On the Connectivity of One-dimensional Vehicular Ad Hoc Networks

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1 RESEARCH PAPER 论文集锦 O the Coectivity of Oe-dimesioal Vehicular Ad Hoc Networs Liao Jiaxi 1,2, Li Yuazhe 1,2, Li Toghog 3, Zhu Xiaomi 1,2, Zhag Lei 1,2 1 State Key Laboratory of Networig ad Switchig Techology, Beijig Uiversity of Posts ad Telecommuicatios, Beijig , Chia 2 EBUPT Iformatio Techology Co., Ltd, Beijig , Chia 3 Techical Uiversity of Madrid, Madrid Spai Abstract: I this paper we aalyze coectivity of oe-dimesioal Vehicular Ad Hoc Networs where vehicle gap distributio ca be approximated by a expoetial distributio. The probabilities of Vehicular Ad Hoc Networ coectivity for differece cases are derived. Furthermore we proof that the odes i a sub-iterval [ z1, z1+δ z] of iterval [0, z], z > 0 where all the odes are idepedetly uiform distributed is a Poisso process ad the relatioship of Vehicle Ad hoc Networs ad oe-dimesioal Ad Hoc etwors where odes idepedetly uiform distributed i [ z1, z1+δ z] is explaied. The aalysis is validated by computig the probability of etwor coectivity ad comparig it with the Mot Carlo simulatio results. Key words: Itelliget Trasportatio Systems (ITS); coectivity; Vehicular Ad Hoc Networs; oedimesioal Ad Hoc Networs I. INTRODUCTION Mobile Ad Hoc NETwors (MANETs) are comprised of self-orgaizig mobile odes that lac a etwor ifrastructure, such as base statios. This techology has bee proposed as a complemetary to the fourth-geeratio wireless etwors[1]. Ad i recet years, there is a growig iterest o the research ad deploymet of MANETs techology for vehicular commuicatio, e.g. the PReVENT project i Europe, IteretITS i Japa, ad Networ o Wheels i Germay. I particular, these etwors have importat applicatios i itelliget trasportatio systems (ITS). The prospective applicatios of VANETs are categorized ito two groups as comfort ad safety applicatios. The first group is expected to improve the passegers comfort ad optimize traffic efficiecy, whereas the

2 Chia Commuicatios secod oe improves drivig safety. For a oe-dimesioal Mobile Ad Hoc Networs (MANETs), assumig all the odes are idepedetly uiform distributed, a few approaches have bee proposed i the literatures for computig the probability of coectivity[2-6]. Desai ad Majuath[2] derived a formula for the probability that the etwor G 1 (all the odes are idepedetly uiform distributed i a closed iterval [0, z], z > 0) is coected. Gore[3, 4] gave some commets o[2] ad derived aother formula for the probability that G 1 is coected with a fixed ode at poit zero. The probability that the oe dimesioal Ad Hoc Networs is composed of at most C clusters is derived by Ghasemi ad Nader- Esfahai i [5]. Referece [6] studies the coectivity of a 1-D ad hoc etwor with a user mobility model ad derives a formula for the coectivity of a source destiatio pair. For a special ad hoc etwors, Vehicular Ad Hoc Networs(VANETs)[7], it is show by Rudac et al.[8] that the vehicle gap distributio ca be approximated by a expoetial distributio λ with vehicle desity µ =, where µ represets EV [ ] the umber of vehicles per uit of distace, λ represets the mea arrival rate of vehicles, E[V] represets the mea vehicular speed. The same coclusio is derived by S. Yousefi et al. i [9]. I this paper, we try to derive some coectivity properties of VANETs i highway scearios. Firstly i Sectio II, we proof that the odes umber located i a sub-iterval of [0, z], z > 0 approximately is a Poisso process ad that uder the coditio that vehicles locate i the iterval (0, L], the positio of these vehicles, which are cosidered as uordered radom variables, are idepedetly ad uiformly distributed i the iterval (0, L]. These coclusios relate the previous wors[2-6] to ours i this paper. Secodly i Sectio III, we get the followig coectivity probability for VANETs i highway scearios: The probability of coectivity of the i th vehicle ad the j th vehicle ( j>i). The probability of coectivity of the vehicles i the iterval [0, L](L> 0). The probability of coectivity of a fixed ode at poit zero ad the geographical poit L(L>0): A example for this sceario is Geocastig which meas deliverig a message from a source ode to odes i a give geographical regio. I other words, the probability is the geographical poit L(L>0) is i multi hop trasmissio rage of the fixed ode at zero. Nodes ad vehicles are alterately used i this paper. II. PRELIMINARIES For a oe-dimesioal ad-hoc etwors G 1, assumig odes are idepedetly uiform distributed i a closed iterval [0, z](z > 0). The probability that there are odes i the iterval [ z1, z1+δz] ( 1 ad z 1, z 1 +Δ z < z) is give by the followig theorem. Theorem 1 The probability that there are odes i the iterval [ z1, z1+δ z] ( 1 ad z1, z1+δ z < z) approximately is a Poisso poit process whe Δ z z ad. µ µ PX ( = ) e (1)! Δz where p =, µ = p. z Proof The probability that a ode locates i the iterval ( 1 ad z 1, z 1 +Δ z < z) is Δz = ( z1 < x1 < z1+δ z) =. So the probability z that there are odes i the iterval [ z1, z1+δ z] ( 1 ) is: PX ( = ) = Cp (1 p) Let p = µ, we get: C p p For large ad p small µ

3 RESEARCH PAPER 论文集锦 µ µ e ( 1)...( + 1) 1 µ Hece, for large ad p small, PX ( = ) e µ µ!. For Vehicular Ad Hoc Networs, we deote the Probability Desity Fuctio (PDF) ad Cumulative Distributio Fuctio (CDF) of the gap distributio as f g (x) ad F g (x) respectively. Based o the coclusio from [8, 9], we have x f ( x) = µ e µ (2) g x Fg ( x) = 1 e µ (3) Obviously the umber of odes which locate i the iterval [ z1, z1+δ z] is a Poisso poit process: ( µ Δz) µ Δz ( = ) = (4)! Theorem 2 Uder the coditio that vehicles locate i the iterval (0, z], the positio of these vehicles, which are cosidered as uordered radom variables, are idepedetly ad uiformly distributed i the iterval (0, z]. Proof The coclusio is derived directly from Theorem 5[10]. Based o the discussio above, we get the coclusio: for oe-dimesioal Vehicular Ad Hoc Networs ad oe-dimesioal Ad Hoc Networs where odes uiformly distributed, the ode umber i a iterval is a Poisso process ad the odes' positio i a iterval is uiformly distributed. Although our aalysis sceario i the ext sectio is Vehicular Ad Hoc Networs, our coclusios ca also be applied i a oe-dimesioal Mobile Ad Hoc Networs where odes uiform distributed. III. PROBABILITY OF CONNECTIVITY I this sectio, we try to calculate the probability of coectivity for oe-dimesioal Vehicular Ad Hoc Networs where vehicle gap distributio is a expoetial distributio. 3.1 Coectivity probability of case 1 Theorem 3 The probability of coectivity of the i th ode ad the j th ode(j > i) is cs1 j i Pco = Fg ( R) (5) where R deotes the trasmissio rage of radio. Proof There are j i gaps betwee the i th ode ad the j th ode. The probability of the distace of ay two adjacet odes less tha R is F g (R). The coclusio is apparetly derived. 3.2 Coectivity probability of case 2 Theorem 4 The probability of coectivity of the vehicles i the iterval [0, L](L > 0) is: 2 co 1 1 ( L jr) j L (6) where u(x) is the uit step fuctio. Proof The probability of coectivity of the vehicles i the iterval [0, L](L > 0) is P = P( X = )* p ( ) cs2 co = 0 c (7) where p c () is the probability of coectivity of odes located i the iterval [0, L]. The positio of these vehicles are uiformly distributed i the iterval (0, L] (see Theorem 2). This leads to p c ()= p c (, L, R) where p c (, L, R) ca be derived by substitutig, L, R ito the formula (8) i paper [2]. So we get: 1 1 ( ) j L jr c (8) j= 0 j L cs2 P co is derived by substitutig formula (8) ito (7) ad the proof is complete. 3.3 Coectivity probability of case 3 Theorem 5 The probability of coectivity of a fixed ode at poit zero ad the geographical poit L(L > 0) is: cs3 j Pco = P( X = ) ( 1) L j= 0 j (9) R 1 ( ) LjR u( LjR) ( L ( j+ 1) R) u( L ( j+ 1) R) L

4 Chia Commuicatios where x is the least iteger that is larger tha x. Proof Defie yi = xi+ 1 xias the distace of two adjacet odes. Followig the otatio of [2-4] ad usig the subscript G for this case, let UG (, L, R ) U G (, L, R) ad U (L) be the volumes of the polytopes S G (, L, R) ad S(, L) i= 0 1 i= 0 (10) (11) S G (, L, R) describes all feasible sequeces of iter-ode distaces for which Networs i case 3 is coected. The probability that fixed ode ad the geographical poit L are coected whe there are odes i the etwors is: U (, L, R) U (, L, R) PG (, L, R) (12) U( L) L! Usig the formula (20) i paper [3], we get: U G L R j= 0 j j ( L jr) u( L jr) ( L ( j+ 1) R) u( L ( j+ 1) R) L (13) By substitutig formula (13) ito (12), p c (, L, R) is derived. To esure the coectivity of the fixed odes ad the poit L, there are at least L 1 odes R i the iterval [0, L]. cs3 Pco = P( X = ) PG (, L, R) (14) L 1 R cs3 P co is derived by substitutig formula (12) ito (14) ad the proof is complete. as a fuctio of R/L for differet values of 1/μ respectively. Because the probability of coectivity of case 1 is simple ad the aalysis results tallies with the simulatio perfectly, the results of case 1 are ot showed. Note that 1/μ is the mea vehicle gap distace. I Figure 1, Figure 2, little 1/μ leads to large coectivity probability. But i Figure 1 whe R/ L<0.35 which meas L is larger tha R/0.35, little 1/μ leads to more odes gap ad ay gap distace larger tha R leads to discoectivity. Also i Figure 1, as R/L icrease, the coectivity probability icrease util reachig their pea value the drop. The reaso for the probability coectivity icrease whe R/L icrease ad before it reach their pea value is straightforward that large R/L meas little L. The reaso the curves declie after their pea value is little L meas little ode locate probability. I Figure 2, as R/L icrease, L decrease. So IV. DISCUSSION I Figure 1 ad Figure 2, cs2 cs3 P co ad P co are plotted

Although our simulatio ad aalysis sceario is Vehicular Ad Hoc Networs, our coclusios ca also be applied i a oe-dimesioal Mobile Ad Hoc Networs where odes uiform distributed based Theorems i Sectio I.

5 RESEARCH PAPER 论文集锦 coectivity probability is icrease. Whe R=L, the coectivity probability is 1 because ode at 0 ca surely cover iterval [0, R]. Sice both the simulatio ad the aalytical curves i Figure 1 ad Figure 2 are close to each other i all the scearios, we coclude that the aalysis is fairly accurate. Although our simulatio ad aalysis sceario is Vehicular Ad Hoc Networs, our coclusios ca also be applied i a oe-dimesioal Mobile Ad Hoc Networs where odes uiform distributed based Theorems i Sectio I. Our wor ca be used to etwors plaig to esure the coectivity of the vehicle ad hoc etwors. For 1/μ=200, 1/μ=300, 1/μ=400 respectively, whe R/L is 0.64, 0.55, 0.47, the coectivity probability reach their pea value 0.801, 0.699, V. CONCLUSIONS I this paper we aalyze coectivity of oedimesioal Vehicular Ad Hoc Networs where vehicle gap distributio ca be approximated by a expoetial distributio ad our coclusios ca also be applied i a oe-dimesioal Mobile Ad Hoc Networs where odes uiform distributed. We will use the probability derived i this paper i routig strategy for our future wors. Acowledgmets This paper was joitly supported by: (1) Natioal Sciece Fud for Distiguished Youg Scholars (No ); (2) Natioal 973 Program (No. 2007CB307100, 2007CB307103); (3) Natioal Natural Sciece Foudatio of Chia (No ); (4) Chiese Uiversities Scietific Fud (BUP- T2009RC0505); (5)Developmet Fud Project for Electroic ad Iformatio Idustry (Mobile Service ad Applicatio System Based o 3G). Refereces [1] Frodigh, M., et al., Future-geeratio wireless etwors. IEEE Perv. Comput. Mag., (5): p [2] Desai, M. ad D. Majuath, O the Coectivity i Fiite Ad Hoc Networs. IEEE Commu. Lett., (10): p [3] Gore, A.D., Commets o "O the Coectivity i Fiite Ad Hoc Networs". IEEE Commu. Lett., (2): p [4] Gore, A.D., Correctio to "Commets o 'O the Coectivity i Fiite Ad Hoc Networs'". IEEE Commu. Lett., (5): p [5] Ghasemi, A. ad S. Nader-Esfahai, Exact probability of coectivity i oe-dimesioal ad hoc wireless etwors. IEEE Commu. Lett., (4): p [6] Chua, H.F., et al., Networ coectivity of oe-dimesioal MANETs with radom waypoit movemet. IEEE Commu. Lett., (1): p [7] Yuazhe, L., et al. A Bechmar for Delay Performace i Spare Vehicular Ad hoc Networs. i Wireless Commuicatios, Networig ad Mobile Computig, WiCOM '09. 5th Iteratioal Coferece o [8] Rudac, M., M. Meice ad M. Lott. O the Dyamics of Ad Hoc Networs for Iter Vehicle Commuicatios (IVC). i Iteratioal Coferece o Wireless Networs ICWN'02, Ju 24-27, Las Vegas, Nevada, USA. [9] Yousefi, S., et al., Aalytical Model for Coectivity i Vehicular Ad Hoc Networs. IEEE Tras. Veh. Techol., (6): p [10] Ross, S.M., Itroductio to Probability Models. 2000, New Yor: Academic. Biographies Liao Jiaxi was bor i 1965, obtaied his PhD degree at Uiversity of Electroics Sciece ad Techology of Chia i He is presetly a professor of Beijig Uiversity of Posts ad Telecommuicatios. He has published hudreds of papers i differet jourals ad cofereces. His research iterests are mobile itelliget etwor, broadbad itelliget etwor ad 3G core etwors

Chia Commuicatios Li Yuazhe was bor i 1979, obtaied his B.S.

D. degree at the State Key Laboratory of Networig ad Switchig Techology, Beijig Uiversity of

His research iterests iclude commuicatios software, Next Geeratio Networ, ad multimedia

Li Toghog was bor i 1968, obtaied his PhD degree from Beijig Uiversity of Posts ad

He is curretly a assistat professor with the departmet of computer sciece, Techical

His mai research iterests iclude resource maagemet, distributed system, middleware, wireless

Zhu Xiaomi was bor i 1974, obtaied his PhD degree from Beijig Uiversity of Posts ad

Now he is a associate professor i Beijig Uiversity of His major is Telecommuicatios ad

6 Chia Commuicatios Li Yuazhe was bor i 1979, obtaied his B.S. degree from Najig Uiversity of Sciece ad Techology, He is curretly worig toward the Ph.D. degree at the State Key Laboratory of Networig ad Switchig Techology, Beijig Uiversity of Posts ad Telecommuicatios. His research iterests iclude commuicatios software, Next Geeratio Networ, ad multimedia commuicatio. Li Toghog was bor i 1968, obtaied his PhD degree from Beijig Uiversity of Posts ad Telecommuicatios i He is curretly a assistat professor with the departmet of computer sciece, Techical Uiversity of Madrid, Spai. His mai research iterests iclude resource maagemet, distributed system, middleware, wireless etwors, ad sesor etwors. Zhu Xiaomi was bor i 1974, obtaied his PhD degree from Beijig Uiversity of Posts ad Telecommuicatios i Now he is a associate professor i Beijig Uiversity of Posts ad Telecommuicatios. His major is Telecommuicatios ad Iformatio Systems. His research iterests spa the area of itelliget etwors ad ext-geeratio etwors with a focus o 3G core etwor ad protocol coversio. He has published over 110 papers, amog which there are more tha 30 fi rst-authored oes, i differet jourals ad cofereces. Zhag Lei was bor i 1974, obtaied his PhD degree from Beijig Uiversity of Posts ad Telecommuicatios i He is curretly a lecturer i Beijig Uiversity of Posts ad Telecommuicatios. His mai research iterests iclude mobile itelliget etwor, 3G etwor ad services

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