State Description of Wireless Channels Using Change-Point Statistical Tests

Transcription

1 3 JOURNAL OF INTERNET ENGINEERING, VOL., NO., JANUARY 27 State Descrpton of Wreless Channels Usng Change-Pont Statstcal Tests Dmtr Moltchanov, Yevgen Koucheryavy, and Jarmo Harju Abstract Wreless channels are characterzed by hghly dynamc tme-varyng nature often manfestng non-statonary behavor. In ths paper we consder the state of a wreless channel n terms of the pecewse covarance statonary sgnal-to-nose rato (SNR) process and parameterze t usng the probablty dstrbuton functon of the receved SNR and lag- autocorrelaton coeffcent of the correspondng autocorrelaton functon (ACF). In order to dscrmnate the state of a wreless channel when ts characterstcs vary n tme we apply methods of statstcal process control. Partcularly, we use exponental weghted movng average (EWMA) change-pont statstcal test to detect shfts n the mean value of SNR process. The proposed approach s tested usng artfcal traces and SNR measurements of IEEE 2.b wreless channel. Obtaned results demonstrate that changes n wreless channel characterstcs can be tmely detected n realtme. The proposed approach s useful n cross-layer envronment where hgher layers often need explct nformaton about the current state of the wreless channel. Index Terms Wreless, non-statonarty, EWMA test. I. INTRODUCTION Due to movement of a moble user and movement of dfferent objects n a rado channel, the propagaton path between the transmtter and the recever may vary from smple lne-ofsght (LOS) to very complex ones. As a result, any metrc used to represent the qualty of a wreless channel s characterzed by tme-dependent stochastc behavor. Insuffcent sgnal-tonose rato (SNR) at a certan nstant of tme may result n ncorrect recepton of the channel symbol possbly leadng to an erroneously receved bt. Technques such as forward error correcton (FEC) and automatc repeat request (ARQ) may allow to recover from these errors locally. However, even n presence of ARQ and FEC, bt errors may stll propagate to hgher layers resultng n loss of protocol data unts (PDU) at those layers. On the other hand, when the channel qualty s locally good, ARQ and FEC may lead to neffcent usage of avalable bandwdth. To ensure acceptable qualty of the wreless channel, some wreless access technologes ncorporate features allowng to control power of the transmtter such that the target bt error rate (BER) s mantaned. Ths capablty, known as power control, may lead to varous undesrable effects ncludng cell breathng n CDMA systems or ncrease n nterference wth neghbor cells n TDMA systems. These effects should be avoded whenever possble. Another way to mantan the acceptable qualty of wreless channels s to dynamcally change correcton capablty of FEC schemes at the data-lnk layer. FEC procedures use proactve approach elmnatng the nfluence of bt errors n advance ntroducng error correcton redundancy. Ths redundancy s exploted at the recever to recover from bt errors. The major advantage of FEC technques s that they do not ntroduce delays allowng unrecovered Manuscrpt receved August 5, 26; revsed November, 26. Authors are wth Tampere Unversty of Technology, Insttute of Communcatons Engneerng, Networks and Protocols group. Postal address: P.O.Box 553, FIN-33, Tampere, Fnland. e-mals: {moltchan,yk,harju}@cs.tut.f nformaton to be lost. There s a number of papers explorng the correctng effect of FEC codes for dfferent condtons of wreless channels (see [7], [] among others). The state of the wreless channel s usually represented n terms of the stochastc process, where the stochastc varable of nterest s ether SNR or a certan functon of t. Bascally, two types of processes are wdely used to descrbe the state of wreless channels. These are SNR (see [9], [2], [] among others) or PDU error processes (see [3], [7], [6] among others). Latter models am at determnng the performance of a wreless channel at a certan layer of the protocol stack n terms of ncorrectly receved PDUs. Although PDU error models can be drectly measured, the major advantage of these models s that they can be derved from the SNR model at the physcal layer [7], []. From ths pont of vew, PDU error models are seen as an extenson of SNR models. Due to ths reason, to represent the state of the wreless channel, SNR process s often suffcent and wll be further used n ths paper. Most papers that consdered performance of nformaton transmsson over wreless channels ether mplctly or explctly assumed covarance statonarty for wreless channel statstcs. Recently, mportant observatons of wreless channel characterstcs have been publshed n [5]. Authors found ther GSM bt error traces to be non-statonary and proposed an algorthm to extract covarance statonary parts. They further used doubly-stochastc Markov processes to model these parts separately. The modelng trace s fnally obtaned by concatenaton. Among other conclusons, authors suggested that a gven bt error trace can be dvded nto a number of concatenated covarance statonary traces. Snce the bt error probablty s a functon of the SNR value, we can expect the same property for SNR observatons. Despte of mportant conclusons made n [5], authors faled to develop theoretcal bass for ther change detecton algorthm. In ths paper n order to dscrmnate the state of the wreless channel n terms of the covarance statonary SNR segments, we propose to use well-known statstcal methodology based on sequental change-pont statstcal tests. The rest of the paper s organzed as follows. In Secton 2 we revew propagaton characterstcs of wreless channels and models used to capture them. Setup of experments that are used to gather SNR statstcs s descrbed n Secton 3. Model for SNR process s ntroduced n Secton. Sequental statstcal tests for detectng changes n parameters of stochastc observatons are revewed n Secton 5. Partcularly, EWMA change-pont statstcal test s proposed there. Numercal results are gven n Secton 6. Conclusons are drawn n the last secton. II. DESCRIPTION OF WIRELESS CHANNELS To determne parameters of a stochastc process, such as mean and varance, based on only one, suffcently large realzaton we usually assume ergodcty for stochastc observatons. The suffcent condton for a stochastc process to be

2 MOLTCHANOV et al.: STATE DESCRIPTION OF WIRELESS CHANNELS USING CHANGE-POINT STATISTICAL TESTS 35 ergodc s lm K() =, where K(), =,,..., s ts ACF. The concept of statonarty s another advantageous property of stochastc processes. It s of paramount mportance n context of stochastc modelng. Practcally, f some observatons are found to be non-statonary, ther stochastc modelng s rarely feasble. A process s sad to be strctly statonary f all ts M-dmensonal dstrbutons are the same. Only few real-lfe processes are strctly statonary. A process s sad to be weakly (covarance) statonary, f the mean of all ts sectons s the same and ACF depends on the tme shft only,.e. K(t, t 2 ) = K(), = t 2 t. III. SETUP OF EXPERIMENTS In ths paper we use SNR observatons of IEEE 2.b wreless local area network (WLAN) channel operatng accordng to dstrbuted coordnaton functon (DCF) of IEEE 2. n DSSS mode at Mbps. In order to gather SNR statstcs we carred out a number of experments n offce envronment as descrbed below. Accordng to the setup experments shown n Fg., there were two enttes nvolved n communcaton. These are the moble node, called tester node, and the access pont (AP). SNR capturng software was nstalled on moble node. The communcaton between AP and the tester node was of nterest. The tester and AP were n non-los (NLOS) and LOS envronments as explaned below. To capture SNR traces we used Network Stumbler software (verson.., []). Ths program s capable of capturng the SNR value, averagng t over.5 second ntervals. corrdor Fg.. 2 AP 3 rooms Testbed for SNR measurements. All experments were carred out n offce envronment as shown n Fg., where reference ponts denote places where measurements were carred out, AP denotes the placement of AP transcever. Note that the tester node, when n ponts and 3, s n NLOS envronment wth respect to the transcever of AP. The dstance between reference pont and AP s shorter than that between reference pont 3 and AP. Reference pont 2 corresponds to LOS envronment. The experments were carred out as follows. Moble staton was at a certan reference pont for some tme and then was moved to another reference pont. SNR was unnterruptedly measured durng the course of each experment. IV. MODEL FOR SNR PROCESS A. Statstcal Characterstcs Consder statstcal characterstcs of the SNR process when a moble staton s n statonary poston. We concentrate our attenton on two statstcal characterstcs. These are the hstogram of relatve frequences of the SNR process and the correspondng ACF. These two propertes of the SNR process were found to produce the major mpact on the performance of hgher layers [7], []. In what follows, results for reference pont are demonstrated. Results for other reference ponts are qualtatvely smlar. Sequences of SNR observatons for two experments are shown n Fg. 2. Respectve hstograms of relatve frequences of these SNR observatons and ther approxmatons by normal probablty densty functons (pdf) are shown n Fg. 3, where f,e ( ), =, 2,... s the relatve frequency correspondng to the s hstogram bn, m s the number of hstogram bns, = (max Y () mn Y ())/m s the length of the hstogram bn, max Y () and mn Y ()) are the maxmum and mnmum values of the SNR, respectvely. Approxmatons shown n Fg. 3 allow us to assume that SNR values are approxmately normally dstrbuted. χ 2 statstcal test carred out wth the level of sgnfcance set to. has shown that statstcal data belongs to the normal dstrbuton. Emprcal normalzed ACFs (NACF) of two SNR traces are shown n Fg. usng sold lnes wth crcles. One may notce that the memory of the process s short and lmted to several lags. In what follows we propose to approxmate such a behavor usng a sngle geometrcal term, e.g. y() = K Y (), =,,..., where K Y () s the lag- autocorrelaton coeffcent. In Fg. approxmatng functons are denoted by sold lnes. These functons exactly capture lag- autocorrelaton coeffcent of the emprcal processes and do not sgnfcantly overestmate or underestmate the autocorrelaton coeffcents for larger lags. In what follows we assume that when a moble staton s n statonary poston, SNR observatons compose realzaton of covarance statonary stochastc process wth normal margnal dstrbuton and non-neglgble postve short-term autocorrelatons. Ths assumpton s supported by frst- and secondorder statstcal characterstcs shown n Fg. 3 and Fg., respectvely. B. Model for SNR Process In order to model SNR traces we propose to use autoregressve process of order one, often denoted by AR(). A process s sad to be autoregressve of order one f t s generated usng the followng recurson Y (n) = φ + φ Y (n ) + ǫ(n), n =,,..., () where φ and φ are some constants, {ǫ(n), n =,,... } are ndependently and dentcally dstrbuted random varables havng the same normal dstrbuton wth zero mean and varance σ 2 [ǫ]. If a process gven by () s covarance statonary we have E[Y ] = µ Y, σ 2 [Y ] = γ Y (), Cov(Y, Y ) = γ Y (), (2) where µ Y, γ Y (), =,,..., are some constants. Mean, varance and covarance of AR() are related to φ, φ and σ 2 [ǫ] as µ Y = φ, σ 2 [Y ] = σ2 [ǫ] φ φ 2, γ Y () = φ γ Y (). (3) Parameters of AR() models can be found as follows φ = K X () φ = µ X ( φ ), () σ 2 [ǫ] = σ 2 [X]( φ 2 ) where K X (), µ X and σ 2 [X] are pont estmates of lag- autocorrelaton coeffcent, mean and varance of SNR observatons, respectvely.

3 36 JOURNAL OF INTERNET ENGINEERING, VOL., NO., JANUARY 27 Y() (a) Experment Y() (b) Experment 2 Fg. 2. Sequences of SNR observatons for two experments. f,e ( ). f,e ( ) (a) Experment (b) Experment 2 Fg. 3. Hstograms of SNR observatons and ther approxmatons. K Y () (a) Experment, lag K Y () (b) Experment 2, lag Fg.. NACFs of SNR observatons and ther approxmatons. V. CHANGE-POINT SEQUENTIAL TESTS A. The Methodology In order to dfferentate between rapd fluctuatons of the SNR around a constant mean and changes n the mean value of the SNR process we propose to use the so-called change-pont sequental statstcal tests. There s a number of change-pont detecton algorthms developed to date. The common approach to deal wth ths task s to use control charts ncludng Shewhart charts [5], cumulatve sum (CUSUM, [], []) charts, or exponentally weghted movng average (EWMA, [3], []) charts. These charts were orgnally developed n statstcal process control (SPC) where they are successfully used to montor the qualty of producton. The dea of control charts s to classfy all causes of devaton of a certan parameter of the process from ts target value nto two groups. These are common causes of devaton and specal causes of devaton. Devaton due to common causes s the effect of numerous dsturbances affectng a gven process. They are nherent part of a process. Specal causes of devaton are not the part of the process, occur accdentally and affect the process sgnfcantly. Control charts sgnal the pont at whch specal causes occur usng two control lmts. If observatons are between them, the process s n-control. If some observatons fall outsde, the process s classfed as out-of-control. For detectng changes n SNR statstcs the followng nterpretaton of causes of devaton s taken. We assume that common causes of devaton are those resultng from smallscale propagaton characterstcs (see [], Ch. 5). Specal causes are those caused by movement of a user ncludng large-scale changes of the dstance between the transmtter and a recever or shadowng of the sgnal by obstacles (see [], Ch. ). The whole procedure s as follows. Intally, a control chart s parameterzed usng estmates of parameters of the SNR process. When change n the controlled parameter occurs, a new process s consdered as n-control and the

4 MOLTCHANOV et al.: STATE DESCRIPTION OF WIRELESS CHANNELS USING CHANGE-POINT STATISTICAL TESTS 37 control chart has to be re-parameterzed accordng to statstcs of ths process. B. Change n the Mean Value Assume that k observatons {Y (n), n =,,..., k } of a certan stochastc process have the same dstrbuton F. In general, the change pont statstcal test refers to testng the null hypothess, H, that a currently observed observaton k has dstrbuton F aganst alternatve hypothess, H, that ths observaton has dstrbuton F. Formally, t s wrtten as H : F Y (k) = F, H : F Y (k) = F, (5) where F Y (k) s the dstrbuton of observaton k. The latter case n (5) represents stuaton when a change occurs n a dstrbuton and can be rewrtten as H : F Y () = F, =, 2,..., k, { F =,,...,k, H : F Y () = F = k. It s often assumed that dstrbutons F and F are known except for some parameters of F. Control charts are used for detectng changes n these unknown parameters. In our case the form of dstrbutons F and F and parameters of F are known n advance (can be estmated from statstcal data) and the unknown parameter s the mean value of F. Therefore, the task s to detect change n the mean value of the SNR process resultng n followng test (6) H : E[Y ()] = µ, =,,..., k, { µ, =,,..., k, H : E[Y ()] =. (7) µ, = k. where µ and µ are the mean values pror and after change, respectvely. The major problem of change-pont statstcal tests s that they often requre observatons to be realzatons of ndependently and dentcally dstrbuted random varables. However, SNR observatons are not necessary ndependent but can be correlated. Autocorrelaton makes classc control charts less senstve to changes n the mean value. For detectng change n the mean of autocorrelated processes two approaches have been proposed. Accordng to the frst approach, control lmts of charts are modfed to take nto account autocorrelaton propertes of observatons. The dea of the second approach s to ft observatons usng a certan tme-seres model of autoregressve ntegrated movng average (ARIMA) type and subsequently control resduals. The shft n the mean value of observatons s transferred to resduals that are montored. If the model fts emprcal data well, resduals are uncorrelated and control charts for ndependent observatons can be used. Performance of change-pont statstcal tests for autocorrelated data has been compared n []. It was shown that the resduals-based approach performs well when the autocorrelaton s negatve. When the autocorrelaton s postve, modfed control charts on ntal observatons perform better. Ths s manly due to the fact that changes n the mean value of orgnal observatons are only partally transferred to resduals and the transfer of the change s dfferent for postve and negatve autocorrelatons []. Due to these reasons, we use the frst approach. C. EWMA Control Charts Let {Y (n), n =,,... } be a sequence of SNR observatons. The value of EWMA statstc at the tme n, denoted by L Y (n), s gven by L Y (n) = γy (n) + ( γ)l Y (n ), () where parameter γ (, ) s some constant. The frst value of EWMA statstcs, L Y (), s usually set to the mean of {Y (n), n =,,... } or, f unknown, to the estmate of mean. As a result, for on-lne real-tme test there should always be a certan warm-up perod nvolvng estmaton of the mean. In () L Y (n) extends ts memory not only to the prevous value but weghts values of prevous observatons accordng to constant coeffcent γ. In () ths prevous nformaton s completely ncluded n L Y (n ). To show t, let us rewrte {L Y (n), n =,,... } statstcs recursvely, startng from L Y () = Y () L Y () = Y (), L Y () = γy () + ( γ)y (), L Y (2) = γy (2) + γ( γ)y () + ( γ) 2 Y (),.... (9) Snce for any constant n the followng holds ( γ) + ( γ) n =, () n γ = t s easy to see that (9) converges to n L Y (n) = γ ( γ) Y (n ) + ( γ) n Y (). () = EWMA charts take central part among other control charts. Although, accordng to (), the most recent value always receves more weght n computaton of L Y (n), the choce of γ determnes the effect of prevous observatons of the process on the current value of EWMA statstcs. Indeed, when γ all weght s placed on the current observaton, L Y (n) Y (n), and EWMA statstcs degenerate to ntal observatons. As a result, EWMA control chart behaves lke Shewhart one. Contrarly, when γ the current observaton gets only a lttle weght, but most weght s assgned to prevous observatons. In ths case, EWMA control chart behaves smlar to CUSUM one. Summarzng, EWMA charts gve more flexblty at the expense of addtonal complexty n determnng one more parameter γ. Assume now that gven observatons {Y (n), n =,,...,N} are taken from strctly statonary process whose all sectons are ndependently and dentcally dstrbuted random varables wth the same dstrbuton, mean E[Y ] and varance σ 2 [Y ]. In fact, () defnes a new stochastc process as a functon of ntal observatons and ths process has dfferent statstcal characterstcs compared to those of {Y (n), n =,, }. Gven that L Y () = E[Y ], the mean of the process s E[L Y (n)] = E[Y ]( ( γ) n ) + ( γ) n E[Y ], () that converges to constant E[L Y ] = E[Y ] = µ Y as n. The varance of {L Y (n), n =,,... } s gven by ( ) γ σ 2 [L Y ] = σ 2 [Y ] ( ( γ) 2n ). (3)

5 3 JOURNAL OF INTERNET ENGINEERING, VOL., NO., JANUARY 27 Usng (3) control lmts for EWMA charts are computed as follows ( ) γ E[Y ] ± kσ[y ] ( ( γ) 2n ), () where k s a desgn parameter whose values are tabulated n the lterature. Accordng to () out-of-control behavor s sgnaled when L Y (n) at some pont n tme s less than (E[Y ] C(n)) or greater than (E[Y ]+C(n)). Note that n () upper and lower control lmts are tme-varyng n nature. However, when n t s easy to see that lm n ( γ ) ( ( γ) 2n = γ and constant control lmts E[Y ] ± kσ[y ] γ 2 γ (5) can be used. Assume now that gven SNR observatons {Y (n), n =,,..., N} are taken from covarance statonary process wth mean E[Y ] and varance σ 2 [Y ] and can be well represented by AR() process n the form Y (n) = φ +φ Y (n )+ǫ(n). If L Y () = E[Y ], observng () t s easy to see that E[L Y ] = E[Y ] = µ when n. When n approxmaton for varance of {L Y (n), n =,,... } s gven by [] σ 2 [L Y ] = σ 2 [Y ] ( γ ) ( + φ ( γ) φ ( γ) ), () where φ s the parameter of AR() process. Control lmts are then gven by ( ) ( ) γ + φ ( γ) E[Y ] ± kσ[y ], (7) φ ( γ) where k s a desgn parameter whose values are tabulated n the lterature. To parameterze EWMA control charts a number of parameters have to be provded. Frstly, parameter γ determnng the declne of the weghts of past observatons should be set. The values of k and γ determne the wdeness of control belts for a gven process wth a certan σ 2 [Y ] and φ. These two parameters affect behavor of the so-called average run length (ARL) curve that s usually used to determne effcency of a certan change detecton procedure. ARL s defned as the average number of observatons of n-control process up to the frst out-of-control sgnal. The ARL s the functon of both k and γ. Dfferent parameters of k and γ for a gven ARL, σ 2 [Y ] and φ are provded n [], [9]. Fnally, E[Y ] and σ 2 [Y ] are not usually known n practce and must be estmated from emprcal data. Therefore, estmates of E[Y ] and σ 2 [Y ] should be used n (7). A. Artfcal Traces VI. NUMERICAL RESULTS Frstly, let us consder performance of the change detecton algorthm usng artfcal traces. Traces were generated as follows. Frst 3 observatons were generated accordng to AR() model wth a certan mean µ. Next 3 observatons were generated accordng to AR() model wth dfferent mean µ. Varance, σ 2, and lag- autocorrelaton coeffcent, K(), were kept constant for each trace. Two subtraces were concatenated to obtan a sngle artfcal trace. Usng ths algorthm we generated two artfcal traces wth the followng parameters: µ =, µ = 5, σ 2 =, K() =. for trace and µ =, µ = 5, σ 2 =, K() =. for trace 2. Tme seres of generated traces are shown n Fg. 5. One may observe that due to the same varance and lag- autocorrelaton coeffcents pror and after the change, the change n the mean value s almost not notceable for both traces. To detect shfts n the mean value EWMA change-pont statstcal test was then appled to these traces. Results of EWMA change-pont statstcal test for dfferent parameters of γ are shown n Fg. 6. The warm-up perod used to compute parameters of emprcal observatons was set to 5 observatons. For all values of γ parameter k was set to 3. However, n general, usage of dfferent values of γ requres dfferent values of k to match a gven ARL. Behavor of n-control ARL for our traces s shown n Fg. 7. Snce ARL s also functon of the varance and the mean value of observatons, these curves are only sutable for presented observatons. Note that k = 3, γ =. corresponds to n-control ARL for trace and 9.6 n-control ARL for trace 2. Parameters k = 3, γ =. corresponds to 53. n-control ARL for the trace and ncontrol ARL for trace 2. However, one can observe from Fg. 6 that changes n the average value of traffc observatons are detected for all values of k and γ and no false alarms occurred. B. SNR Measurements Let us now apply the proposed EWMA test to SNR observatons. Two SNR traces were obtaned as explaned below. Accordng to the course of the frst experment, the tester node was statonary at the reference pont (see Fg. ) for approxmately 3 mnutes and then was moved to the reference pont 2, where t was also statonary for a long tme. The full tme-seres of SNR observatons s shown n Fg. (a). One can see that the change n the mean value of SNR observatons occurs around th observaton. Accordng to the second experment, the tester node was n the statonary poston at the reference pont for approxmately mnutes. Then, t was moved to the reference pont 3 skppng the reference pont 2. The node was then at the reference pont 3 for a long perod of tme. The full tme seres of SNR observatons s shown n Fg. (b). The change n the average value of SNR observatons occurs around th observaton. Results of EWMA change-pont statstcal test for trace are shown n Fg. 9(a), Fg. 9(b). The warm-up perod used to compute parameters of emprcal observatons was set to 5 SNR observatons (25 seconds). Note that for all γ the parameter k was set to 3. It was found that usage of tabulated (theoretcal, [9]) values of k and γ correspondng to relatvely small values of n-control ARL (< ) usually leads to many out-of-control sgnals even when the tester node s n the statonary state. One of the reasons s that the montored process may not be exactly covarance statonary and may occasonally contan some extreme observatons (outlers). These observatons are of local sgnfcance only and may not affect the future evoluton of the montored process. Snce the test s real-tme n nature, each out-of-control sgnal starts a new warm-up perod. Durng ths perod new average value and control lmts are estmated and the process cannot be montored. From ths pont of vew we should detect only those changes that occur for sure. Durng experments, t was found that value k = 3 provdes a good choce for any value of γ n the range γ (.,.). One can observe from the Fg. 9 that the change n the average value of SNR

6 MOLTCHANOV et al.: STATE DESCRIPTION OF WIRELESS CHANNELS USING CHANGE-POINT STATISTICAL TESTS 39 Y() (a) Trace : µ =, µ = 5, K() =. Y() (b) Trace 2: µ =, µ = 5, K() =. Fg. 5. Tme seres of artfcally generated traces. L Y (), C mn (), C max () L Y (), C mn (), C max () 2 (a) Trace, γ = (c) Trace 2, γ =. L Y (), C mn (), C max () L Y (), C mn (), C max () 2 (b) Trace, γ = (d) Trace 2, γ =. Fg. 6. EWMA charts for generated traces. ARL, tme 6 ARL, tme Gamma =. Errors for Gamma =. k Gamma =. Errors for Gamma =. (a) Trace : µ =, σ 2 =, K() = Gamma =. Errors for Gamma =. k Gamma =. Errors for Gamma =. (b) Trace 2: µ =, σ 2 =, K() =.2 Fg. 7. In-control ARL curves for generated traces.

7 JOURNAL OF INTERNET ENGINEERING, VOL., NO., JANUARY 27 Y() Y() AP 2 AP (a) Trace (b) Trace 2 Fg.. Tme seres of SNR observatons. L Y (), C mn (), C max () (a) EWMA chart, γ =. L Y (), C mn (), C max () (b) EWMA chart, γ =. Fg. 9. SNR observatons and EWMA chart for reference ponts and 2. L Y (), C mn (), C max () (a) EWMA chart, γ =. L Y (), C mn (), C max () (b) EWMA chart, γ =. Fg.. SNR observatons and EWMA chart for reference ponts and 3. ARL, tme 6 ARL, tme Gamma =. Errors for Gamma =. k Gamma =. Errors for Gamma =. (a) Trace : µ =, σ = 5, K() = Gamma =. Errors for Gamma =. k Gamma =. Errors for Gamma =. (b) Trace 2: µ = 2, σ = 3, K() =. Fg.. In-control ARL curves for our measurements.

8 MOLTCHANOV et al.: STATE DESCRIPTION OF WIRELESS CHANNELS USING CHANGE-POINT STATISTICAL TESTS observatons s successfully detected for all values of k and γ and no false alarms occurred. Results of EWMA change-pont statstcal test for trace 2 are shown n Fg. (a), Fg. (b). For all values of γ parameter k was set to 3. The warm-up perod was constant and set to 5 observatons. One can observe from Fg. that the change n the average value of SNR observatons s successfully detected for all values of k and γ and no false alarms occurred. Behavor of n-control ARL for our traces and dfferent values of γ and k s shown n Fg.. VII. CONCLUSION To dscrmnate the state of a wreless channel n terms of the covarance statonary SNR processes we proposed to use EWMA change-pont statstcal test. To represent SNR process when a moble user s n the statonary state we use AR() model. EWMA test s appled to detect a pont at whch the mean value of the SNR process changes. Presented results demonstrate that the proposed test s able to quckly detect ponts at whch change n the mean value of SNR observatons occur. Although some modfcatons are requred, we beleve that the proposed approach can be also used for detectng changes n parameters of PDU error observatons. Possble applcatons of the proposed algorthm nclude real-tme cross-layer parameters adaptaton system, where the proposed test may sgnal ponts at whch change n wreless channel statstcs occur. In response to these sgnals, layers of the protocol stack of the wreless channel may perform approprate actons ncludng changes n the transmtted sgnal strength, error correcton capablty of the FEC code, type of the ARQ scheme, etc. The choce of the smoothng parameter that affects wdeness of control lmts s stll an open ssue. Our experments have shown that theoretcally-computed parameters correspondng to relatvely small values of n-control ARL may lead to many out-of-control sgnals. Ths s due to a number of reasons ncludng mperfect modelng, naccurate estmaton of parameters, presence of outlers, etc. Optmal length of warmup s another queston that remans open. In ths context, the am of the future work s to optmze the proposed test for on-lne mplementaton. REFERENCES [] D. Apley and J. Sh, The GLRT for statstcal process control of autocorrelated processes, IIE Transactons, vol. 3, 999, pp [2] J. Arauz and P. Krshnamurthy, Dscrete raylegh fadng channel modelng, Wreless Communcatons and Moble Computng, vol., no., July 2, pp [3] E. Glbert, Capacty of a burst-nose channel, Bell Systems Techncal Journal, vol. 39, 96, pp [] Y. Y. Km and S. Q. L, Capturng mportant statstcs of a fadng/shadowng channel for network performance analyss, IEEE JSAC, vol. 7, no. 5, May 999, pp. 9. [5] A. Konrad, B. Zhao, A. Joseph, and R. Ludwg, Markov-based channel model algorthm for wreless networks, Wreless Networks, vol. 9, no. 3, May 23, pp [6] A. Kopke, A. Wllg, and H. Karl, Chaotc maps as parsmonous bt error models of wreless channels, n Proc. IEEE INFOCOM 23, San Francsco, USA, March 23. [7] D. Moltchanov, Y. Koucheryavy, and J. Harju, Cross-layer modelng of wreless channels for IP layer performance evaluaton of delay-senstve applcatons, Computer Communcatons, vol. 29, no. 7, Aprl 26, pp. 27. [] D. Moltchanov, Y. Koucheryavy, and J. Harju, Loss performance model for wreless channels wth autocorrelated arrvals and losses, Computer Communcatons, vol. 29, no. 3, August 26, pp [9] A. Neskovc, N. Nescovc, and G. Paunovc, Modern approaches n modelng of moble rado systems propagaton envronment, IEEE Communcaton Surveys, Thrd Quarter 2, pp. 2. [] Network stumbler, Verson.., Accessed on..25. [] E. Page, A test for a change n a parameter occurrng at an unknown pont, Bometrka, vol. 2, 955, pp [] T. Rappaport, Wreless communcatons: prncples and practce, 2nd ed. Prentce Hall, 22. [3] S. Roberts, Control chart tests based on geometrc movng averages, Technometrcs, vol., 959, pp [] W. Schmd and A. Schone, Some propertes of the EWMA control chart n the presence of autocorrelaton, Ann. Stats., vol. 25, no. 3, 997, pp [5] W. Shewhart, Statstcal Method from the Vewpont of Qualty Control, 939. [] W. Shmd, CUSUM control schemes for gaussan processes, Statstcal Papers, vol. 3, 997, pp [7] W. Turn and M. Sondh, Modelng error sources n dgtal channels, IEEE JSAC, vol., March 993, pp [] J. Wernga, Control charts for montorng the mean of AR() data, Unversty of Gronngen, Department of Econometrcs, Faculty of Economc Scences, Avalable at: Accessed on [9] J. Wernga, Statstcal process control for serally correlated data, PhD Thess, Unversty of Gronngen, Department of Econometrcs, Faculty of Economc Scences, Avalable at: Accessed on..25. Dmtr Moltchanov s a Senor Research Scentst n the Insttute of Communcatons Engneerng at the Tampere Unversty of Technology, Fnland. He receved the M.Sc. and Canddate of Scence degrees from State Unversty of Telecommuncatons, St.Petersburg, Russa n 2 and 22, respectvely, and the PhD degree from Tampere Unversty of Technology, Tampere, Fnland n 26. Hs research nterests nclude performance evaluaton and optmzaton ssues of wreless and wred IP networks, traffc nature n the Internet, ad hoc and P2P networks. Dmtr Moltchanov serves as a TPC member n a number of nternatonal conferences. He authored more than publcatons. Yevgen Koucheryavy s a Senor Research Scentst n the Insttute of Communcatons Engneerng at the Tampere Unversty of Technology, Fnland. He receved the M.Sc. and Canddate of Scence degrees from State Unversty of Telecommuncatons, St.Petersburg, Russa, and the PhD degree from Tampere Unversty of Technology, Tampere, Fnland. Before jonng the Tampere Unversty of Technology, he spent fve years wth R&D LONIIS n St.Petersburg, Russa, where he held varous techncal and manageral postons. Hs research nterests nclude QoS provson technques for wreless/moble communcatons, network performance evaluaton, traffc measurement and teletraffc theory. He s the General/Program Char of several nternatonal conferences. He s the Char of COST 29 Acton Traffc and QoS Management n Wreless Multmeda Networks. He authored over 5 publcatons n journals and nternatonal conferences. Jarmo Harju receved hs M.Sc. from Helsnk Unversty of Technology n 979 and Ph.D. n mathematcs from the Unversty of Helsnk n 9. Durng 95-9 he was a senor researcher at the Telecommuncatons Laboratory of the Techncal Research Center of Fnland, workng wth the development of protocol software. In he was professor of data communcatons at Lappeenranta Unversty of Technology. Snce 996 he has been professor of telecommuncatons n the Insttute of Communcatons Engneerng at Tampere Unversty of Technology, Fnland, where he s leadng a research group concentratng on network archtectures and QoS ssues.