Modeling the Reliability of Packet Group Transmission in Wireless Network

Transcription

1 Modeling he Reliabiliy of Packe Group Transmission in Wireless ework Hao Wen, Chuang Lin, Hongkun Yang, Fengyuan Ren, Yao Yue Tsinghua Universiy, Beijing China Cornell Universiy, USA {wenhao, clin, yanghk, Absrac Mos previous models abou wireless channel are proposed o capure long-ime channel characerisics. However, when we combine group ransmission wih error correcing mechanisms, he seady-sae probabiliies are no longer accurae enough o depic he shor-ime loss saes. In his paper, we propose a new analyical model for group ransmission which can capure influences of iniial channel sae and group lengh on ransmission reliabiliy. The model offers us a significan insigh ino loss characerisics of group ransmission, which is essenial o design reliable wireless proocols. Finally o illuminae he srengh of our model, we also apply our model o compare he reliabiliy performance of mulipah ransmission wih single pah ransmission. Index Terms Reliabiliy, packe group ransmission, Gilber model, correlaed wireless channel I. ITRODUCTIO Driven by indusrial and scienific applicaions, packe-loss or bi-error modeling of wireless nework has recenly araced much aenion from academia. For unpredicable wireless channels, a lo of models are proposed []-[4], which offer us a significan insigh ino characerisics of he underlying wireless channels. In mos previous models, a compleely accurae analysis of he error process could be complicaed and ofen only he long-ime saisics can be compued. For insance, Markov models are generally employed o characerize error processes. The seady-sae probabiliies of a Markov model, which represen he long-run proporion of he ime spen in each sae, are key parameers in modeling. However, when we adop packe group ransmission based on Erasure coding [5], [6] or FEC [7], he homogeneousness among packes is modified and seady-sae probabiliies are no accurae enough o depic he loss even. For example, in a channel wih seady loss probabiliy 0., isolaed loss and burs loss make no difference for he reliabiliy of mass ransmission (e.g. more han housands of packes) while having a grea impac on he reliabiliy of group ransmission. As shown in Fig., based on he loss probabiliy, he corresponding Erasure coding [M4: R] can recover all isolaed losses bu only some burs losses. Manuscrip received Sep 8h, 007. Fig.. Group Transmission using Erasure Coding Hence in unreliable wireless nework, he correlaion beween packes in one group has o be considered and he shor-ime saisic is indispensable o design an efficien and reliable mechanism. Especially in low-rae Wireless Sensor eworks, he daa raffic generaed by one sensor may be of very low inensiy bu burs raffic may be riggered by a se of sensors due o a common even. Thus he design based on he long-ime saisics is no longer accurae and our work is moivaed by he need o undersand he shor-ime loss behavior in wireless nework. In his paper we propose a new model, based on he Gilber heory [8], o compue of he loss probabiliy of packe group. Our conribuion can be summarized as follows: ) Exend he Gilber model o capure he influence of iniial channel saes on ransmission reliabiliy. Since mos previous models are uilized for long-ime saisics, hey paid lile aenion o he influence of iniial channel sae. However, when considering he shor-lengh group ransmission, we can no longer ignore ha poin. Their imporance is demonsraed wih heoreical analysis. ) Wih he help of he new model, we provide a heoreical sudy abou how o define reliable proocols based on packe group ransmission. In radiional one-by-one ransmission, large numbers of packes can ensure he accuracy of long-ime saisic. However, in low-rae wireless neworks (e.g. WSs) ha reuire high reliabiliy, correlaions inside group are no negligible, which makes long-ime saisics lose accuracy. Therefore in his paper, we deliberaely explore he relaionship beween reliabiliy and group lengh. 3) To illuminae he value of our model, we also apply our model o mulipah scheme and discover ha mulipah is no always more reliable han single pah ransmission. The res of his paper is oulined as follows. In Secion II we

2 briefly inroduce he basic wireless channel model, which is a prereuisie heoreical foundaion for he whole paper. Then we provide a new analyical model o represen reliabiliy of packes in groups under differen parameers in Secion III. The heoreical evaluaion is carried ou in Secion IV. In Secion V we compare mulipah scheme wih single pah scheme based on our new model. Conclusions are given in Secion VI. II. PRELIMIARIES Errors or losses occur on he wireless channel due o various impairmens including inerference and mobiliy, which exhibi some degree of correlaions. Markov models have been widely used o characerize loss behavior [], []. Considering he racabiliy and accuracy in low-rae nework of he Gilber Model [9], i is adoped in his paper as a basic model. In he Gilber Model (Fig. ), p is he ransiion probabiliy of going from a non-loss sae o a loss sae and is he probabiliy of going from a loss sae o a non-loss sae. -, also called condiional loss probabiliy (clp) is he probabiliy ha he nex packe is again los, provided he previous one has been los. Fig.. The Gilber model The saionary probabiliies of he Gilber model represen he long-ime proporion of he ime spen in each sae. Once he ransiional probabiliies are known, we can compue saionary probabiliy π 0 for non-loss sae and π for loss sae, which is also called he uncondiional loss probabiliy (ulp): p π 0, π. () p + p + In oher words, π 0 and π also represens he mean arrival and loss probabiliy. From colleced nework raffic races, we can easily obain rained parameers p and. III. THEORETICAL MODEL Since radiional mehods (e.g. Gilber model) use long-ime saionary probabiliies o represen packe-loss or bi-error processes, i is hard o characerize he dynamic facors in heerogeneous group ransmission. In his secion we propose a new analyical model, which can capure he influence of iniial channel sae and group lengh on ransmission reliabiliy. For packes ransmied in one group, each packe can ransmi successfully or suffer loss even. We define {s(),..} as he sochasic process o represen ransmission sae of packes, hen packes series model is consruced in Fig. 3. Here he ransiional probabiliies p and are boh he same as he Gilber model, represening one-sep sae ransiion probabiliy beween loss sae and non-loss sae. We define probabiliies for loss and non-loss sae of i-h pa- Fig. 3. Packes series model -cke as: αi Psi {() }, βi Psi {() 0},( i [, ]) () Specifically, we se he iniial probabiliies for he firs packe as α a, β a (3) From he model in Fig. 3, i is observed ha αk p αk (4) βk p βk Then we can ge αk α k 0 A,( k, A I) (5) βk β p where A. p Le E l be he mean loss number and E n be he mean arrival number, which can be calculaed as El αi α i A En i β i i β (6) To simplify he above formula, we uilize Jordan ormal Form of A as A S J S (7) where 0 J 0 p p p+ p+ and S. p+ p+ Based on he Jordan ormal Form, we can rewrie (6) as El α i S J S E (8) n i β Owing o he fac ha 0 i J ( p i 0, (9) p+ we have El a a α (0) En a a β where

3 3 Fig. 4. Iniial Channel Sae a Fig. 6. Iniial Channel Sae a0.5 Fig. 5. Iniial Channel Sae a0.7 ( p + p a p( + ( p + ) a ( + ( p + ) a p ( p p+ a Then he mean number of loss packes E l can be calculaed as El αa+ βa ( + ( p ) (( a) p a + p. () And he mean loss probabiliy of one group can be expressed as ( + ( p ) (( a) p a + p η. () Obviously from (), we can figure ou he mean loss probabiliy in packe group ransmission is deermined by ransiional probabiliies (i.e. p and, he group lengh and he iniial probabiliies a. Fig. 7. Iniial Channel Sae a0 And as he group lengh approaches infiniy, he limi of η is p( p + p lim η π, (3) which is exacly he mean loss probabiliy π in (). This resul can denoe ha our model is fine-granulariy applicaion of he Gilber model. IV. UMERICAL RESULTS In his secion, he model proposed above is uilized o analyze he reliabiliy of packes group ransmission under differen condiions. Due o lengh consrain, here we only discuss he case of saionary packe loss probabiliy π 0.5 wihou loss of generaliy. Under his saionary probabiliy, he loss resuls under differen iniial channel sae (a, 0.7, 0.5 and 0) are shown from Fig. 4 o Fig. 7. Fig. 4 shows loss probabiliies of group ransmission for wo channel saes (namely, differen ransiional probabiliies p and under he same iniial channel sae. I is observed ha in such a bad iniial channel condiion (a), he reliabiliy of group ransmission is exraordinary unaccepable when group lengh is smaller han 5. However wih he increase of, he

4 4 loss probabiliies of group ransmission gradually coincide wih he saionary loss probabiliy 0.5. Addiionally, we also noe from Fig. 4 ha he smaller he ransiional probabiliies are, he slower he change of loss probabiliies are. For example, he loss resul of p0. and 0.3 alers more slowly han ha of p0.5 and This conclusion resuls from ha small ransiional probabiliies, which means high correlaion beween packes, will enhance he channel memory. Similar wih Fig. 4, he resuls from Fig. 5 also represen he influence of group lengh under iniial channel sae. Compared wih resuls when a, a beer iniial channel sae a0.7 will resul in beer reliabiliy wih he same group lengh. In Fig. 6, i is obviously ha when iniial channel condiion a is eual o saionary packe loss probabiliy π, he resul of our model is exacly he same wih he long-ime saisic value. And no maer how long he group lengh is, he reliabiliy of group ransmission is always sable. To furher invesigae he influence of he iniial channel condiion on group reliabiliy, we repor resuls in Fig. 7 under an ideal iniial channel sae (a0). Due o he channel memory, he reliabiliies of group ransmission behave beer han seady sae, especially when he group lengh is small. Furhermore we can also noice ha he smaller ransiional probabiliies are, he more apparen he channel memory is. From above analysis, i is clearly observed ha our new model does reflec he influence of he iniial channel sae and he group lengh. Our heoreical resuls verify ha in order o obain high reliabiliy of group ransmission in bursy or unsable wireless nework, we need increase he group lengh or add redundan packe o enhance resisance agains loss evens. And he design based on long-ime saisics canno guaranee reliabiliy in complicaed and unpredicable wireless nework. V. MULTIPATH STUDY To illuminae he value of our model, we apply our model o mulipah ransmission in his secion. The applicaion of mulipah echniue in wireless nework seems naure, since i may diminish he effec of unsable wireless links o increase reliabiliy. Thus mos previous work adop mechanism which combine pah redundancy (i.e. muliple disjoin pahs) and daa redundancy (e.g. Erasure codes or FEC) o provide reliable ransmission [7], [0]-[]. However based on our model, we can discover some facs ha are negleced before. In our reliabiliy comparison we assume pahs are available for packes ransmied from a source o a desinaion node. Each pah has he same ransiional probabiliies, which resul in he same long-ime seady sae, and differen iniial channel sae a i. Wihou loss of generaliy, i is assumed ha a a... a. (4) Addiionally, load balancing is implemened in mulipah ransmission o ensure packes are allocaed evenly in each single pah. Using our model, we can ge he following heorem: Theorem : The mulipah scheme euals or is beer han he wors single-pah one when ransmiing large numbers of packes. Proof: Based on (), he loss probabiliies of packes in single-pah ransmission can be represened as ( ( p ) ( ai p) + p ( ). (5) ( p + Owing o he linear relaionship beween a i and η i, we can ge η( ) η ( )... η ( ). (6) Using mulipah ransmission wih load balancing, packes are allocaed evenly in single-pah, hen he loss probabiliies of packes in mulipah ransmission is ( ) ( ) η i i ( ) ηmul η ( ). (7) When he group lengh approaches infiniy, here exis lim η ( ) lim η ( ). (8) Refer o (7) and (8), we finally obain η η. (9) ( ) mul This heorem does explain why i is generally agreed ha mulipah ransmission can diminish he effec of unsable wireless links o increase reliabiliy. However, when we se iniial channel sae a i as he same fixed value a for all muliple pahs, we discover anoher fac. Theorem : The mulipah scheme is no always more reliable han he single-pah ransmission. Proof: Since a a a... a, he loss probabiliies of packes in single-pah ransmission can be represened as ηsig η( ), (0) and he loss probabiliies in mulipah ransmission is ( ) ( ) i i ηmul η( ). () Then difference of reliabiliy is Δ η ηsig ηmul ( p a )(( p + ( p ). () ( p + For convenience of discussion, we se 3 and rewrie () as Δ η ηsig ηmul 3 ( p a )(( p 3( p + ). (3) 3 I is no difficul o prove ha when p, [0,], we have 3 ( ) p 3( p 0 +. (4) Here he value of p a is he key deerminan o he final resul. Specifically when p a, Δη 0 which means he mulipah ransmission is more reliable; Or else single-pah scheme provides more reliable packe ransmission.

5 5 Fig. 8. a0.4, 4 Fig. 0. a0.4, -ing o he correlaion effec, namely channel memory, he mulipah scheme behaves even worse ha single-pah under bad iniial channel sae. ACKOWLEDGMET The auhors wish o hank Chuanpin Fu for her helpful commens. Fig. 9. a0.5, 4 From Fig. 8 and Fig. 9, i is clearly observed he influence of iniial channel sae on resul. We call he area above conour line 0 as single-pah superior region and he area below as mulipah superior region. From he movemen of conour line 0 in wo figures, we can find ha worse iniial channel sae (a0.5) will cause smaller mulipah superior region, which is o verify ha due o he effec of correlaion, he mulipah scheme is no always more reliable han he single-pah ransmission. Finally as approaches infiniy, he limi of ( ) ( ) Δ η is lim Δ η lim η lim η 0. (5) When comparing Fig. 8 and Fig. 0, we can figure ou increase of will reduce he reliabiliy difference of wo schemes. VI. COCLUSIO In his paper, we propose a simple bu racable model o analyze he loss probabiliy for packe group ransmission. Our resuls show ha in bad iniial channel saes, high correlaion among packes in group will largely deeriorae he reliabiliy performance. However, wih he increase of he group lengh, he impac of correlaion gradually diminishes. Finally we apply our model o mulipah ransmission and reveal ha ow- REFERECES [] M. Zorzi and R. R. Rao, On he Saisics of Block Errors in Bursy Channels, IEEE Transacions on Communicaions, vol. 45, no. 6, June 997, pp [] A. Konrad, B. Y. Zhao, A. D. Joseph, and R. Ludwig, A Markov-based Channel Model Algorihm for Wireless eworks, ACM Wireless eworks Journal (WIET), vol. 9, 003, pp [3] A. Köpke, A. Willig, and H. Carl, Chaoic Maps as Parsimonious Bi Error Models of Wireless Channels, in Proceedings of he nd Conference on Compuer Communicaions (Infocom 03), San Francisco, March 003. [4] S. A. Khayam and H. Radha, Linear-Complexiy Models for Wireless MAC-o-MAC Channels, ACM Wireless eworks (WIET) Journal (WIET), vol., no. 5, Sepember 005. [5] S. Kim, R. Fonseca, and D. Culler, Reliable ransfer on wireless sensor neworks, in Proceeding of he Firs IEEE Communicaions Sociey Conference on Sensor and Ad Hoc Communicaions and eworks ( Secon 04), 004, pp [6] Hao Wen, Chuang Lin, Fengyuan Ren, Yao Yue and Xiaomeng Huang, Reransmission or Redundancy: Transmission Reliabiliy in Wireless Sensor eworks, in Proceeding of he Fourh IEEE Inernaional Conference on Mobile Ad-hoc and Sensor Sysems (Mass 07), Pisa, Ocober 007. [7] Pear Djukic and Shahrokh Valaee, Reliable packe ransmissions in mulipah roued wireless neworks, IEEE Transacions on Mobile Compuing, vol. 5, no. 5, May 006, pp [8] E.. Gilber, Capaciy of a Burs oise Channel, Bell. Sys. Tech. Journal, vol. 39, Sepember 960, pp [9] Maya Yajnik, Sue Moon, Jim Kurose, and Don Towsley, Measuremen and modeling of he emporal dependence in packe loss, in Proceedings of he 8h Conference on Compuer Communicaions (Infocom 99), ew York, 999, pp [0] E. Ayanoglu, I. Chih-Lin, R. D. Gilin, and J. E. Mazo, Diversiy Coding for Transparen Self-Healing and Faul-Toleran Communicaion eworks, IEEE Trans. Comm., vol. 4, no., ov [] A. Tsirigos and Z.J. Haas, Analysis of Mulipah Rouing-Par I: The Effec on he Packe Delivery Raio, IEEE Trans. Wireless Comm., vol. 3, no., Jan. 004, pp [] A. Tsirigos and Z.J. Haas, Analysis of Mulipah Rouing, Par : Miigaion of he Effecs of Freuenly Changing ework Topologies, IEEE Trans. Wireless Comm., vol. 3, no., Mar. 004, pp