Comuter Modelling and ew Technologies, 5, Vol.9, o., 3-39 Transort and Telecommunication Institute, Lomonosov, LV-9, Riga, Latvia MATHEMATICAL MODELLIG OF THE WIRELESS COMMUICATIO ETWORK M. KOPEETSK Deartment of Comuter Engineering, Holon Academic Institute of Technology, Golomb 5, Holon 58 E-mail: mkoeets@.net.il, Tel: 97-3-5653, Fax: 97-3-565 The aer is devoted to study and evaluation of the erroneous ackets flow on the hysical layer of a wireless communication network. A mathematical model of the erroneous ackets assed to the communication channel has been structured and analysed. The statistical estimation of the erroneous ackets number is resented and discussed in the aer. Keywords: Wireless communication channel, erroneous ackets, error bursts.. Introduction Most communication systems are sensitive to random errors and synchronization failures. The Physical Layer analysis of any Comuter Communication network is very imortant. This is because many different roblems concerning network execution and utilization are caused by errors and failures on the Physical Layer. Wireless communication networks are not the excetion. Moreover, the roblem of achieving high reliability and error tolerance is urgent in modern communication technologies such as mobile Internet Protocol (IP) or General Packet Radio Service (GPRS) networks []. A number of roblems arise in a wireless communication channel context. These roblems are caused by the following reasons. The first reason is related to non-reliability of the radio channel. As a matter of fact, a sufficiently high Bit Error Rate (BER) and very high synchronization failure robability as oosed to the qualitative wire and esecially fiber otic channels characterize these channels. BER may 3 be rather within. The second reason is that BER in the radio channel is not a constant value. As soon as the reliability of the wireless channel deends on several external reasons, BER is essentially a time function. However, in recent years there has been an intensive interest in wireless channels. A number of aers devoted to the imact of burst errors on the network reliability and on acket synchronization have aeared [, 3], among others.. General Princiles for Determining the Probabilistic Model of the Wireless Channels A various range of BER within a given Virtual Connection characterizes wireless communication channels. Therefore, it is reasonable to create an aroriate robabilistic model of the wireless channel. The urose of this aer is to analyse the general number of erroneous ackets during a secific time interval. It should be mentioned that BER and Packet Error Rate (PER) do not change during a acket transmission in the resented model. The PER change is exected after the ending of a given acket transmission. Mention that PER is a BER function in a network model. Let us assume that there is n* various channel states. Each channel state is defined by a secific PER. Actually, is a discrete random variable. For each channel state i, is equal to i. Let s assume that the current channel state corresonding to the given acket transmission is. Then. For the () next acket transmission there exists a finite robability ( ) of non- changing of the current channel n* ( n*),( i state. Let us denote the robabilities set,...,,... ). ( i) () ( i) Each value secifies a robability that within a next acket transmission a channel state will be i. Actually, a simle Markov chain may describe the discrete rocess that determines the current channel state [4]. The chain is defined by the robability transition matrix Z z ) and by the initial robabilistic ( i) ( i () 3
vector π ( ) [...]. Evidently, dim(z)n*. The Markov chain is homogeneous. The homogeneity is caused by the following reasons: The acket flow in the communication channel is stationary. The acket transmission time is assumed to be finite and constant. It should be remarked that acket transmission time variance and in the acket rocessing time variance are omitted. The concise evaluation of the erroneous ackets number causes significant difficulties, which are related to the variability of the robability. Therefore, the roosed mathematical model is directed to the robabilistic evaluation of the researched characteristics. 3. The Distribution of the Erroneous Packets umber in Case n* Assume that a single change of the value is robable during a transmission of several ackets (Figure ). () c Figure. The channel state robability as a function of a transmitted acket number c in the case n*. The robability of the change of the channel state is equal to. It can be assumed that the channel state was changed after a transmission of (c-) ackets. Then the revious (c-) ackets are characterized by the PER while the next (-c+) cells are characterized by the PER. The ( ) ( ) following discrete random variables y and y are introduced: y y,..., c y y } } c,.... } } c () The discrete random variables () () c y and () c y are defined. c Since the random variables under consideration are discrete and integer, it is reasonable to determine the Generating Functions (GF) [4, 5] of the s abstract variable: π / c c+ x () c c x x} s () c x x} s [ [ ( ] x c+ ( ]. c, π / c () () Since the random variables c and c are indeendent for a fixed value (c-), then the Generating () () Function of the random variable + is determined as c c c ( ) ( c / c / c / c π ( s ) π ( s ) π ( s ) [ ( s )] [ ( s )]. () () c+ 33
The random variable c determines the erroneous ackets number under the condition that the channel state was switched on after the (c-) acket transmission. The GF π / c ( s) characterizes conditional distribution of the erroneous ackets number under the following condition: c-l+. Here l denotes the number of ackets, which have been transmitted in channel state. The GF corresonding to the unconditional distribution is generated as π { π / l P( l)} {[ [ l ( ] l ( ] l [ ( ( { P() + P() +... + ) ( ( l ( ] P( l)} P( )}. P(l) is the random variable l distribution [6]. Clearly, the sum from the equation () is essentially the GF ( of the random variable l with the argument, which substitutes the argument s. Consequently, ( ( π [ ( ] π l. ( (3) The equation (3) contains information about by means of the items P(l). The first item of π (s) () [ ( ] determines GF of the binomial (Bernoulli) distribution with the argument s. Let us take u the limit transition that is investigated if. Such situation is relevant if the number of transmitted ackets is sufficiently large. For this urose the following rates are introduced: () a, a, d. () () The random variable that is characterized by the binomial distribution with the GF π s) [ ( ], is also introduced. The urose is to determine ( () da ( s) lim π lim [ a ( ] e. Considering the PER switching mechanism let s find the limit of the second item of the π (s). Clearly, () () P() ( ), P() ( ),..., P( ), P( ). Hence, ( π l ( ( () ( () ) () ) ( () ( ) ( )( ) s s ( ) ( ) s s () ( ( ) () ( + ( ( ) () () () ( + + ( + + ( () ). ( By carrying out all actions related to the determining of the lim π l, we conclude that ( dv( s) d da ( ) ( ) s d e e π s lim π e e +, (5) v + vs where v a a. In summary, any order initial distribution moment of the random variable under consideration is determined on the π ( s ) base. As a matter of fact, exact determination of the erroneous ackets number is ossible if n and the only single switch of the channel mode is robable. () () (4) 34
4. Analysis of the Erroneous Packets umber in Case n*> In a general case, if during a secific interval S, the number of channel modes switches is more than one, as well as in the case n*3, the distinct calculation of π ( s ) is rather comlicated. The roblem discussion is resented for n*3. It can be assumed that the channel switching rocedure is deicted in Figure. Q 3 Q () ( ) Q () ( ) () c c c Figure. The channel state robability as a function of a transmitted cell number c in case n*3. The following assumtions are made:. The only two leas of the robability are ossible during a secific time interval S.. On the first stage,. It means that no error takes lace during the transmission of (c-) ackets. 3. The two channel state switches are ossible with the robabilities and after the transmission of ( c ) and ( c ) ackets resectively. As a result the grah (c) includes the tree areas: Q, Q and Q 3. The channel state switch robability in the area Q or Q is variables () y and or consequently. In the area Q 3 this robability is equal to. The random () y are defined as in the revious case. Then π π l [ ( ] ( ) + ( ( (6) + ( π l[ ( ] π l[ ( ] ( ))). ( + ( According to the assumtion, no acket error is ossible in the range [,,c-]. The arguments of the GFs π l{ f ( s )} essentially are the functions of s. Therefore π l{ f } are formed from (4) by means of ( the argument changing of f. Clearly, the deendence π (s) on is reflected in ( the argument change. Such deendence is absent in (6) in the obvious form. The considerable calculation difficulties arise during a limit transition when. The function π as well as in the revious d case contains the item e. Contrary to the revious model, π ( ) is not reresented as the multilication of the GFs number. However, this roblem can be solved by means of numerical methods based on the comuter calculations. The resented Table demonstrates the mathematical exectation and variance of the erroneous ackets number considering the arameters,,,,. s 35
TABLE. Mathematical exectation E and variance V of the non-synchronous cells number.. E 8 6. 7 V 8.5 8 6. 8.9 6 8 6.8.3 5 4.5. 8 8 5. Evaluation of the Erroneous Packets umber in Case of any n* and It should be stated that in a more general case it is not reasonable to study the erroneous ackets number by means of the discrete distributions. Therefore, the transition from the discrete random variable to the continuous one is required. Hence, study of the discrete random variable distribution is relaced by the evaluation of the continuous random variable U distribution moments. The researched characteristic U takes all ossible values in the range [, ]. Moreover, the and U mathematical U exectations are equal: E E. It should be assumed that the transition condition of the Central Limit Theorem takes lace, i.e. E U, is sufficiently large. According to the general rinciles of the research methods of continuous random variables, The Moment Generating Function (MGF) is introduced M U + Uη ( η) e ϕ( U ) du, where ϕ(u ) is the robability density of the random variable U. Evidently, GFs and MGFs are connected by common features. Analyzing the roblem under consideration M U (η ) is formed from π (or π ) by the η argument change s e. Consequently, based on the theorem of continuity and on the theorem of uniqueness [], the limit M lim M U (η) defines the U limit distribution function (or robability density η function) in the unique way if. The general theorem of continuity sets corresondence between the limit distribution functions and the MGFs of a certain argument. As a matter of fact, in the continuous distribution case, any initial or central U distribution moment of the -th order is not equal to the corresonding moment of the discrete distribution if. However, the following inequality is correct: µ U µ,. (7) Actually this inequality is caused by the following reasons: in the general case of the U continuous distribution for the analysis of the ( c ) function not a single realization with a given robability, but the random variable w distribution must be taken into account. Actually, w takes w m values that denote the channel switch from the current state to the m state for the next acket transmission (Figure 3). n* Obviously, the normalizing condition takes lace: P ( w m ). m The roosed mathematical method is based on the change of all random rocess realizations that corresond to the channel state b switch to the set of the other ossible channel states, by the average realization. This realization is secified by the channel state switch robability b n* m m b g, where g is the average robability of the channel state m. This fact immediately exlains the inequality (7). m m 36
ν 3 () () (3) ( ν ) c c c Figure 3. The channel state robability as a function of a transmitted acket number c in the case of any n* The case of n* will be discussed further. The transition from π ( ) (5) to the corresonding MGF is executed by means of argument change s e : η s M U ( η) e η da ( e ) e d η dv( e ) e e + v + vs d. (8) Assuming that is rather small, the first item tends to. Each -th order initial distribution moment is derived by means of the -th order derivative in the oint η. Hence the generated distribution function determines the low limit of any order initial (or central) distribution moment. In order to study the more general deendence ( c ) (Figure 3) the following limitations are introduced: d, a (conditions ). The first condition ( d ) takes lace in the case when the channel state switch (or switches) occurs with a sufficiently high robability if the cells transmission time is rather large. The second condition ( a ) covers the case when the channel switches are rather rare. Then the random variable U is transformed into the random variable B such as M ( η ) lim ( η). Based on the solution B M U d, a methods of the analogous roblems in Probability Theory in relation to the continuous distributions [5], the B random variable is reresented as the linear combination of the indeendent random variables B l. The Bl number deends uon the ( c ) ums. In fact for the model reresented by Figure the following equation is correct: B B ab. It is assumed that the acket error in the channel state is imossible. Here the B random variable follows the normal distribution with the mathematical exectation and the variance E ( B ) V ( B ) ad. The B random variable also follows the χ distribution with two degrees of freedom, i. e., the B robability density function is β / ϕ( B ) e. In the same way in the case n*3 from the revious aragrah B B ab ( a a ) B3. Here, a. 37
B is characterized by a normal distribution with the arameters E B ) V ( B ) a. Finally, B and ( d B 3 follow the χ distribution with two degrees of freedom. Analogously in the general case (Figure 3) the random variable is transformed into the continuous random variable B under the limit transition: B B aυ B ( a a ) B3 ( a a ) B +. υ υ υ υ υ (9) Here B is characterized by the normal distribution with the arameters f E( B ) V ( B ) aυ d, a f,. f f Consequently, the MGF of the erroneous ackets number, which is aroximated by the random variable B, is determined as the MGF items multilication. Indeed the B items distributions are known. If all items in (9) are of the close orders, then the B distribution tends to the normal distribution. This fact is confirmed by the Launov theorem: Assuming that the random variable z is resented as z z + z + zk, where z z, z,, zk are the indeendent random variables and k growth is unlimited. The random variable z k distribution function is F k ( z ) f ( z )dz. k z 3 E( zi E( zi )) i If lim k 3 / ( V ( z)), (**) z E( z) then lim ( ). ( ) F k z Φ k V z t y Here Φ( t) e dy is the Lalace function. Therefore, if the limit (**) takes lace and the π number of k items is sufficiently large, then the z random variable distribution is closed to the normal distribution. Otherwise, if among the items ( aυ a f ) f there are items greatly different from the others, then the B random variable is aroximated by the sum [6]: B B + hb, h const. Each initial B -th B B order distribution moment µ is calculated as d ( ) B µ M B s. Finally, µ is substituted by the ds U µ r evaluation in the inequalities (7). Conclusions The following remarks comlete the aer. It is ossible to create the similar mathematical model in the case when acket errors are not () neglected in the range [, c ]. The above mentioned range corresonds to the robability. The monotonous growth of the and values by the transmitted ackets number increment is assumed. evertheless, this assumtion is not necessary. Moreover, there are no limitations on the (c) or ( c ) functions ums if the conditions ( ) take lace. Actually, based on equation (9) in the non-monotonous ( c ) case the υ is equal to the maximal f robability. It should be mentioned that f value corresonds to a given channel state switch. Evidently, the items order in the sum is meaningless. Finally, the limit transition is legitimate if the transmitted cells number is sufficiently large. Several roblems are considered for future work. Based on the real data, the modified decoding algorithm that aroriates to the non-reliable noisy wireless environment should be chosen and alied in the modern mobile IP or GPRS technologies. Thus, the PER in the wireless network should be decreased. Consequently, the network reliability should be significantly imroved in comarison with the existing models. s 38
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