Markov Modulated Punctured Autoregressive Processes for Traffic and Channel Modeling *

Transcription

1 Markov Modulated Puctured Autoregressive Processes for Traffic ad Chael Modelig * Trista Pei-chu Che ad Tsuha Che + Departmet of Electrical ad Computer Egieerig Caregie Mello Uiversity, PA 1513 {peichu, tsuha}@adrew.cmu.edu Abstract We preset a ew stochastic process called the puctured autoregressive (AR) process, ad use it to model both the variable bit rate (VBR) video traffic ad the wireless chael dyamics. To model the VBR video traffic, we propose to use puctured autoregressive processes modulated by a doubly Markov process. The doubly Markov process models the state of each video frame while the autoregressive processes describe the umber of bits of each frame at each state. The puctured autoregressive process cosiders the timig iformatio betwee frames of the same state ad thus gives better modelig performace. The model captures the log-rage depedecy (LRD) characteristics as well as the shortrage depedecy (SRD) characteristics of the video traffic. Queuig behavior of the puctured autoregressive process is also closer to the real video traffic tha the covetioal autoregressive process. I additio to video traffic modelig, we also apply the same model to wireless chael dyamics. The chael error rate is modeled as a sigle Markov modulated puctured autoregressive process. The sythetic chael error rate geerated by the puctured autoregressive process performs closer to the real chael error rate tha the oe geerated by the covetioal autoregressive process. I. Itroductio To both the video service providers ad the etwork desigers, it is importat to have a good model for the video traffic. A good model for video traffic allows for better admissio cotrol, schedulig, etwork resource allocatio policies, etc., that guaratee a desired quality of service (QoS) as well as a better utilizatio of the etwork resources. A good model captures essetial characteristics of the real video traffic. The sythetic trace geerated by such a model ca be used to test the etwork performace uder a certai, for example, admissio cotrol policy. Therefore, the etwork desigers ca desig a etwork that is more friedly to the video traffic ad thus delivers a better video service. I additio to the video traffic modelig, it is also crucial to model the wireless chael dyamics. I wired etworks, the trasmissio is early error free. However, i the wireless etwork, the chael error rate fluctuates depedig o the oise, distace, speed of the mobile uit, multi-path iterferece, etc. A good model of the wireless chael dyamics, i particular the chael error rate i this paper, ca be used to provide a better error cotrol mechaism for the data packets as well as a better wireless etwork desig. For example, we ca test differet Automatic Repeat Request (ARQ) protocols with sythetic chael errors geerated by the chael modeler to select a ARQ scheme that suits the best. Because of the importace of both the variable bit rate (VBR) video traffic modelig ad the wireless chael dyamic modelig, may models have bee proposed. To evaluate differet models, there are geerally three criteria to be take ito accout (1) A good model should capture the statistical properties of the real trace. A trace is defied as a sequece of data we ited to model. I the case of VBR video traffic modelig, a trace is a sequece of umbers, each represets the umber of bits to ecode each Group of Blocks (GOB)/frame/Group of Pictures (GOP). I the case of wireless chael modelig, a trace is a sequece of umbers, each represets the chael bit error rate (BER) at differet time istat. The statistical properties should iclude those that are related to the log-rage depedecy (LRD) of the trace as well as those that are related to the short-rage depedecy (SRD) of the trace [1]. * Work supported i part by Idustrial Techology Research Istitute. + Correspodig author: Tsuha Che, Electrical ad Computer Egieerig, Caregie Mello Uiversity, Pittsburgh, PA 1513, Tel: (41) , Fax: (41) , tsuha@cmu.edu.

2 () The sythetic video trace should be similar to the real video trace i terms of the queuig behavior. Likewise, the sythetic trace of the wireless chael should be similar to the real trace i terms of the QoS behavior. (3) The model should be simple ad easy to be aalyzed. Related discussio o how to evaluate the performace of the models ca be foud i []-[4]. Video traffic modelig is challegig i several aspects. First, video is usually ecoded with VBR to be adaptive to the video cotet. Secod, depedig o differet codig schemes, the video trace has very differet properties. Popular video codig schemes iclude H.63 [5] ad MPEG-4 [6]. The video frame ca be Itra (I), Predictive (P), or Bidirectioally predictive (B) ecoded. The GOP structure, which is defied as frames eclosed by two I frames icludig the leadig I frame, ca be fixed or dyamic. A fixed GOP structure that is commoly used is IBBPBBPBBPBB. Existig work for video traffic modelig icludes DAR [7], which fails to capture the LRD property of the video traffic. Models such as [8]-[11] are costraied by a fixed GOP structure. No-statistical methods as [1][13] are usually more difficult to aalyze. The wavelet-based model [14] ad the autoregressive process (AR) based model [15] do ot capture the dyamic ature of the video traffic icely. I geeral, it is preferred to use a Markov chai like process to model the dyamic ature of the video traffic. Models such as [16] are too complex. We propose to build a model based o the work doe by [17], which models the video traffic as a doubly Markov process with AR processes iside each Markov state. However, the model of [17] does ot use the timig iformatio betwee frames of the same Markov state. I this paper, we explicitly use the timig iformatio betwee frames of the same Markov state ad refer to the ew model as a doubly Markov process modulated puctured AR process. It is show that the proposed model outperforms the model of [17] i terms of statistics, both SRD ad LRD, queuig behavior, ad has the same complexity i terms of the umber of model parameters. Covetioal chael aalyses model the chael statistics with physical layer parameters such as the sigal to oise ratio (SNR), speed of the mobile uit, etc. Most of the models cocetrate o matchig the probability distributio fuctio (pdf) of the sythetic trace with the real trace of the chael ad do ot match the LRD property such as the Hurst parameter [3]. I additio, a chael model with a perfect statistical property match to the real chael does ot guaratee similar behavior to the packets trasmitted o top of which. It is more importat to capture the behavior of the packets sedig over the chael. Some related models based o Markov process ca be foud i [18]. More related work i modelig the chaels ca be foud i [19][]. We propose i this paper a Markov process modulated puctured AR process to model the error rate of the chael. It is show that the puctured AR process performs better tha the covetioal AR process. This paper is orgaized as follows. Sectio II itroduces the core cocept of this paper puctured AR process as opposed to the covetioal AR process. I Sectio III, we preset the doubly Markov process modulated puctured AR process as well as the covetioal o-puctured scheme. Both models are show side by side with performace comparisos. I Sectio IV, we apply the puctured AR process to model the wireless chael. The o-puctured scheme is also preseted ad compared. Sectio V cocludes our work. II. Puctured Autoregressive Modelig A covetioal autoregressive (AR) process x is defied as follows: x = ρx 1 + σ 1 ρ e, = 1,, L (1) where is the time idex of the process, ρ describes the depedecy of the sample at time with the previous sample at time 1, σ is the variace of the process x, ad e is a Gaussia radom variable with mea ad variace 1 to characterize the radom ature of the process. Cosider the case where more tha oe AR processes are iterleaved together such as show i Figure 1. At time istat 1, AR process x takes place; at time istat, AR process y takes place; ad so o.

3 Covetioal method to trai two sets of AR parameters of sequeces x ad y is by splittig the sigle process i Figure 1 to two separate processes as show i Figure 1 ad Figure 1 (c). Each oe of the processes represets the traiig sequece of x or y regardless of the time idex associated with each sample. For example, the sequece i Figure 1 is used as if samples are ~ x ~ 1x ~ x ~ ~ ~ ~ ~ ~ 3x4x x 5x6 x7 x () 8 9 where ~ x 1 = x1, ~ x = x3, ad so o. x 1 y x 3 x 4 x 5 x 6 y 7 y 8 x 9 y 1 x 11 x 1 x 13 y 14 Figure 1. Two iterleaved autoregressive processes x ad x 1 x 3 x 4 x 5 x 6 x 9 x 11 x 1 x 13 y y 7 y 8 y 1 y 14 autoregressive process x ; (c) autoregressive process (c) y : the iterleaved process; To sythesize samples usig this model, two separate AR processes are geerated with parameters traied by x~ ad y~. Sythetic samples geerated by parameters traied by x~ are take oe after the other to form the fial sythetic process. Sythetic samples geerated by parameters traied by y~ are formed i a similar way. I brief, o sythetic samples are skipped. It ca be see that the covetioal method does ot take ito accout the timig iformatio of the two processes i both the traiig ad sythesis stages. To cosider the timig iformatio, we propose to trai ad sythesize the AR processes as follows (Figure ). First split the sigle process i Figure to two separate processes as show i Figure ad Figure (c). Notice that the timig iformatio is utilized while leavig the sample of x blak if at some particular time istace the origial process is with the other process y (Figure ). Similarly, the sample of y is blak if at some time istace the origial process is with the other process x (Figure (c)). x 1 y x 3 x 4 x 5 x 6 y 7 y 8 x 9 y 1 x 11 x 1 x 13 y 14 Figure. Two iterleaved autoregressive processes x ad y. x 1 _ x 3 x 4 x 5 x 6 x 9 _ x 11 x 1 x 13 _ _ y y 7 y 8 _ y 1 _ y 14 (c) y : the iterleaved process; puctured y. autoregressive process x ; (c) puctured autoregressive process To trai the puctured AR processes, we fid the value ρ ~ of the process ~ x ~ 1x ~ x ~ ~ ~ ~ ~ ~ 3x4x x 5x6 x7 x8 9. We also costruct the histogram of the sample spacig. For example i Figure, the histogram has values [5,, 1] at spacig [1,, 3]. Recall that i the simplest case where all the samples are adjacet to each other, the histogram has values [8,, ] at spacig [1,, 3]. The ρ of the process i Figure ca be foud by solvig the followig equatio: The value of ρ + ρ + 1ρ ~ = ρ ( ) ρ Y i Figure (c) ca be solved i the same way. (3)

4 To sythesize samples usig this model, two separate AR processes are geerated by ρ ad ρ Y. The fial sythetic process is formed by some meas of multiplexig/modulatio of the two sythetic processes. Samples are ot take oe after the other but with cosideratio of how separate the samples of the same type are. I brief, some sythetic samples are skipped. Puctured AR processes ca be modulated by differet processes. I this paper, we specifically cosider the Markov process. III. Variable Bit Rate Video Traffic Modelig 1. Doubly Markov modulated AR processes: Puctured ad Covetioal Before we proceed to the proposed model for VBR video traffic, let us briefly describe the method proposed by [17] with which we will compare (Figure 3). The model comprises of two layers of Markov process. Without loss of geerality, we cosider two frame types, I ad P, i this paper. The doubly Markov process models I ad P frame trasitios, as well as differet frame activities. The outer Markov process describes how I ad P frames trasit. The frame type ca be further categorized ito differet activity levels. Frames of higher activity level cosume more umber of bits, while frames of lower activity level cosume less umber of bits. The ier Markov process describes how the frames of differet activity levels trasit. This model is ot costraied by a fixed GOP structure. There are six states i total, amely, I frame i high activity level, I frame i medium activity level, I frame i low activity level, P frame i high activity level, P frame i medium activity level, ad P frame i low activity level. Each state is modeled as a AR process with differet AR parameters. We ca cosider this as a slightly more complex process tha the oe described i Figure 1 of Sectio II. There are two states i Figure 1 of Sectio II while there are six states i this model. The AR parameters are traied i the way described i Figure 1 of Sectio II i a covetioal o-puctured maer. The sythetic trace is geerated by procedures described i Sectio II as well. To simply the model, the ier Markov process of I frames is characterized by iitial probabilities of three activity levels oly. Sice I frames are usually far apart i a video sequece, they do ot eed to modeled as a Markov process. Such simplificatio will have similar performace as the full model. We call this model Method 1, as show i Figure 3, for later discussio. We propose to model the VBR video traffic as a doubly Markov modulated puctured AR process. This ew model explicitly cosiders the timig iformatio betwee two frames of the same state. Agai, this is a slightly more complex process tha the oe described i Figure. There are two states i Figure of Sectio II while there are six states i this model. The AR parameters are traied i the way described i Figure of Sectio II i a puctured maer. The sythetic trace is geerated by procedures described i Sectio II as well. We call this model Method, as show i Figure 3, for later discussio. AR(1) H I State M H P State M Puctured AR(1) H I State M H P State M L L L L Figure 3. Models for VBR video traffic Method 1: Doubly Markov modulated AR process; Method : Doubly Markov modulated puctured AR process.. Experimet We ow compare the performace of the o-puctured Method 1 with the proposed puctured Method. To evaluate the performace of the models, we cosider four performace metrics: (1) first order statistics by meas of the quatile -quatile (Q-Q) plot; () secod order statistics by meas of the autocorrelatio fuctio (ACF); (3) LRD property by meas of the Hurst parameter from the rage/stadard deviatio (R/S) plot; ad (4) queuig behavior by meas of the packet loss rate ad the queuig delay. Defiitios of the performace metrics ca be foud i []-[4].

5 The experimet settig is as follows. Two differet types of TV programs are recorded: ews as show i Figure 4 ad talk show as show i Figure 4. The two TV programs are ecoded usig video compressio codec H.63 to geerate the real video traces. Both of them are ecoded with frame rate of 15 frames/sec ad with duratio of 3 miutes each. The video trace of the clip ews is show i Figure 5 ad with differet scales. The video traces are the fed ito both models: Method 1 ad Method to geerate sythetic traces. The performaces of the two models are evaluated. Figure 4. Test videos: ews; talk show 4 x 14 3 x 14 bitrate (bits/frame) secods bitrate (bits/frame) secods Figure 5. Sample traces from the TV program ews : a secod trace; a secod trace. The performace compariso is summarized i Table 1. MSE refers to mea square error compared to the real trace. It ca be see that the proposed Method outperforms Method 1 i all five aspects. The MSE improvemet is computed by ( Method ) ( Method 1) / ( Method 1). Detailed discussio associated with each performace metric will be preseted later. Table 1 Summary of performace compariso betwee modelig methods Method 1 ad Method Q-Q plot ACF Hurst parameter Real.53 Method e e (98e-4 i SE) Method.1747e+7.454e (3.85e-4 i SE) packet loss rate e e-4 queuig delay e e-4 MSE improvemet 85.5% 63.56% 96.1% 1.88% 6.87% Figure 6 shows the performace of both models i terms of first ad secod order statistics. The first order statistics i Figure 6 is show by the Q-Q plot. The Q-Q plot is costructed by a pair of cumulative distributio fuctios (CDF). The closer oe CDF is to the other CDF i oe pair, the more the curve will look like a straight lie y = x. I Figure 6, a dotted straight lie is plotted as a referece. We have two pairs of CDF to compare: the sythetic trace by Method 1 with respect to the real video trace ad the sythetic trace by Method with respect to the real video trace. The dashed-dotted curve i Figure 6 refers to the first pair. The solid curve i Figure 6 refers to the secod pair. It is show that the curve of Method is closer to the referece dotted straight lie tha the curve of Method 1.

6 The secod order statistics i Figure 6 is show by the ACF. The dotted curve i Figure 6 refers to the ACF of the real video trace. The dashed-dotted curve i Figure 6 refers to the ACF of the sythetic trace by Method 1. The solid curve i Figure 6 refers to the ACF of the sythetic trace by Method. It is show that the curve of Method is closer to the referece curve tha the curve of Method x 1 4 Real Method 1 Method x x Real Method 1 Method lag Figure 6. First ad secod order statistics of the sythetic traces geerated by Method 1 ad Method with respect to the real video trace of the clip ews. First order statistics: Q-Q plot; Secod order statistics: ACF. The LRD property i Figure 7 is show by meas of the Hurst parameter, which is the slope of the liear regressio lie of the poits i a R/S plot. Figure 7 shows the R/S plot of the real video trace. The liear regressio lie is show as a dotted lie. Figure 7 shows the R/S plot of the sythetic trace by Method 1. Figure 7 (c) shows the R/S plot of the sythetic trace by Method. It is show that the sythetic trace by Method has closer Hurst parameter value to the real video trace tha the sythetic trace by Method 1. R/S i log R/S i log R/S i log scale N i log scale N i log scale N i log (c) Figure 7. LRD properties of three traces by Hurst parameter from the R/S plots: real video trace; sythetic trace by Method 1; (c) sythetic trace by Method The queuig behavior of the traces is evaluated by meas of the packet loss rate ad the queuig delay. Packet loss rate ad queuig delay are measured at differet drai rates ad buffer sizes. The etwork is a leaky bucket with drai rate R ad time to drai M R, where M is the buffer size. The queuig performace of the real trace is show i Figure 8. The queuig performaces of the sythetic traces by both Method 1 ad Method have similar look. However, the sythetic trace by Method has smaller MSE i both the packet loss rate ad the delay tha the sythetic trace by Method 1 (Table 1).

7 Figure 8. Queuig behavior of the real video trace: packet loss rate; queuig delay. IV. Wireless Chael Modelig I this paper, we model i particular the bit error rates (BER) of the chael at differet time istaces. Ulike other chael modelig work, which models the chael with iformatio such as the SNR of the received sigal, speed of the mobile uit, chael modulatio ad codig scheme, etc., we model the chael solely from the trace of the bit error rates. To evaluate the models, we ot oly examie the statistics of the sythetic traces but also the queuig behaviors of the traces with ARQ as the error cotrol scheme. We preset two models with variatios i the uderlyig AR processes: puctured ad opuctured. 1. Markov modulated AR processes: Puctured ad No-Puctured The BER at ay time istace ca be categorized as high, medium, or low. The three error rate states: high, medium, ad low, are cosidered as three states i a Markov process with certai iitial probabilities ad trasitio probabilities. At each state, the BER is modeled as a AR process. There are three AR processes modulated together by a sigle Markov process. The Markov modulated opuctured ad puctured AR processes are illustrated i Figure 9. We call the covetioal opuctured Markov modulated AR process Method 1 ad proposed Markov modulated puctured AR process Method. The traiig ad sythesis procedures are described i Sectio II with umber of AR processes three. AR(1) Puctured AR(1) High Medium High Medium Low Low Figure 9. Models for the chael Method 1: Markov modulated AR process; Method : Markov modulated puctured AR process. Experimet A sample trace of BER i time is show i Figure 1, which is a Rayleigh fadig chael [1].

8 .5.4 error rate secods Figure 1. Sample trace of the wireless chael i BER over time. The performace compariso is summarized i Table 1. It ca be see that the proposed Method outperforms Method 1 i the Hurst parameter ad the queuig behavior. Method does ot perform as well as Method 1 i terms of first ad secod order statistics. Sice we are more iterested i the wireless etwork performace, we should cocetrate more o the queuig behavior. Detailed discussios associated with each performace metric will be preseted later. Table Summary of performace compariso betwee modelig methods Method 1 ad Method Q-Q plot ACF Hurst parameter Real.77 Method e (.39 i SE) Method e (. i SE) packet loss rate queuig delay 9e-4 13e e e-4 MSE improvemet % % 48.7% 78.6% 7.65% Figure 11 shows the first ad secod order statistics of the sythetic traces geerated by Method 1 ad Method. The straight dotted lie i Figure 11 is the referece. The dashed-dotted curve i Figure 11 refers to the first pair of the sythetic trace by Method 1 respect to the real trace. The solid curve i Figure 11 refers to the secod pair of the sythetic trace by Method respect to the real trace. It is show that the curve of the sythetic trace geerated by Method strays further away from the referece lie tha the curve of the sythetic trace geerated by Method 1. The dotted curve i Figure 11 refers to the ACF of the real trace. The dashed-dotted curve i Figure 11 refers to the ACF of sythetic trace by Method 1. The solid curve i Figure 11 refers to the ACF of sythetic trace by Method. I Figure 11, both ACF curves of Method 1 ad Method are close to the ACF curve of the real trace Real Method 1 Method Real Method 1 Method lag Figure 11. First ad secod order statistics of sythetic traces geerated by Method 1 ad Method with respect to the real trace of the wireless chael. First or der statistics by Q-Q plot; Secod order statistics by ACF.

9 Figure 1 shows the R/S plot of the real trace. The liear regressio lie is show as the dotted lie. Figure 1 shows the R/S plot of trace by Method 1. Figure 1 (c) shows the R/S plot of trace by Method. It is show that the sythetic trace by Method has closer Hurst parameter value to the real trace tha the sythetic trace by Method R/S i log R/S i log R/S i log scale N i log scale N i log scale N i log (c) Figure 1. R/S plots of three traces: real trace; sythetic trace by Method 1; (c) sythetic trace by Method. If ARQ is applied as the error cotrol mechaism, we ca derive the data packet service rate of the packet [19][]. The packet error rate PER is defied as: [ ] PER = 1 1 BER (4) where is the size of a packet i bits. The data packet service rate is the: R c = [ 1 PER]µ (5) where 1 µ is the frame iterval. Note that the frame here does ot refer to the video frame (Figure 13). frame trasmitter receiver NACK Figure 13. ARQ frame structure. We ca further aalyze the queuig behavior of the data packet service rate derived from the BER trace. The queuig performace of the real trace is show i Figure 14. The queuig performaces of the sythetic traces of both Method 1 ad Method have similar look. However, the sythetic trace by Method has smaller MSE i both the packet loss rate ad the delay tha the sythetic trace by Method 1 (Table ). O ACK time Figure 14. Queuig behavior of the real trace of the wireless chael: packet loss rate; queuig delay.

10 V. Coclusio We proposed a ew puctured AR processes to model both the video traffic ad the wireless chael. The puctured AR processes are modulated by Markov processes. The puctured AR processes explicitly cosider the timig iformatio betwee samples of each state. Thus, it outperforms the covetioal approach i VBR video traffic modelig as well as the wireless chael modelig. A good set of performace metrics are experimeted showig the ovelty of the proposed model i differet aspects, especially i the queuig behavior. VI. Referece [1] Heejume Ah, Jae-Kyoo Kim, Sog Chog, Bara Kim, ad Bog Dae Choi, A Video Traffic Model based o the Shiftig-Level Process: the Effects of SRD ad LRD o Queueig Behavior, INFOCOM, pages [] D. M. Lucatoi, M. F. Neuts, ad A. R. Reibma, Method for Performace Evaluatio of VBR Video Traffic Models, IEEE/ACM Trasactios o Networkig, Vol., Issue, pages , April [3] O. Rose, Statistical Properties of MPEG Video Traffic ad Their Impact o Traffic Modelig i ATM Systems, Proceedigs of th Coferece o Local Computer Networks, pages , [4] P. R. Jelekovic, A. A. Lazar, ad N. Semret, The Effect of Multiple Time Scales ad Subexpoetiality i MPEG Video Streams o Queueig Behavior, IEEE Joural o Selected Areas i Commuicatios, Vol. 15, No. 6, pages , August [5] Video Codig for Low Bit Rate Commuicatio, ITU-T Recommedatio H.63 Versio, Ja [6] Motio Pictures Experts Group, Overview of the MPEG-4 stadard, ISO/IEC JTC1/SC9/WG11 N459, [7] D. Heyma, A. Tabatabai, ad T. V. Lakshma, Statistical Aalysis ad Simulatio Study of Video Coferece Traffic i ATM Networks, IEEE Trasactios o Circuits ad Systems for Video Techology, Vol., No. 1, March 199. [8] iya Wag, Souhwa Jug, ad J. S. Meditch, VBR Broadcast Video Traffic Modelig- A Wavelet Decompositio Approach, GLOBECOM 97, pages , [9] Zhag Yogqiag, Che ili, iog Bigqu, A New VBR Model of MPEG Streams o ATM Networks, Proceedigs of 1998 Iteratioal Coferece o Commuicatio Techology, [1] A. M. Dawood ad M. Ghabari, Cotet-Based MPEG Video Traffic Modelig, IEEE Trasactios o Multimedia, Vol. 1, pages 77-87, March [11] H. Liu, N. Asari, ad Y. Q. Shi, The Tale of a Simple Accurate MPEG Video Traffic Molel, ICC 1, pages [1] iog Simi, Liag Jisog, Yag Zhimi, ad Lei Zhegmig, Video Traffic Modelig Based o RBF Networks, Proceedigs of 1998 Iteratioal Coferece o Commuicatio Techology, [13] Q. Liag, J. M. Medel, MPEG VBR Video Traffic Modelig ad Classificatio Usig Fuzzy Techiques, IEEE Trasactios o Fuzzy Systems, Vol. 9, No. 1, pages , February 1. [14] S. Ma, C. Ji, Modelig Video Traffic Usig Wavelets, ICC 98, pages , [15] Derog Liu, E. I. Sara, ad Wei Su, Nested Auto-regressive Processes for MPEG-Ecoded Video Traffic Modelig, IEEE Trasactios o Circuits ad Systems for Video Techology, Vol. 11, No., pages , February 1. [16] M. M. Kruz, A. M. Makowski, Modelig Video Traffic usig M/G/ Iput Processes: A Compromise betwee Markovia ad LRD Models, IEEE Jourals o Selected Areas i Commuicatios, Vol. 16, Issue 5, pages , [17] Deepak S. Turaga ad Tsuha Che, Hierarchical Modelig of Variable Bit Rate Video Sources, Packet Video 1. [18] William Turi ad M. Moha Sodhi, Modelig Error Sources i Digital Chaels, IEEE Joural o Selected Areas i Commuicatios, Vol. 11, No. 3, pages , April [19] Youg Yog Kim ad Sa-Qi Li, Capturig Importat Statistics of A Fadig/Shadowig Chael for Network Performace Aalysis, IEEE Joural o Selected Areas i Commuicatios, Vol. 17, No. 5, pages [] Youg Yog Kim ad Sa-Qi Li, Modelig Multipath Fadig Chael Dyamics for Packet Data Performace Aalysis, Wireless Networks, Vol 6, pages , Dec.. [1] T. S. Rappaport, Wireless Commuicatios: Priciples ad Practice, Pretice Hall 1996.