unity is mae up of zero or more collision perios followe by a successful frame transmission. Consier now the more general case, that if a chain of p successive flui unities; such a chain can be calle a MAC unity (Fig. ) an enote by {T 1,, T p }. C S C S C S C S IDLE PERIOD T 1 T p Fig.. We have enote by S k the number of ile slots before the k-th collision or the successful transmision, 1 k N, in a flui unity. Denote by L = T T p, the length of the above consiere MAC unity. By a similar reasoning, by consiering T i like inepenent ranom variables which have the same istribution, one can show that the characteristic function L(ω ) of L satisfies the functional relation L(ω ) = S t (λ λ L(ω )).T(ω ), (14) where S t is the total number of ile slots in that MAC unity, uner the hypothesis that the number of transmission attempts is Poisson istribute with a parameter λ > 0.

7 On some statistical inicators of the traffic in wireless local area networks 19 Put u = λ λ L(ω ), hence L(ω ) = S t (u(ω )).T(ω ); by erivating two times with respect to ω, one obtain : L L T = St (u(ω )).( λ ).T(ω ) + St (u(ω )). an ω u ω ω L ω = u L S t (u(ω )).λ t ω. L T(ω ) λ St (u(ω )). u ω.t(ω ) L T L T λ St (u(ω )).. + St (u(ω )). ( λ ). + u ω ω u ω ω T S t (u(ω )). ω. Put Z = St (u(ω )) an W = S u t (u(ω )). By making in the u above relations an keeping into account that for any ranom variable X, one has X X(ω ) = 1, ν 1 (X) = j. X(ω ) an ν (X) =, one ω ω gets : ν 1 (L) = Z( λ ν 1 (λ )) ν 1 (T) an ν (L) = W.λ.( ν 1 (L)) λ.z. ν (L) + λ.z. j 1.ν 1 (L). j 1 ν 1 (T) + Z + + λ. j 1.ν 1 (L). j 1 ν 1 (T) + ν (T), whence (1 + λ Z). ν 1 (L) = ν 1 (T) an (1 + λ Z). ν (L) = W.λ.( ν 1 (L)) λ.z. ν 1 (L). ν 1 (T) + ν (T). Thus, we have prove: PROPOSITION 4. The mean an the variance of the length of a MAC unity are given by : T L = ; (15) 1+ λ Z Var L = 1 λ T. Var T 3 1+ λ Z (1 + λ Z) (1 + W + Z ). (16) These relations can be interprete in terms of the physical reality.

8 0 Şeban Alexanru Stănăşilă R E F E R E N C E S [1] H. Kim, Jennifer C. Hou, Improving Protocol Capacity for UDP / TCP Traffic, IEEE Journal on Selecte Areas in Communications, vol., no. 10, , ec [] Ş. Al. Stănăşilă, Transfer control protocol (TCP) over ATM networks, Matrix Rom, [3] W.R.Stevens, TCP /IP The protocols,vol. 1, Aison Wesley, [4] A.S. Tanenbaum, Reţele e calculatoare, (e. a III-a), E Agora, 1998.