Throughput Capacity of Multi-Channel Hybrid Wireless Network with Antenna Support

Transcription

1 Appl. Math. If. Si. 8 No Applied Mathematis & Iformatio Siees A Iteratioal Joural Throughput Capaity of Multi-Chael Hybrid Wireless Networ with Atea Support Che Li Shool of Computer ad Iformatio Shaghai Seod Polytehi Uiversity Shaghai 0109 Chia Reeived: 4 Jul. 013 Revised: 8 Nov. 013 Aepted: 9 Nov. 013 Published olie: 1 May. 014 Abstrat: Capaity is oe fudametal problem i wireless etwors. Diretioal ateas to wireless etwors a derease iterferee ad irease spatial reuse. Thus it is a great tehology to improve throughput apaity of wireless etwors. I the paper the upper ad lower bouds of multi-hael hybrid wireless etwor with atea supported are ivestigated. Both simplified atea model ad hybrid atea model are addressed. Moreover two salig regimes are idetified based o the growth of the umber of haels relative to that of iterfaes. Keywords: Capaity diretioal atea multi-hael hybrid 1 Itrodutio Capaity is oe of the most importat problems i wireless ad ho etwors whih provides the theoretial guidelie to its improvemet [1]. Moreover it depeds o may aspets of etwors suh as etwor arhiteture etwor topology traffi patter etwor ode desity umber of haels used for eah ode trasmissio power level ad ode mobility[3]. I the ladmar paper [4] Gupta ad Kumar showed that the throughput apaity obtaiable by eah ode for a radomly hose destiatio is bits/s assumig that umber of mobile odes are plaed radomly a somewhat disappoitig results. Ispired by this wor may researhers have ivestigated the apaity issue for wireless etwors uder various ostraits ad tried to fid out the method to improve the throughput apaity [5]. Exploitig mobility is oe way to improve throughput apaity [67]. Base statio is aother method. I [8] it was show that to ahieve better ivestmet i the ifrastruture is eeded. Kozat ad Tassiulat [9]showed that for radom ad ho etwors with ifrastruture support where the ratio of wireless odes to base statio is bouded above by some ostat a throughput apaity of aot be obtaied. Aother method to improve the throughput apaity of wireless etwors is to use multiple haels. Kyasaur ad Vaidya [10] studied the impat of the umber of haels ad iterfaes per ode o the apaity of multi-hael ad ho etwors as the umber of ad ho odes ireases. Moreover Bhadari ad Vaidya [11] studied oetivity ad asymptoti trasport apaity of the multi-hael pure ad ho etwors with hael swithig ostraits. Reetly diretioal ateas have emerged as a promisig tehology due to the higher spatial reuse ratio [5 1]. The apaity of sigle hael ad ho etwors usig diretioal ateas was ivestigated i [1]. [5] explored the throughput apaity of radom sigle hael diretioal etwors with oe-hop relay shemes. Base statio ad diretioal ateas are ombied osidered i [13]. The apaity of multi-hael wireless etwors with multiple diretioal ateas is studied i [14]. However few studies osidered the diretioal atea radiatio patter impat o multi-hael hybrid wireless etwors whih we all as MC-HDA etwors ad multi-hael hybrid wireless etwors with omidiretioal ateas alled as MC-HOA etwors. Furthermore the real diretioal atea impat o MC-HDA etwors has ot bee ivestigated. To the best of our owledge there is o theoretial aalysis o the throughput apaity of suh etwors. The paper oetrates o fidig the apaity bouds for a MC-HDA etwor ad explorig the beefits of this Correspodig author hel@sspu.edu. 014 NSP Natural Siees Publishig Cor.

2 1456 C. Li: Throughput Capaity of Multi-Chael Hybrid Wireless... etwor. Our otributios a be summarized as follows: *We formally idetify MC-HDA etwors that haraterize the features of multi-hael hybrid etwors with multiple diretioal ateas at eah ode. The apaity of MC-HDA etwors has ot bee studied before. I [] oly MC-HOA etwors are ivestigated. *We drive the upper ad lower bouds o the apaity of MC-HDA etwors uder simplified diretioal ateas model ad pratial diretioal ateas whih osider the side lobe ad ba lobe effet of diretioal ateas. *Our theoretial results show that itegratig diretioal ateas with multi-hael hybrid etwors a irease etwor oetivity ad redue iterferee resultig i improved etwor apaity. Impliatios from aalytial results are also give. The remaider of the paper is orgaized as follows: Defiitio ad otatio used i the paper are summarized i Setio atea ad other models are desribed i Setio 3 the the upper ad lower bouds are preseted i Setio 4 ad 5. We summarize our wor i Setio 6. Defiitio ad Notatio We use the followig otatio to represet asymptoti bouds: f = Og meas there exists some ostat α ad iteger N suh that f αg for >N. f=og meas that lim f/g=0. f=ωg meas that g=o f. f=ωg meas that g=o f. f=θg meas that f=og ad g= O f. The traffi betwee a soure-destiatio pair is referred to as a flow. As i [] we say that per flow apaity is λ if eah flow i the etwor a be guarateed a throughput of at least λ ad the etwor apaity is defied to the aggregate throughput over all the flows i the etwor λ. 3 Model MC-HDA etwors osist of base statios ad ad ho odes. All these odes ad statios are uiformly distributed o the surfae of a uit torus. Ad ho odes aloe form a oeted topology graph with high probability. I other words ay pair of ad ho odes a ommuiate with eah other alog the paths that ross oly the ad ho odes with probability lose to oe. Base statios are oeted through wired ad alterative wireless lis with relatively high badwidth. Moreover Fig. 1: Simplified atea model Fig. : Hybrid atea model base statios oly relay traffi ad do ot geerate their ow data. Ad ho odes are also allowed to utilize the ifrastruture etwor omposed by base statios. These ad ho odes are apable of trasmittig ad reeivig W bits/s via all haels ad the base statios a also ommuiate with ad ho odes by these ommo haels. Ay ode is equipped with m iterfaes ad eah iterfae is assumed to be apable of trasmittig or reeivig data o ay hael. For simpliity the permutatio traffi model is adopted where implies that eah ode is the soure of exatly oe flow ad eah ode is the destiatio of exatly oe flow [10 11]. As [9] Eah ad ho ode geerates a flow with rate λ for a radom destiatio. ad are the umber of ad ho odes ad base statios respetively. We assume that the umber of ad ho odes per base statio is bouded ad lim /=β where β Atea model I this paper two ids of diretioal atea model are used. Oe is the simplified model where the diretioal atea gai is assumed to be withi a speifi agle θ θ is the beamwidth of the atea. The gai outside the beamwidth is assumed to be zero whih is show i Fig.1. At ay time the atea beam a oly be poited to a ertai diretio [1 14]. We all suh MC-HDA etwors usig simplified atea model as MC-HDAS etwors. Aother is the pratial diretioal atea so that we use a hybrid atea model [1]. I the model the effet of side lobes ad ba lobes are osidered whih is show i Fig. The mai lobe of beamwidth θ is haraterized as a setor. Side lobes ad ba lobes form a irle. We defie parameter s as the ratio of the radius of the irle ad the setor whih is geerally less tha 1. We all suh MC- HDA etwors with hybrid atea model as MC-HDAH etwors. 014 NSP Natural Siees Publishig Cor.

3 Appl. Math. If. Si. 8 No / Iterferee model The protool model proposed by Gupta ad Kumar [4] is exteded to ilude diretioal ateas i the paper. It is the reeiver-based iterferee model proposed i [14]. Our model also oly osiders both diretioal trasmissio ad diretioal reeptio. Suppose ode X i trasmits over the mth hael to a ode X j. The this trasmissio is suessfully reeived by ode X j for every other ode X simultaeously trasmittig over the same hael ad the guard zoe δ > 0 the followig oditio holds. { X X j 1+δ X i X j or X s beam does ot over ode X j X i also deotes the loatio of a ode. 3.3 Chael model The total data rate is divided equally amog the haels ad the the data rate supported by ay oe of the haels is W/ 4 Upper boud The apaity of MC-HDA etwors both MC-HDAS ad MC-HDAH is limited by two ostraits ad eah of them is used to obtai a boud o the etwor apaity. The miimum of the two bouds the bouds deped o ratio of the umber of haels to the umber of iterfaes m is a upper boud o the etwor apaity. Costrait 1.Iterfae Costraits: It a be show that the protool model for iterferee requires that eah trasmissio osumes a ertai area ad two simultaeous trasmissios our o the same hael ad are suessfully reeived the area must be disjoit. Therefore we a boud the umber of areas. Lemma 1.The apaity of MC-HDAS etwors is bits/se ad that of MC-HDAH etwors is λ = O bits/se. λ = O + s log Proof.for MC-HDAS etwors the oditio iterferee θ zoe is portio of that whe omiidiretioal π ateas are used at both eds [4 14]. Sie for omiidiretioal ateas Therefore the iterferee θ zoe is r i is the trasmissio radiusfor eah π π r i 4 trasmissio o hael i 1 i. Hee umber of simultaeous trasmissios o hael i must be smaller tha 16π. As a result give the average umber of hops ri h i for trasmissios o hael i per ode flow apaity λ i o hael i followig iequality holds. i λ i h i 16π W ri 1 It is proved i [49] that the trasmissio radius r i must at least satisfy the followig boud for havig a oeted graph with probability oe. log i r i π i Usig the iequality ad the fat h i ad there are haels i etwors with probability of oe the upper boud holds uder ay routig ad shedulig sheme. For per flow apaity λ 16π W 3 log So the etwor apaity of MC-HDAS etwors is proved λ = O Next the MC-HDAH etwor is osidered. Sie for eah trasmissio the oditioal iterferee area is πr s + 1 s through the same proof tehique the etwor apaity of MC-HDAH etwors is λ = O bits/se. + s log Costrait.Iterferee ostraits: The apaity of MC-HDA etwors is also limited by the umber of simultaeous trasmissios supported by iterferees i the etwor. Lemma.The apaity of MC-HDA etwors is O m Proof.Aordig to the hael model eah iterfae a trasmit at a rate of W bits/s ad eah ode has m iterfaes so the total etwor apaity is bouded by O m Combiig with the two bouds the etwor apaity is bouded by O mi m bits/s for MC-HDAS etwors ad O mi bits/s for m + s log MC-HDAH etwors. Theorem 1.The upper boud of the apaity of MC-HDA etwors are as follows: 1Whe m is O θπ log the etwor apity is O bits/s over a MC-HDAS etwor; whe m is O + s log bits/s the etwor apaity is O bits/s over a MC-HDAH + s log etwor. Whe m is Ω θπ log the etwor apaity is O m bits/s over a MC-HDAS etwor; whe m is Ω + s log bits/s the etwor apaity is Ø m bits/s over a MC-HDAH etwor. While there may be other ostraits o apaity as well the ostraits we osider is suffiiet to provide a tight boud. 014 NSP Natural Siees Publishig Cor.

4 1458 C. Li: Throughput Capaity of Multi-Chael Hybrid Wireless... 5 Lower boud I the setio we provide a ostrutio of the routig sheme ad shedulig strategy to establish a lower boud of the etwor apaity. We first deal with MC-HDAS etwors ad provide a ostrutio of a routig sheme ad a trasmissio shedule for 1 etwor ad the exted the result to a m etwor by usig Lemma i [10]. Fig. 3: Routig sheme 5.1 Cell ostrutio. The surfae of the uit torus is divided ito square ells usig a square grid ad we deote eah of area by α similar to that used i [10]. I partiular we set α = max 100log. The trasmissio radii of eah ode are set to 8α thereby satisfyig the requiremet that ay ode i oe ell a ommuiate with ay other ode i its eighborig ells ad also to guaratee that ad ho odes form a oeted graph with probability oe. We eed to boud the umber of ad ho odes ad base statios that are preset i eah ell. Lemma 3. Both the umber of base statios ad ad ho odes belogig to ay ell are asymptotial i the same order i.e.θα[]. Proof.We provide a proof based o Vapi-Chervoeis VC-theory.Beause base statios are radomly loated o the uit torus the probability that ay base statio belogs to a speifi ell is Θα. The set of axis-parallel square D are ow to have VC-dimesio 3. Similar to the proof i [4] We set ε = σ = 50log/ to satisfy the VC-theory ad the we have Prob sup N d α 50log d D > 1 50log Where N d is the umber of base statios i Cell d. Therefore with high probability the followig result holds: a 50log N d a 50log beause lim /=β where β 0 so a β a β 50log1+1/β 1+β N d + 50log1+1/β 1+β provided α>50log/. Sie α = max 100log so the oditio is always satisfied ad we obtai that the umber of base statios belogs to ay ell is i.e.θα. Followig the same proof proedure we a also obtai the same result for ad ho odes. Hee we a olude that both the umber of base statios ad ad ho odes belogig to ay ell are asymptotial i the same order i.e.θα. If a ode i Cell A iterferes with aother trasmissio i Cell B this ell is alled as iterferig ell. The we prove that the umber of iterferig ells aroud a ell is a ostat whih is idepedet of a ad. Lemma 4.The umber of ells that iterfere with the trasmissios of a speifi ell is bouded above by a ostat 1 whe 1 = 81+δ [14]. 5. Routig strategy. There are two routig strategy [] whih is show i Fig.3: oe if destiatio ode is loated i the same ell as soure ode the paets are trasmitted to its destiatio ode diretly; otherwise paets are firstly trasmitted to the base statio i its ow ell that is with the least umber of flows load. Next the base statio forwards the data to that base statio that is loated at the same ell with the destiatio ode ad with the least umber of flows. Lastly the base statio trasmits the data to the destiatio ode diretly. I the routig sheme the flows through base statios are balaed; otherwise some base statio may beome the traffi bottlee. Thus whe a flow eeds to pass through a ell from amogst all base statios i the ell it is assiged to the oe whih has the least umber of flows assiged o it so far. The followig lemma is to boud the umber of flows assiged to ay base statio i ay ell. Lemma 5.The umber of flows that are assiged to ay oe base statio i ay ell is O1 with high probability []. 5.3 Trasmissio shedulig.from lemma 4 the umber of ells that iterfere with the trasmissios of a speifi ell is bouded above by a ostat 1 that oly depeds o δ. Thus we have a 014 NSP Natural Siees Publishig Cor.

5 Appl. Math. If. Si. 8 No / shedulig of legth slots that a alloate a ell-slot for eah ell i a roud robi fashio. I eah ell-slot the orrespodig ells are ative. We divide oe seod ito O ell-olor slots ad eah ell-slot has a legth of Ω seods. Next the trasmissio shedulig is resposible for geeratig a withi-ell shedulig to esure that at ay istat there is o more tha oe trasmissio o ay give hael ad o ode expeted to trasmit or reeive more tha oe paet i the ell. We first assig a hael for every ode i the ell that is ready to trasmit paets from available haels. There are Θα odes i ay ell with high probability from Lemma.3i.e. at most 3 α odes for some ostat 3 that a be umbered from 1 to 3 α. A ode umbered q is allowed to trasmit o hael q mod +1. We further divide eah ell-olor slot ito mii-slots. The we build a shedule for eah trasmissio over a hael i the ative ell. We ostrut a oflit graph i two steps. Firstly reate a separate vertex for eah trasmissio i the ell ad eah vertex for eah trasmissio has two properties: the assiged hael ad two edpoits of the trasmissio. Seodly oet two verties by a edge if they satisfy two requiremets: the same hael or at least oe same edpoit. The shedulig problem thus redues to obtaiig a vertex-olorig of this graph. It a be easily see that the degree of the oflit graph is O α + 1=O α. It is well ow that a graph with maximum degree e a be vertex-olored with at most e + 1 olors. Therefore the oflit graph a be vertex-olored with at most 3α olors. Thus the ell-slot is further 1 divided ito Ω equal legth mii-slots ad 3 α all flows get a slot for trasmissio. Eah hael a trasmit at the rate of W Thus ombied with ell-slot ad mii-slot for eah flow W λ=ω α/ W = Ω bits/s 3 α/ sie 3 α/ 3 a/+1 we have λ=ω W 3α+ Due to the asymptoti order either a or will domiate the deomiator of λ. Hee the etwor apaity is equal to λ = Ω Mi W α The we obtai that whe = O θπ log λ = Ω bits/s; ad whe =Ω θπ log λ=ω For MC-MDAH etworthe oditio iterferee zoe is s + 1 s portio of that whe omidiretioal ateas are used ompared to 4π portio for MC-HDAS etwor. The surfae of a uit torus the a be divided ito square ells usig a square grid ad we the set α = max 100log s + 1 s to satisfy the oetivity requiremet for MC-MDAH etwor. Thus aordig to the same ostrutio proesses the throughput apaity of MC-HDAH etwors is equal to Ω Mi + s α bits/s The we obtai that whe is O + s log bits/s the etwor apaity is Ω bits/s; Whe is + s log Ω + s log bits/s the etwor apaity is Ω m bits/s over a MC-HDAH etwor. Based o Lemma. i [10] we have the followig theorem for the m multi-hael hybrid wireless etwor: Theorem. The etwor apaity of a m multi-hael hybrid wireless etwor with atea support is as follows: 1Whe m is O θ π log the etwor apaity is Ω bits/s over a MC-HDAS etwors; Whe m is O + s log bits/s the etwor apaity is Ω + s log bits/s over a MC-HDAH etwors. Whe m is Ω θ π log the etwor apaity is Ω m bits/s over a MC-HDAS etwors; Whe m is Ω + s log bits/s the etwor apaity is Ω m bits/s over a MC-HDAH etwors. The upper ad lower bouds have the same order so the bouds are tight. 6 Colusio I the paper we address the throughput apaity of multi-hael hybrid wireless etwor with atea support. Both MC-HDAS etwor with simplified atea model ad MC-HDAH etwor with hybrid atea model are dealt with. We have foud that usig diretioal ateas i multi-hael hybrid etwors ot oly a ehae etwor oetivity but also redues 014 NSP Natural Siees Publishig Cor.

6 1460 C. Li: Throughput Capaity of Multi-Chael Hybrid Wireless... iterferees. Combiig diretioal ateas with multi-hael hybrid wireless etwor a ahieve sigifiat improvemet o the etwor performae. Aowledgemet The first author aowledges the fiaial support by Natioal Natural Siee Foudatio of Chia Grat No Shaghai Natural Siee Foudatio of Chia Grat No. 1ZR Iovatio Program of Shaghai Muiipal Eduatio Commissio Grat No. 13ZZ141 ad the ostrutig program of the ey breedig disiplie i Shaghai Seod Polytehi Uiversity Computer Appliatio Tehology No.XXKPY1301. The authors are grateful to the aoymous referee for a areful heig of the details ad for helpful ommets that improved this paper. Referees [1] J. Wag L. H. Kog ad M. Y. Wu Capaity of Wireless Ad ho Networs Usig Pratial Diretioal Ateas Proeedigs of IEEE Wireless Commuiatios ad Networig Coferee [] L. Che S. T. Wei Throughput apaity of hybrid multihael wireless etworsaeu-iteratioal Joural of Eletrois ad Commuiatios [3] A. I X. Wag W. Wag Wireless mesh etwors: A survey Computer Networs [4] P. Gupta ad P. R. Kumar The apaity of wireless etwors IEEE Trasatios o Iformatio Theory [5] P. Li C. Zhag Y. G. Fag The apaity of Wireless Ad ho Networs Usig Diretioal Ateas IEEE Trasatios o Mobile Computig [6] M. Grossglauser D. Tse Mobility ireases the apaity of ad ho wireless etwors IEEE/ACM Trasatios o Networig [7] A. Gamal J. Mamme B. Prabhaar D. ShahThroughput delay trade-off i wireless etwors Proeedigs of the IEEE Ifoom [8] B. Liu Z. Liu D. Towsley O the apaity of hybrid wireless etwors Proeedigs of IEEE Ifoom [9] U. Kozat L. TassiulatThroughput apaity of radom ad ho etwors with ifrastruture support Proeedigs of ACM mobiom [10] P. Kyasaur N. Vaidya Capaity of multi-hael wireless etwor: impat of umber of haels ad iterfaes Proeedigs of ACM mobiom Gologe Germay [11] V. Bhadari N. Vaidya Coetivity ad Capaity of multi-hael wireless etwors with hael swithig ostraits Tehial Report Uiversity of Illiois at Urbaa-Champaig USA 006. [1] Y. Su Y. Pei S. Kalyaarama O the Capaity Improvemet of Ad Ho Wireless Networs Usig Diretioal AteasProeedigs of MobiHo [13] G. L. Zhag Y. Y. Xu X. B. Wag M. Guizai Capaity of Hybrid Wireless Networs with Diretioal Atea ad Delay CostraitIEEE Trasatios o Commuiatios [14] H. N. Dai K. W. Ng R. W Wog ad M. Y. Wu O the apaity of multi-hael Wireless Networs Usig Diretioal AteasProeedigs of the IEEE Ifoom Li Che reeived the PhD degree i Computer Applied Tehology at Togji UiversityShaghai Chia. He is a assoiate professor with the departmet of omputer siee ad tehology. His researh iterests are i the areas of vehiular ad ho etwor wireless sesor etwor ad yber-physial system. He has published tes of researh artiles i reputed iteratioal jourals or oferees of iformatio siees. 014 NSP Natural Siees Publishig Cor.