Distribution of the Maximum Waiting time of a Hello Message in Ad hoc Networks

Transcription

1 Iteratioal Joural of Comuter Alicatios ( ) Volume 47 No4 Jue istributio of the Maximum Waitig time of a Hello Message i Ad hoc Networs Karima Adel-Aissaou LAMOS Laboratory Uiversity of Béjaia jamil Aissai LAMOS Laboratory Uiversity of Béjaia Nathalia jellab LANOS Laboratory Uiversity of Aaba ABSTRACT This aer is a cotributio to the mathematical modelig of a imortat elemet i Ad hoc etwors: Hello message A queuig system is itroduced to model the hello message recetios at give ode Here the customers are the Hello messages ad the service is defied as the rocessig of these messages The modelig i questio allows us to study the maximum waitig time of Hello messages before their treatmet i the recetio ode Uder the hyothesis that Hello messages arrive accordig to a oisso rocess ad by usig the theory of radom wals we obtai the closed form exressio of the maximum waitig time We validate our aroach by the simulatio study Geeral Terms Your geeral terms must be ay term which ca be used for geeral classificatio of the submitted material such as atter Recogitio Security Algorithms et al Keywords Ad hoc etwor Hello message radom wal waitig time factoriatio methods INTROUCTION Mobile Ad Hoc Networs (MANTs) are emerged as a category of wireless etwors which use the radio lis ad are able to fuctio with self-orgaiig ad self-cofigurig ad without a fixed ifrastructure [] The utility of the ifrastructure i the wireless etwors with ifrastructure is relaced by the cotrol messages called Hello messages i the Ad hoc etwors Uder the assumtio that these cotrol messages idicate a reliable commuicatio with a source it is ossible to determie the aths The basic rotocol which rooses the use of these messages to maitai the coectivity is iitially described i OSF (Oe Shortest ath First) [] The odes trasmit regularly the Hello messages to idicate their reseces to their eighborhood The Hello messages lay a very imortat role i the data trasmissio betwee the Ad hoc odes ad also i their localiatio [3]-[4] May asects of Ad hoc etwors have bee studied or are uder ivestigatio by the iteratioal research commuity For examle i [5] the authors give a survey i which they show how the messages ca be efficietly dissemiated i differet tyes of MANTs I [3] the authors roof that the effectiveess of routig rotocols usig the Hello messages deeds o the sie of these messages the trasmissio frequecy ad o the lifetime of a etry i the routig table These criteria are very imortat but little research has bee directed i this way Study of the recet literature reveals that a exact mathematical modelig of Ad hoc etwors is also gaiig attetio [6]-[] Some iterestig situatios i Ad hoc etwors (li coectivity maiteace Hello rotocols) are discussed i []-[] This aer is a cotributio to the mathematical modelig of a imortat characteristic of Ad hoc etwors: Hello message We focus o the distributio of the maximum waitig time of a Hello message We have to model the recetio of Hello messages by a Ad hoc ode as a queuig system where the customers are the hello messages ad the service is defied as the treatmet of these messages by the recetio ode By usig the radom wal theory we fid the distributio of the maximum waitig time Our results ca be used to determie the TTL (Time to Live) of a etry i the routig table or to adat the trasmissio frequecy of Hello messages The aer is orgaied as follows: I sectio we reset a mathematical modelig of the recetio of a Hello message as a queueig system Sectio 3 is devoted to aalye the maximum waitig time distributio To validate our aroach some simulatio results (erformed uder NS-) are obtaied ad comared with the aalytic oes Cocludig remars are give i sectio 4 MATHMATICAL MOLING Notatios ad some reewal relatios Let be the sequece of ideedet ad idetically distributed radom variables Cosider S t mi S t ( t S Let be the reewal rocess so for all S t ) S be the i radom wal e be the characteristic fuctio of the radom variable efie also max S S max S S S S S S S mi S mi

2 Iteratioal Joural of Comuter Alicatios ( ) Volume 47 No4 Jue mi : S if S mi : S if S mi S S S mi : S if for all S Factoriatio methods The methods i questio rovide the ossibility to fid the characteristic fuctio of the joit distributio of the radom variables ad They allow also a factoriatio of the characteristic fuctio whe ad Im ( is a real umber) From [3] we have the followig results: Theorem [3] Whe ad Im we have C C C () where C ( C comoet of factoriatio: ) is the ositive (egative) is C ex e S i e C ex S is i C ex e S e Note that the fuctio Im ( Im whe Im ( Im Theorem [3] C ( C ) is aalytic whe ) ad cotiuous ad bouded ) The characteristic fuctio of the radom variable S max is give by e i S if () i e For the alicatio uroses it is iterestig to ow the distributio of the radom variable S max Moreover it is ossible to establish the relatio betwee the distributio of S with that of ad as well as with the factoriatio comoets of the fuctio To oe this ed cosider the exressio () Whe obtais i C e (3) e is where I the case of exoetial distributio of the radom variable (the distributio of does ot deed o the distributio of ) x x be ay the characteristic fuctio () becomes e is ( i b x ) x (4) if (5) Moreover we have the followig distributio of the cosidered radom variable S: S x S x e x (6) The robability ca be obtaied with the hel of (3) ad (5): Here C i C i

3 Iteratioal Joural of Comuter Alicatios ( ) Volume 47 No4 Jue i e i ad (8) where is the uique ositive solutio of the equatio i (9) 3 Queuig model We cosider a sigle server queuig system with ifiite waitig sace Here the server is a articular ode ad the customers are Hello messages arrivig to this ode Suose that Hello messages arrive by time itervals W (the W first message arrives at time the secod after W etc) The sequece W cotais the ideedet ad idetically ex distributed (accordig to a exoetial law i occurrece) radom variables O the arrival of the -th Hello message the duratio of coectivity with the trasmitter ode is We cosider this duratio as the service life of the -th Hello message If at the time of arrival of this message the ode is occuied by the service of aother Hello message (comig from aother ode) the Hello message i questio is laced i waitig We see the distributio of the waitig time T of the -th Hello message The waitig time of the (+)-th Hello message is equal to 3 ALICATION Assume that W Uder this assumtio we have the followig set of equatios T max T ad T Its solutio is give by T S mi S S S S Ideed S Verify that S max S If S the (or S S ) If T S S the (7) S (or =) Therefore Moreover S max S S S S S S is distributed as max S S max Sice lim the limitig waitig time distributio is the distributio of the radom variable S (described i sectio ): T x T x max x S x lim This distributio exists if W It is ifiite if W or W but W The sequece cotais the ideedet ad idetically distributed radom variables Suose that the sequeces W ad are ideedet Sice the iter-arrival time is exoetial oe: W x e x x x W x y df yw x y e x e df y e e x F is the distributio fuctio of the service life Theorem 3 x y Let x e x e The ad x e x ad the distributio of maximum waitig time of a Hello message is: x x T x e () roof The roof is based o the factoriatio methods (see sectio i articular the exressios (4) ad (6)) If the distributio of have is also exoetial with rate we x e x e e x By usig (7)-(9) we fid that 3

4 Iteratioal Joural of Comuter Alicatios ( ) Volume 47 No4 Jue Corollaire For the radom variable (or T) if the sequece W (as well as the sequece ) is exoetial with rate (with rate ) ad uder the coditio that W we have x e x The distributio of the maximum waitig time of a Hello message is: To validate our study we have imlemeted a simulatio rogram i C++ For the followig values of arameters: duratio of the simulatio Tmax=s µ= λ= we have costructed two curves: oe of which corresods to the aalytic solutio (obtaied with the hel of ()-()) ad the secod reresets the simulatio outcomes (see figure ) It is easy to see that the two curves coicide i articular for time t<34 From the oit t=34 the differece aears more ercetibly This may be due to the fact that we have omitted the clusterig ad the sie of the etwor as well as we have ot ut a limit o the queue x x T x e () Fig : Comariso 4 CONCLUING RMARKS Ad hoc etwors comosed of wireless mobile odes ca freely ad dyamically self-orgaie ad are without reexistig ifrastructure I these commuicatio systems maitaiig of the coectivity is a very big roblem for which Hello messages are a solutio I this wor we modeled the arrival of the Hello messages at a ode as a radom wal By usig the aroach develoed by Borovov (978) we obtaied the exact exressio of the waitig time distributio of a Hello message before its treatmet by the recetio ode With the hel of this result it is ossible to calculate the TTL (Time To Live) of a message ad the iterval Hello (two essetial arameters i the routig rotocols) 5 RFRNCS [] Basagi S Coti M Giordao S ad Stojmeovi I 4 Mobile Ad Hoc Networig I ress Joh Wiley [] Moy J 994 OSF - Oe Shortest ath First RFC 583 [3] Chaers I ad Bedig-Royer M The utility of hello messages for determiig li coectivity roceedig the 5th Iteratioal Symosium o Wireless ersoal Multimedia Vol () [4] Igelrest F ad Mitto N ad Simlot-Ryl 7 A Turover based Adative HLLO rotocol for Mobile Ad Hoc ad Sesor Networs roceedig of the 5th 4

5 Iteratioal Joural of Comuter Alicatios ( ) Volume 47 No4 Jue I Iteratioal Symosium o Modelig Aalysis ad Simulatio of Comuter ad Telecommuicatio Systems (MASCOTS 7) 9-4 [5] Al Habali A Ibrahim M Simo V Varga ad Carreras I 8 A Survey of Message iffusio rotocols i Mobile Ad hoc Networs rocedig of the 3rd Iteratioal Coferece o erformace valuatio Methodologies ad Tools [6] Groeevelt R Nai ad Koole G 5 The message delay i Ad hoc etwors erformace valuatio Joural Vol(6) N -4-8 [7] Al Habali A ad Nai ad Altma 8 erformace of ad hoc etwors with two-ho relay routig ad limited acet lifetime (exteted versio) erformace valuatio Joural Vol(65) [8] ia J etit J ad Sera M 998 Radom geometric roblems o [ ] Radomiatio ad Aroximatio Techiques i Comuter Sciece vol 58 of Lecture Notes i Comuter Sciece Sriger-Verlag Berli [9] Nemeth G ad Vattay G Giat clusters i radom ad hoc etwors codmat/35 [] Newma M J Strogat S H adwatts J Radom grahs with arbitrary degree distributios ad their alicatios cod-mat/735 [] Gome C ad Cuevas A ad aradells J ) AHR: a two-state adative mechaism for li coectivity maiteace i AOV roceedig of the d iteratioal worsho o multi-ho mobile Ad Hoc etwors: from theory to reality 98- [] Girua V ad Sighal M 5 Hello rotocols for ad hoc etwors : Overhead ad accuracy trade-offs roceedig of the Sixth I Iteratioal Symosium o a World of Wireless Mobile ad Multimedia Networs (WoWMoM 5) Vol () [3] Borovov A A 978 Theory of robability Naua Moscow 5