On Capacity and Delay of Multi-channel Wireless Networks with Infrastructure Support

Transcription

1 O Capaity ad Delay of Multi-hael ireless Networks with Ifrastruture Support Hog-Nig Dai, Member, IEEE, Raymod Chi-ig og, Seior Member, IEEE, Hao ag, Member, IEEE Abstrat I this paper, we propose a ovel multi-hael etwork with ifrastruture support, alled a MC-IS etwork, whih has ot bee studied i the literature To the best of our kowledge, we are the first to study suh a MC-IS etwork Our proposed MC-IS etwork has a umber of advatages over three existig ovetioal etworks, amely a sigle-hael wireless ad ho etwork (alled a SC-AH etwork, a multihael wireless ad ho etwork (alled a MC-AH etwork ad a sigle-hael etwork with ifrastruture support (alled a SC-IS etwork I partiular, the etwork apaity of our proposed MC-IS etwork is log times higher tha that of a SC-AH etwork ad a MC-AH etwork ad the same as that of a SC-IS etwork, whereis the umber of odes i the etwork The average delay of our MC-IS etwork is log/ times lower tha that of a SC-AH etwork ad a MC-AH etwork, ad mi{c I,m} times lower tha the average delay of a SC- IS etwork, where C I ad m deote the umber of haels dediated for ifrastruture ommuiatios ad the umber of iterfaes mouted at eah ifrastruture ode, respetively Our aalysis o a MC-IS etwork equipped with omi-diretioal ateas oly has bee exteded to a MC-IS etwork equipped with diretioal ateas oly, whih are amed as a MC-IS- DA etwork e show that a MC-IS-DA etwork has a eve lower delay of 2π θ C I ompared with a SC-IS etwork ad our MC-IS etwork For example, whe C I = 2 ad θ = π 2, a MC-IS-DA a further redue the delay by 24 times lower that of a MC-IS etwork ad redue the delay by 288 times lower tha that of a SC-IS etwork I INTRODUCTION How to improve the etwork performae, i terms of the etwork apaity ad the average delay, has bee a key issue i reet studies [] Covetioal wireless etworks typially osist of odes that share oe sigle hael for ommuiatios It is foud i [2], [3] that i a radom ad ho etwork with odes, eah ode has a throughput apaity of Θ(/ log (where is the total etwork badwidth ad the average delay of this etwork is Θ( /log he the umber of odes ireases, the per-ode throughput dereases ad the average delay ireases Oe major reaso is that all the odes withi the etwork share the same medium he a ode trasmits, its eighborig odes i the same Copyright ( 205 IEEE Persoal use of this material is permitted However, permissio to use this material for ay other purposes must be obtaied from the IEEE by sedig a request to pubs-permissios@ieeeorg H-N Dai is with Faulty of f Iformatio Tehology, Maau Uiversity of Siee ad Tehology, Maau SAR ( hdai@ieeeorg R C- og is with Departmet of Computer Siee ad Egieerig, The Hog Kog Uiversity of Siee ad Tehology, Clear ater Bay, Kowloo, Hog Kog SAR ( raywog@seusthk H ag is with Faulty of Egieerig ad Natural Siees, Norwegia Uiversity of Siee & Tehology i Aalesud, Norway ( hawa@tuo hael are prohibited from trasmittig to avoid iterferee Besides, multi-hop ad short-raged ommuiatios are preferred i this etwork i order to miimize the iterferee ad ahieve the high etwork apaity [2] However, the multi-hop ommuiatios ievitably lead to the high edto-ed delay Furthermore, every ode equipped with a sigle iterfae aot trasmit ad reeive at the same time (ie, the half-duplex ostrait e ame this sigle-hael ad ho etwork as a SC-AH etwork Oe approah to improve the etwork performae is to use multiple haels istead of a sigle hael i a wireless etwork The experimetal results of [4] [9] show that usig multiple haels a sigifiatly improve the etwork throughput Oe possible reaso for the improvemet is that usig multiple haels a separate multiple ourret trasmissios i frequey domais so that the iterferee a be mitigated Aother reaso is that multiple simultaeous trasmissios/reeptios are supported by multiple etwork iterfaes mouted at a wireless ode, osequetly leadig to the improved frequey reuse ad the ireased throughput However, it is show i [2] [8] that eah hael (or up to O(log haels must be utilized by a dediated iterfae at a ode i order to fully utilize all the haels simultaeously so that the etwork apaity a be maximized he the oditio is ot fulfilled, the apaity degrades sigifiatly Besides, the average delay of a MC-AH etwork is also Θ( /log, whih ireases sigifiatly with the ireased umber of odes e all this multi-hael wireless ad ho etwork as a MC-AH etwork Reet studies [0] [5] ivestigated the performae improvemet by addig a umber of ifrastruture odes to a wireless etwork Speifially, as show i [0], [4], deployig ifrastruture odes i the wireless etwork a sigifiatly improve the etwork apaity ad redue the average delay But, every ode i suh a etwork equipped with a sigle iterfae aot trasmit ad reeive at the same time Besides, oly oe sigle hael is used i suh a etwork e all this sigle-hael etworks with ifrastruture support as a SC-IS etwork I this paper, we propose a ovel multi-hael etwork with ifrastruture support that overomes the above drawbaks of existig etworks This etwork osists of ommo odes, eah of whih has a sigle iterfae, ad ifrastruture odes (or base statios, eah of whih has multiple iterfaes Both ommo odes ad base statios a operate o differet haels This multi-hael wireless etwork with ifrastruture support is alled a MC-IS etwork that has the followig harateristis

2 2 TABLE I COMPARISON ITH OTHER EXISTING IRELESS NETORKS Pure Ad Ho Ad Ho with Ifrastruture Sigle Chael SC-AH etworks SC-IS etworks [2], [3] [0] [7] Multiple Chaels MC-AH etworks MC-IS etworks [4] [9] (this paper Eah ommo ode is equipped with a sigle etwork iterfae ard (NIC Eah base statio is equipped with multiple NICs There are multiple o-overlappig haels available Eah NIC at either a ommo ode or a base statio a swith to differet haels quikly Base statios are oeted via a wired etwork that has muh higher badwidth tha a wireless etwork Eah ommo ode with a sigle NIC a ommuiate with either aother ommo ode or a base statio, where a ommuiatio with aother ommo ode is alled a ad-ho ommuiatio ad a ommuiatio with a base statio is alled a ifrastruture ommuiatio But, a ommo ode supports oly oe trasmissio or oe reeptio at a time Besides, it aot simultaeously trasmit ad reeive (ie, it is i a half-duplex mode Eah base statio with multiple NICs a ommuiate with more tha oe ommo ode I additio, a base statio a also work i a full-duplex mode, ie, trasmissios ad reeptios a our i parallel Our proposed MC-IS etworks have provided a solutio to the ew appliatios, suh as Devie-to-Devie (D2D etworks [8], wireless sesor etworks (SNs, smart grid ad smart home [9], [20] For example, the theoretial aalysis o the throughput ad the delay of MC-IS etworks a be used to aalyze the performae of the overlaid D2D etworks (refer to Setio VII-C for more details Table I ompares our proposed MC-IS etworks with other existig etworks, where oe a observe that MC-IS etworks a fully exploit the beefits of both MC-AH etworks ad SC-IS etworks ad a potetially have a better etwork performae (i terms of the etwork apaity ad the delay tha other existig etworks However, to the best of our kowledge, there is o theoretial aalysis o the apaity ad the average delay of a MC-IS etwork The goal of this paper is to ivestigate the performae of a MC-IS etwork ad to explore the advatages of this etwork The primary researh otributios of our paper are summarized as follows ( e formally idetify a MC-IS etwork that haraterizes the features of multi-hael wireless etworks with ifrastruture support To the best of our kowledge, the apaity ad the average delay of a MC-IS etwork have ot bee studied before (2 e propose a geeral theoretial framework to aalyze the apaity ad the average delay e show that other existig etworks a be regarded as speial ases of our MC-IS etwork i our theoretial framework Besides, we fid that our MC-IS etworks are limited by four requiremets (to be defied i Setio IV simultaeously but the existig etworks are oly limited by subsets of them (ot all of them This meas that studyig the performae of our MC-IS etworks is more hallegig but it is more useful ad realisti to osider four requiremets simultaeously sie they exist aturally i real life appliatios (3 Our proposed MC-IS etwork has a lot of advatages over existig related etworks I partiular, a MC-IS etwork a ahieve the optimal per-ode throughput, whih is log times higher tha that of a SC-AH etwork ad a MC-AH etwork ad the same as that of a SC-IS etwork, while maitaiig the smallest delay, whih is log/ times lower tha that of a SC-AH etwork ad a MC-AH etwork, ad mi{c I,m} times lower tha that of a SC-IS etwork The performae improvemet maily owes to the multiple NICs at a base statio, ompared with a sigle NIC at a base statio i SC-IS etworks As a result, our MC-IS etworks have a better performae tha SC-IS etworks though the theoretial aalysis is also more ompliated tha that of SC-IS etworks (4 e also exted our MC-IS etworks with the osideratio of usig diretioal ateas istead of omidiretioal ateas Speifially, all aforemetioed etworks (ie, SC-AH etworks, MC-AH etworks, SC- IS etworks ad our MC-IS etworks are equipped with omi-diretioal ateas but the exteded MC- IS etworks have both the base statios ad all ommo odes equipped with diretioal ateas e ame the exteded MC-IS etworks as MC-IS-DA etworks e show that a MC-IS-DA etwork a have a eve lower delay of ompared with both a MC-IS etwork 2π θ ad a SC-IS CI etwork, where θ is the beamwidth of a diretioal atea mouted at the base statio (usually θ < 2π Cosider the ase of C I = 2 ad θ = π 2 that is feasible i Millimeter-ave systems [2] A MC-IS- DA a further redue the delay by 24 times lower tha that of a MC-IS etwork ad redue the delay by 288 times lower tha that of a SC-IS etwork The remaider of the paper is orgaized as follows Setio II presets a survey o the related studies to our MC-IS etwork e preset the models used i this paper i Setio III Setio IV the summarizes our mai results e ext derive the apaity ad the delay otributed by ad ho ommuiatios i a MC-IS etwork i Setio V Setio VI presets the apaity ad the delay otributed by ifrastruture ommuiatios i a MC-IS etwork e exted our aalysis with the osideratio of diretioal ateas as well as the mobility ad provide the impliatios of our results i Setio VII Fially, we olude the paper i Setio VIII II RELATED ORKS e summarize the related works to our study i this setio The first etwork related to our proposed MC-IS etwork is a SC-AH etwork A SC-AH etwork has a poor performae due to the followig reasos: (i the iterferee amog multiple ourret trasmissios, (ii the umber of simultaeous trasmissios o a sigle iterfae ad (iii the multi-hop ommuiatios [2], [3]

3 3 The seod etwork related to our MC-IS etwork is a MC- AH etwork, i whih multiple haels istead of a sigle hael are used Besides, eah ode i suh a etwork is equipped with multiple NICs istead of sigle NIC This etwork has a higher throughput tha a SC-AH etwork beause eah ode a support multiple ourret trasmissios over multiple haels However, this etwork suffers from the high delay ad the ireased deploymet omplexity The average delay of a MC-AH etwork is the same as that of a SC- AH etwork, whih ireases sigifiatly with the umber of odes The deploymet omplexity is maily due to the oditio [8] that eah hael (up to O(log haels must be utilized by a dediated iterfae at a ode so that all the haels are fully utilized simultaeously he the oditio is ot fulfilled, the apaity degrades sigifiatly The third etwork related to our MC-IS etwork is a SC- IS etwork [0] [7], [22] It is show i [0], [4] that a SC-IS etwork a sigifiatly improve the etwork apaity ad redue the average delay However, a ifrastruture ode i suh a etwork equipped with a sigle iterfae aot trasmit ad reeive at the same time (ie, the half-duplex ostrait is still efored Thus, the ommuiatio delay i suh a SC-IS etwork is still ot miimized Besides, suh SC-IS etworks also suffer from the poor spetrum reuse The fourth etwork related to our MC-IS etwork is a multi-hael wireless mesh etwork with ifrastruture support (alled a MC-Mesh-IS etwork [23] [28], whih is the evolutio of multi-hael multi-iterfae wireless mesh etworks (alled a MC-Mesh etwork [29], [30] A MC- Mesh-IS etwork is differet from our MC-IS etwork due to the followig harateristis of a MC-Mesh-IS etwork: (i a typial MC-Mesh-IS etwork osists of mesh liets, mesh routers ad mesh gateways while a MC-IS etwork osists of ommo odes ad ifrastruture odes (ii differet types of ommuiatios exist i the multi-tier hierarhial MC-Mesh-IS etwork, whih are far more ompliated tha a MC-IS etwork For example, there are ommuiatios betwee mesh liets, ommuiatios betwee mesh gateways, ad ommuiatios betwee a mesh gateway ad a mesh router (iii a MC-Mesh-IS etwork uses wireless liks to oet the bakboe etworks (orrespodig to the ifrastruture etwork i a MC-IS etwork As a result, the assumptio of the ulimited apaity ad the iterfereefree ifrastruture ommuiatios i a MC-IS etwork does ot hold for a MC-Mesh-IS etwork (iv the traffi soure of a MC-Mesh-IS etwork is either from a mesh liet or from the Iteret while the traffi always origiates from a MC-IS etwork Therefore, the aalyti framework o the apaity ad the delay of suh MC-Mesh-IS etworks is sigifiatly differet from that of a MC-IS etwork I this paper, we aalyze the apaity ad the delay of a MC-IS etwork Although parts of the results o the aalysis o the apaity ad the delay otributed by ad ho ommuiatios have appeared i [3], our aalysis i this paper sigifiatly differs from the previous work i the followig aspets Fig X34 Base statio Commo ode Data flow X8 X4 B X24 X X20 X32 X5 X8 X3 X26 X6 B2 Data flow 4 X0 X5 X9 X2 Data flow 2 X X4 X7 Data flow 2 Network topology of a MC-IS etwork X25 B6 X35 X3 X33 X29 B3 X23 X36 X28 X27 X2 X30 X3 Ad ho ommuiatios Ifrastruture ommuiatios Data flow 3 e derive the apaity ad the delay of a MC-IS etwork otributed by ifrastruture ommuiatios i this paper while [3] oly addresses the apaity ad the delay otributed by ad ho ommuiatios e fully ivestigate the apaity ad the delay of a MC- IS etwork with osideratio of both ifrastruture ommuiatios ad ad ho ommuiatios Speifially, we also aalyze the average delay ad the optimality of our results, all of whih have ot bee addressed i [3] e also ompare our results with other existig etworks, suh as a SC-AH etwork, a MC-AH etwork ad a SC-IS etwork ad aalyze the geerality of our MC-IS etwork i this paper e exted our aalysis with osideratio of usig diretioal ateas i a MC-IS etwork Disussios o the mobility are also preseted i this paper III MODELS e adopt the asymptoti otatios [32] i this paper e the desribe the MC-IS etwork model i Setio III-A Setio III-B ext gives the defiitios of the throughput apaity ad the delay A MC-IS Network Model Take Fig as a example of MC-IS etworks I this etwork, ommo odes are radomly, uiformly ad idepedetly distributed o a uit square plae A Eah ode is mouted with a sigle iterfae that a swith to oe of C available haels Eah ode a be a data soure or a destiatio All the odes are homogeeous, whih meas that they have the same trasmissio rage I additio, there are b ifrastruture odes, whih are also alled base statios iterhageably throughout the whole paper e assume that b a be expressed as a square of a ostat b 0 (ie, b 2 0 where b 0 is a iteger i order to simplify our disussio Eah base statio is equipped with m iterfaes ad eah iterfae is assoiated with a sigle omi-diretioal atea, whih a operate o oe of C haels The plae A is evely partitioed ito b equal-sized squares, whih are alled BS-ells Similar to [0], [4], [5], we also assume that a base statio is plaed at the eter of eah BS-ell Ulike a ode, a base statio is either a data soure or a destiatio ad it oly helps forwardig data for odes All the base statios are oeted through a wired etwork without apaity ostrait ad delay ostrait B5 X7 B4 X2 X9 X22 X6

4 4 There are two kids of ommuiatios i a MC-IS etwork: (i Ad ho ommuiatios betwee two odes, whih ofte proeed i a multi-hop maer; (ii Ifrastruture ommuiatios betwee a ode ad a base statio, whih spa a sigle hop A ifrastruture ommuiatio osists of a uplik ifrastruture ommuiatio from a ode to a base statio, ad a dowlik ifrastruture ommuiatio from a base statio to a ode I the followig, we desribe two major ompoets for etwork ommuiatios The first ompoet is the routig strategy The seod ompoet is the iterferee model Routig Strategy: I this paper, we osider the H-maxhop routig strategy, i whih, if the destiatio is loated withi H (H hops from the soure ode, data pakets are trasmitted through ad ho ommuiatios Otherwise, data pakets are forwarded to the base statio through ifrastruture ommuiatios (ie, the uplik ifrastruture ommuiatio The base statio the relays the pakets through the wired etwork After the pakets arrive at the base statio that is losest to the destiatio ode, the base statio the forwards the pakets to the destiatio ode (ie, the dowlik ifrastruture ommuiatio Take Fig as the example agai Data flow starts from ode X to ode X 6 i the multi-hop ad ho maer sie ode X 6 is withi H hops from ode X ith regard to Data flow 2, sie destiatio ode X 28 is far from soure ode X 36, data pakets are trasmitted from soure ode X 36 to its earest base statio B 3 first ad the are forwarded through the wired etwork till reahig base statio B 5 that fially seds the data pakets to destiatio ode X 28 The H-max-hop routig strategy a avoid the problem that arises by usig the k-earest-ell routig strategy i the ase of two odes ear the boudary of two adjaet BS-ells For example, Data flow 4 as show i Fig startig from ode X 0 to destiatio ode X 25 will be trasmitted i oe hop by ad ho ommuiatios aordig our H-max-hop routig strategy However, i the k-earest-ell routig strategy [0], ode X 0 has to trasmit to its earest BS (ie, B 3 first ad the B 3 forwards the data pakets through the wired etwork till they reah B 2, whih is the earest BS to ode X 25 This problem may result i ieffiiet use of badwidth resoures It is obvious that whe there is a uplik ommuiatio, there is always a dowlik ommuiatio e the divide the total badwidth of bits/se ito three parts: ( for ad ho ommuiatios, (2 I,U for uplik ifrastruture ommuiatios ad (3 I,D for dowlik ifrastruture ommuiatios Sie I,U is equal to I,D, it is obvious that = + I,U + I,D = +2 I,U To simplify our aalysis, we use I to deote either I,U or I,D Correspodig to the partitio of the badwidth, we also split the C haels ito two disjoit groups ad C I, i whih haels are dediated for ad ho ommuiatios ad C I haels are dediated for ifrastruture ommuiatios Thus, C = + C I Besides, eah base statio is mouted with m NICs, whih serve for both the uplik traffi ad the dowlik traffi It is obvious that the umber of NICs servig for the uplik traffi is equal to the umber of NICs servig for the dowlik traffi So, m must be a eve umber 2 Iterferee model: I this paper, we osider the iterferee model [2], [8], [0] [2], [4] he ode X i trasmits to ode X j over a partiular hael, the trasmissio is suessfully ompleted by ode X j if o ode withi the trasmissio rage of X j trasmits over the same hael Therefore, for ay other ode X k simultaeously trasmittig over the same hael, ad ay guard zoe > 0, the followig oditio holds dist(x k,x j (+ dist(x i,x j where dist(x i,x j deotes the distae betwee two odes X i ad X j Note that the physial iterferee model [2] is igored i this paper sie the physial model is equivalet to the iterferee model whe the path loss expoet is greater tha two (it is ommo i a real world [2], [33] The iterferee model applies for both ad ho ommuiatios ad ifrastruture ommuiatios Sie ad ho ommuiatios ad ifrastruture ommuiatios are separated by differet haels (ie, ad C I do ot overlap eah other, the iterferee oly ours either betwee two ad ho ommuiatios or betwee two ifrastruture ommuiatios B Defiitios of Throughput Capaity ad Delay The otatio of throughput of a trasmissio from a ode X i to its destiatio ode X j is usually defied as the umber of bits that a be delivered from X i to X j per seod The aggregate throughput apaity of a etwork is defied to be the total throughput of all trasmissios i the etwork The per-ode throughput apaity of a etwork is defied to be its aggregate throughput apaity divided by the total umber of trasmissios (or all odes ivolved i trasmissios I this paper, we maily oetrate o the per-ode throughput apaity ad the average delay, whih are defied as follows Defiitio : Feasible per-ode throughput For a MC-IS etwork, a throughput of λ (i bits/se is feasible if by ad ho ommuiatios or ifrastruture ommuiatios, there exists a spatial ad temporal sheme, withi whih eah ode a sed or reeive λ bits/se o average Defiitio 2: Per-ode throughput apaity of a MC-IS etwork with the throughput of λ is of order Θ(g( bits/se if there are determiisti ostats h > 0 ad h < + suh that lim P(λ = hg( is feasible = ad lim if P(λ = h g( is feasible < I this paper, the per-ode throughput apaity of a MC-IS etwork is expressed by λ = λ a +λ i, where λ a ad λ i deote the throughput apaity otributed by the ad ho ommuiatios ad the ifrastruture ommuiatios, respetively Besides, we use T, T A, T I to deote the feasible aggregate throughput, the feasible aggregate throughput otributed by ad ho ommuiatios, ad the feasible aggregate throughput otributed by ifrastruture ommuiatios, respetively Defiitio 3: Average Delay of a MC-IS etwork The delay of a paket is defied as the time that it takes for the paket to reah its destiatio after it leaves the soure [3] After averagig the delay of all the pakets trasmitted i

5 5 Case C = O( F H = og ( G A C =Ω( F Case 2 C =Ω( F A ad A 2 C = O( F A 2 H =Ω( H = og ( H =Ω( G H = og ( 2 2 H =ΩG ( 3 3 Case 3 Sub-ase Sub-ase 2 Sub-ase 2 Sub-ase 22 Sub-ase 3 Sub-ase 32 (Iterfae-bottleek(Coetivity (Iterfae-bottleek(Iterferee (Iterfae-bottleek(Destiatio-bottleek Fig 2 All possible sub-ases osidered the whole etwork, we obtai the average delay of a MC-IS etwork, deoted by D The average delay of a MC-IS etwork is expressed by D = D a +D i, where D a ad D i deote the delay otributed by ad ho ommuiatios ad the delay otributed by ifrastruture ommuiatios, respetively To derive the average delay i this paper, we osider the fluid model proposed by A El Gamal et al i [3] I this model, the paket size is allowed to be arbitrarily small so that the time take for trasmittig a paket may oly oupy a small fratio of oe time slot, implyig that multiple pakets a be trasmitted withi oe time slot The fluid model a be easily exteded to the ase of the paket with ostat size as show i [34] Note that we do ot out the delay aused by the ifrastruture ommuiatios withi the wired etwork Besides, we also igore the queuig delay i this model I order to ompare the optimality of our results with the existig oes, we itrodue the optimal per-ode throughput apaity λ opt, whih is the maximum ahievable per-ode throughput apaity, ad the optimal average delay D opt, whih is the average delay whe the optimal per-ode throughput apaity λ opt is ahieved IV MAIN RESULTS e first preset the four requiremets that limit the apaity of a MC-IS etwork i Setio IV-A Setio IV-B the gives the mai results A Four Requiremets e have foud that the apaity of a MC-IS etwork is maily limited by four requiremets simultaeously: (i Coetivity requiremet - the eed to esure that the etwork is oeted so that eah soure ode a suessfully ommuiate with its destiatio ode; (ii Iterferee requiremet - two reeivers simultaeously reeivig pakets from two differet trasmitters must be separated with a miimum distae to avoid the iterferee betwee the two trasmissios for the two reeivers; (iii Destiatio-bottleek requiremet - the maximum amout of data that a be simultaeously reeived by a destiatio ode; (iv Iterfae-bottleek requiremet - the maximum amout of data that a iterfae a simultaeously trasmit or reeive Besides, eah of the four requiremets domiates the other three requiremets i terms of the throughput of the etwork uder differet oditios o ad H Our fidigs are sigifiatly differet from the previous studies i SC-AH etworks, MC-AH etworks ad SC-IS etworks, whih are limited by oly subsets of the four requiremets For example, the apaity of SC-AH etworks ad SC-IS etworks is limited by Coetivity requiremet ad Iterferee requiremet as show i [2] ad [0] while the apaity of MC-AH etworks is limited by Coetivity requiremet, Iterferee requiremet ad Iterfae-bottleek requiremet [8] As a result, our aalysis o a MC-IS etwork is far more hallegig tha those i the previous studies More speifially, as show i Fig 2, a be partitioed ito 3 ases: Case orrespodig to the ase whe = O(F, Case 2 orrespodig to the ase whe = Ω(F ad = O(F 2, ad Case 3 orrespodig to the ase whe = Ω(F 2, where F = log ad F 2 = ( loglog(h2 log log(h 2 log 2 Uder eah of the above ases, H a be partitioed ito two sub-ases Uder Case, H is partitioed ito 2 sub-ases, amely Sub-ase ad Sub-ase 2 Sub-ase is whe H = o(g ad Sub-ase 2 is whe H = Ω(G, where G = 3/log 2 3 Uder Case 2, H is partitioed ito 2 subases, amely Sub-ase 2 ad Sub-ase 22 Sub-ase 2 is whe H = o(g 2 ad Sub-ase 22 is whe H = Ω(G 2, where G 2 = 3 C 6 A /log 2 Uder Case 3, H is partitioed ito 2 sub-ases, amely Sub-ase 3 ad Sub-ase 32 Subase 3 is whe H = o(g 3 ad Sub-ase 32 is whe H = Ω(G 3, where G 3 = 2/log 2 Fig 2 shows all possible sub-ases we osider Eah requiremet domiates the other at least oe sub-ase uder differet oditios as follows Coetivity Coditio: orrespodig to Sub-ase 2 i whih Coetivity requiremet domiates Iterferee Coditio: orrespodig to Sub-ase 22 i whih Iterferee requiremet domiates Destiatio-bottleek Coditio: orrespodig to Subase 32 i whih Destiatio-bottleek requiremet domiates Iterfae-bottleek Coditio: orrespodig to Subase, Sub-ase 2, or Sub-ase 3, i whih Iterfae-bottleek requiremet domiates B Summary of Results e summarize the mai results as follows Throughput ad Delay for a MC-IS etwork Theorem : The per-ode throughput λ for a MC-IS etwork has four regios as follows i he Coetivity Coditio is satisfied, λ = Θ ( ( H log + Θ mi{ b, bm C I } I, where λa = Θ ( H log ad λi = Θ ( mi{ b, bm C I } I ; ii he ( Iterferee Coditio is satisfied, λ = Θ ( Θ C 2 A H log 2 C 2 A H log 2 + Θ(mi{ b, bm C I } I, where λ a = ad λ i = Θ ( mi{ b, bm C I } I ; iii he ( Destiatio-bottleek Coditio is satisfied, λ = 2 loglog(h Θ 2 log + Θ(mi{ b H log 2 log(h 2 log, bm C I } I, ( 2 loglog(h where λ a = Θ 2 log ad λ i = Θ ( mi{ b, bm C I } I ; H log 2 log(h 2 log

6 6 iv he ( Iterfae-bottleek Coditio is satisfied, λ = Θ H 2log A +Θ(mi{ b, bm C I } I, where λ a = ( Θ H 2log A ad λ i = Θ ( mi{ b, bm C I } I Theorem 2: The( average delay( of all pakets i a MC-IS etwork is D = Θ H 3 log +Θ mi{c I,m}, where D a = ( ( Θ H 3 log ad D i = Θ mi{c I,m} 2 Overview of Our Proof Sie ad ho ommuiatios ad ifrastruture ommuiatios are arried i two disjoit hael groups ad C I, we will derive the bouds o the apaity ad the delay otributed by the two ommuiatios separately I partiular, we first obtai the bouds o the the apaity otributed by ad ho ommuiatios i Setio V More speifially, we will derive the upper bouds o the apaity by osideratio of the aforemetioed four requiremets ad the prove the lower bouds by ostrutig the ells, desigig routig sheme ad TDMA sheme properly Although our approah is the itegratio of the previous studies o SC-IS etworks [4] ad MC-AH etworks [8], our solutio is o-trivial due to the followig reasos: (i the apaity of MC-IS etworks is limited by the aforemetioed four oditios simultaeously while those of SC-IS etworks ad MC-AH etworks are oly limited by subsets of the four oditios; (ii as a result, we eed to redesig the ell ostrutio, the routig sheme ad the shedulig sheme based o various fators (suh as H, ad, whih are ot straight-forward Details about our proof o ad ho ommuiatios will be give i Setio V e will ext derive the apaity otributed by ifrastruture ommuiatios i Setio VI Similarly, we eed to ostrut BS-ells, desig routig sheme ad TDMA sheme i this phrase while these ostrutios are differet from those of ad ho ommuiatios The omplete proof of Theorem ad Theorem 2 will be give i Setio VI 3 Geerality of MC-IS Networks Our proposed MC-IS etwork offers a more geeral theoretial framework tha other existig etworks I partiular, other etworks suh as a SC-AH etwork [2], a MC-AH etwork [8], ad a SC-IS etwork [4] a be regarded as speial ases of our MC-IS etwork uder the followig searios (A A SC-AH etwork is a speial ase of our MC-IS etwork: The theoretial bouds i the SC-AH etwork [2] are osistet with our bouds whe our ofiguratio is set to the oe for the SC-AH etwork Speifially, the ofiguratio is that H is set to Θ( /log, =, = ad I = 0 I that ofiguratio, the total badwidth is assiged for ad ho ommuiatios ( = ad I = 0, there is a sigle hael available ( = orrespodig to that of a SC-AH etwork [2] (B A MC-AH etwork is a speial ase of our MC-IS etwork: The theoretial bouds i the MC-AH etwork [8] are osistet with our bouds show i Theorem, whe our ofiguratio is set to the oe for the MC-AH etwork, i whih H is set to Θ( /log, orrespodig to that of a MC-AH etwork [8] I partiular, we have the followig ases: Case I: whe = O(log ad H = Θ( /log (Coetivity Coditio is satisfied, the per-ode throughput λ = Θ(/ log ad the average delay D = Θ( /log, whih mathes the result of a MC- AH etwork [8]; Case ( ( II: whe = Ω(log ad = O loglog(h 2 2 log log(h 2 log, ad H = Θ( /log (Iterferee Coditio is satisfied, the per-ode throughput λ = Θ(/ ad the average delay D = Θ( /log, whih mathes the result of a MC-AH etwork [8]; ( ( Case III: whe = Ω loglog(h 2 2 log ad H = log(h 2 log Θ( /log (Destiatio-bottleek Coditio is satisfied, the per-ode throughput λ = Θ( loglog log ad the average delay D = Θ( /log, whih mathes the result of a MC-AH etwork [8] Note that we do ot osider the apaity otributed by ifrastruture ommuiatios i the above four ases (C A SC-IS etwork is a speial ase of our MC-IS etwork: Similarly, the theoretial bouds i the SC-IS etwork [4] are osistet with our bouds whe our ofiguratio is set to the oe for the SC-IS etwork I partiular, we have the followig ases: Case I: whe = ad H = Ω( 3/log 2 3 (Coetivity Coditio is satisfied, λ = Θ( a H log + b i ad D = Θ( H3 log +, whih mathes the result of a SC-IS etwork [4]; Case II: whe = ad H = o( 3/log 2 3 (Iterfae-bottleek Coditio is satisfied, λ = Θ(H 2log a C a + mi{ b, bm C i } I ad D = Θ( H3 log +, whih mathes the result of a SC-IS etwork [4] 4 Optimality of Results e aalyze the optimality of the per-ode throughput apaity λ ad the average delay D of a MC-IS etwork Speifially, the aalysis is ategorized ito two ases: ( whe λ a domiates λ i ; (2 whe λ i domiates λ a Case : whe λ a domiates λ i (ie ad I / 0 e obtai the maximum per-ode throughput apaity as the followig sub-ases: (i λ = Θ ( ( H log with Coetivity oditio; (ii λ = Θ with C 2H log 2 ( 2 loglog(h Iterferee oditio; (iii λ = Θ 2 log CH log 2 log(h 2 log with Destiatio-bottleek oditio; (iv λ = Θ ( H 2 log C with Iterfae-bottleek oditio I all the above ( sub-ases, we always have the average delay D = Θ H 3 log The results imply that we should assig most of hael badwidth to ad ho ommuiatios i order to obtai the maximum apaity ad the miimum delay However, we show ext that the above results are ot optimal ompared with Case 2 Case 2: whe λ i domiates λ a (ie I /2 ad / 0 I this ase, the maximum per-ode throughput apaity λ = Θ( b ad the average delay D = Θ ( mi{c I,m} It implies that whe whe λi domiates λ a,

7 7 D( log mi( CI, m 2π CI θ A log B C ' C SC-AH MC-AH SC-IS MC-IS MC-IS-DA Fig 3 Capaity ad delay regios uder differet etworks The sales of the axes are i terms of the orders i λ( ateas oly to a MC-IS etwork equipped with diretioal ateas oly, whih are amed as a MC-IS-DA etwork, we a obtai a eve lower delay of as show i poit 2π θ CI C, where θ is the beamwidth of a diretioal atea mouted at the base statio (usually θ < 2π Cosider the same ase of C I = 2 ad θ = π 2 that is feasible i most of mmave systems [2] A MC-IS-DA a further redue the delay by 24 times lower that of a MC-IS etwork ad redue the delay by 288 times lower tha that of a SC-IS etwork Details o this exteded work will be addressed i Setio VII to maximize the apaity, most of the hael badwidth should be assiged for ifrastruture ommuiatios At this time, ireasig the umber of base statios a sigifiatly improve the etwork apaity More speifially, if b = Ω(, the λ = Θ(, whih is sigifiatly higher tha those i Case This is beause the multi-hop ad ho ommuiatios may lead to the apaity loss due to the higher iterferee of multiple ad ho ommuiatios Meawhile, the miimum average delay D i this ase is bouded by Θ ( mi{c I,m} mi{c I,m}, where is a ostat ad is idepedet of ( It is obvious that mi{c = o I,m} Θ ( H 3 log sie H is determied by the umber of odes Ituitively, we have muh lower delay tha that of Case The reaso behid this lies i the higher delay brought by the multi-hop ommuiatios i Case I summary, MC-IS etworks have the optimal per-ode throughput apaity λ opt = Θ( ad mi{c I,m} the optimal average delay D opt = Θ( e ompare our results with other etworks (a MC-AH etwork, a SC-IS etwork, ad a SC-AH etwork i terms of the optimal per-ode throughput λ ad the optimal average delay D As show i Fig 3, a MC-IS etwork a ahieve the optimal per-ode throughput λ opt = Θ( (poit C i Fig 3, whih is log times higher tha that of a MC- AH etwork ad a SC-AH etwork (poit A i Fig 3, ad the same as that of a SC-IS etwork (poit B i Fig 3, implyig that there is o degradatio i the optimal per-ode throughput of a MC-IS etwork Besides, a MC-IS etwork a ahieve the smallest delay Θ ( mi{c I,m} (poit C i Fig 3 whe the optimal perode throughput apaity λ = Θ( is ahieved It is show i [3] that i a SC-AH etwork ad a MC-AH etwork, the ireased apaity pays for the higher delay due to the multi-hop trasmissios However, a MC-IS etwork ad a SC-IS etwork a overome the delay pealty by trasmittig pakets through ifrastruture, iside whih there is o delay ostrait Furthermore, a MC-IS etwork a ahieve a eve shorter delay tha a SC-IS etwork by usig multiple NICs at eah base statio, whih a support multiple simultaeous trasmissios Speifially, as show i Fig 3, a MC-IS etwork (poit C has a delay redutio gai of mi{c I,m} over a SC-IS etwork (poit B For example, a MC-IS etwork with C I = m = 2, i whih we assig a dediated iterfae for eah hael, has a delay 2 times lower tha a SC-IS etwork Besides, whe we exted our aalysis o a MC-IS etwork equipped with omi-diretioal V CAPACITY CONTRIBUTED BY AD HOC COMMUNICATIONS e first derive the upper bouds o the etwork apaity otributed by ad ho ommuiatios i Setio V-A ad the preset the ostrutive lower bouds o the etwork apaity otributed by ad ho ommuiatios i Setio V-B, whih have the same order of the upper bouds, implyig that our results are tight eext give the aggregate throughput apaity i Setio V-C A Upper Bouds o Network Capaity Cotributed by Ad Ho Commuiatios The etwork apaity otributed by ad ho trasmissios i a MC-IS etwork, deoted by λ a, is maily affeted by ( Coetivity requiremet, (2 Iterferee requiremet, (3 Destiatio-bottleek requiremet ad (4 Iterfaebottleek requiremet e first derive the upper bouds o the per-ode throughput apaity uder Coetivity Coditio Before presetig Propositio, we have Lemma to boud the expetatio of the umber of hops deoted by h Lemma : The expetatio of the umber of hops h is bouded by Θ(H Proof e first deote P(h = i by the probability of the evet that a paket traverses h = i hops Aordig to the H-max-hop routig sheme, P(h = i is essetially equal to the probability that a paket traverses at most h = i hops with the exlusio of the evet that a paket traverses o more tha h = i hops, where i > 0 Thus, P(h = i is equal to the ratio of the area of a disk with radius (i r( to the area of a disk with radius i r(, where r( is the distae of a hop As a result, P(h = i = (i2 (i 2 πr 2 ( πi 2 r 2 ( e the have h = E(h = H i= i (i 2 (i 2 πr 2 ( πh 2 r 2 ( Sie i[i 2 (i 2 ] i Eq ( are the series of hexagoal umbers, the Eq ( a be simplified as follows h = ( 6H(H +(4H H 2 = 4H3 +3H 2 H 6H 2 (2 It is obvious that h is a futio of H as show i Eq (2 The limit of h(h as H approahes is lim H h(h = Θ(H, whih a be diretly derived from the defiitio of the asymptoti otatio Θ( ad Eq (2

8 8 e the have Propositio that bouds the per-ode throughput apaity otributed by ad ho ommuiatios uder Coetivity oditio, Propositio : he Coetivity requiremet domiates, the per-ode throughput apaity otributed by ad ho ommuiatios is λ a = O ( H 3 log 2 Proof e first alulate the probability that a ode uses the ad ho mode to trasmit, deoted by P(AH, whih is the probability that the destiatio ode is loated withi H hops away from the soure ode Thus, we have P(AH = πh 2 r 2 ( Sie eah soure geerates λ a bits per seod ad there are totally soures, the total umber of bits per seod served by the whole etwork is required to be at least P(AH h λ a e ext prove that P(AH h λ a k is bouded by 2 (r( 2 A Deote the maximum umber of simultaeous trasmissios o a partiular hael by N max As proved i Lemma 54 i [2], N max is upper bouded k by 2 (r(, where k 2 > 0 is a ostat, idepedet of Note that eah trasmissio over the hael is of / bits/se Addig all the trasmissios takig plae at the same time over all the haels, we have that the total umber of trasmissios i the whole etwork is o more k tha CA k 2 (r( 2 = = 2 (r( 2 A bits/se Therefore, k we have P(AH h λ a 2 (r( 2 A Combiig the above results with Lemma yields λ a k 2 r 2 ( πh 3 r 2 ( k2a H 3 r 2 (, where k 2 is a ostat Besides, to guaratee that the etwork is oeted with high probability (whp, we require r( > log π [2] Thus, we have λ a k3a H 3 log 2, where k 3 is a ostat e the derive the upper bouds o the per-ode throughput apaity uder Iterferee Coditio Propositio 2: he Iterferee requiremet domiates, the per-ode throughput ( apaity otributed by ad ho ommuiatios is λ a = O C 2 A H3 log 3 2 Proof e preset a proof of the boud i Appedix A Before provig the upper bouds o the throughput apaity uder Destiatio-Bottleek oditio, we have Lemma 2 to boud the umber of flows towards a ode uder the H-maxhop routig sheme Lemma 2: The maximum umber of flows towards a ode uder the H-max-hop routig sheme is D H ( = ( Θ log(h 2 log loglog(h 2 log whp Proof Let N i ( i be a radom variable defied as follows, { if soure ode i trasmits to its destiatio ode; N i = 0 otherwise Let N t be a radom variable represetig the total umber of soure odes trasmittig i ad ho mode e have N t = i= N i Thus, the expeted umber of ( soure odes trasmittig i ad ho mode is E(N t = E i= N i = i= E(N i Sie f(n i = = P(AH = πh 2 r 2 ( ad r( eeds to be Θ( log/ to esure that the etwork is oeted, we have E(N i = πh 2 r 2 ( + 0 ( πh 2 r 2 ( = πh 2 r 2 (, ie, E(N i = Θ(πH 2 log Therefore, E(N t = πh 2 log = πh2 log Reall the Cheroff bouds [35], we have for ay δ > 0, P(N t > ( + δπh 2 log < ( πh e δ (+δ (+δ 2 log ; for ay 0 < δ <, P(N t < ( δπh 2 log < e πh2 log δ 2 /2 I summary, for ay 0 < δ <, we a obtai P( N t πh 2 log > δπh 2 log < e επh2 log, where ε > 0 Thus, whe, the total umber of soure odes trasmittig i ad ho mode is Θ(H 2 log whp Besides, it is proved i [36] that the maximum umber of flows towards ay give ode i a radom etwork with N odes, deoted byd(n, is upper bouded byθ ( logn loglogn, whp Combiig the two results leads to the above result e the prove the upper bouds o the per-ode throughput apaity uder Destiatio-bottleek Coditio Propositio 3: he Destiatio-bottleek requiremet domiates, the per-ode throughput ( apaity otributed by ad 3 2 loglog(h ho ommuiatios is λ a = O 2 log H 3 log 3 2 log(h 2 log Proof Sie eah ode has oe iterfae that a support at most A ad Sie eah ode has at most D H ( flows uder the H-max-hop routig sheme, the data rate of the miimum rate flow is at most A C, where D AD H( H( is bouded by ( Θ log(h 2 log loglog(h 2 log by Lemma 2 After alulatig all the data rates at eah ode times with the traversig distae, we have P(AH λ a h r( A e the have λ a πh 3 r 3 log(h ( 2 log log log(h 2 log D H( D H(P(AHhr( This is beause h = Θ(H ad P(AH = πh 2 r 2 ( are derived i Lemma ad ( i the log proof of Propositio, respetively Sie r( = Θ as proved i [2], we the prove the result Fially, we prove the upper bouds o the per-ode throughput apaity uder Iterfae-bottleek Coditio Propositio 4: he Iterfae-bottleek requiremet domiates, the per-ode throughput apaity otributed by ad ho ommuiatios is λ a = O( A Proof I a MC-IS etwork, eah ode is equipped with oly oe iterfae, whih a support at most A data rate Thus, λ a is also upper bouded by A Note that this result holds for ay etwork settigs B Costrutive Lower Bouds o Network Capaity Cotributed by Ad Ho Commuiatios e the derive the lower boud o the etwork apaity by ostrutig a etwork with the orrespodig routig sheme ad shedulig sheme whe eah requiremet is osidered The derived orders of the lower bouds are the same as the orders of the upper bouds, meaig that the upper bouds are tight I partiular, we first divide the plae ito a umber of equal-sized ells The size of eah ell is properly hose so that eah ell has Θ(a( odes, where a( is the area of a ell (Se V-B e the desig a routig sheme to assig

9 9 a ell a( Fig 4 Plae divided ito a umber of ells Fig 5 oe seod 2 3 CA- CA mii-slot edge-olor slot TDMA trasmissio shedule the umber of flows at eah ode evely (Se V-B2 Fially, we desig a Time Divisio Multiple Aess (TDMA sheme to shedule the traffi at eah ode (Se V-B3 Cell Costrutio: e divide the plae ito /a( equal-sized ells ad eah ell is a square with area of a(, as show i Fig 4 The ell size of a( must be arefully hose to fulfill the three requiremets, ie, the oetivity requiremet, the iterferee requiremet ad the destiatiobottleek requiremet { I partiular, similar to [8], we set a( = mi { max 00log, log3 2 C 2 A Time }, log 3 2 log(h 2 log 3 2 loglog(h 2 log Note that the iterfae-bottleek requiremet is idepedet of the size of a ell The maximum umber of odes i a ell a be upper bouded by the followig lemma Lemma 3: If a( > 50log, the eah ell has Θ((a( odes whp Proof Please refer to [8] e ext hek whether all the above values of a( are properly hose suh that eah ell has Θ((a( odes whp whe is large eough (ie, Lemma 3 is satisfied It is obvious that 00log > 50log ad log 3 2 } > 50log C 2 A sie we oly osider i Coetivity Coditio ad Iterferee Coditio Besides, greater tha 50log > 50log with large sie log 3 2 log(h 2 log 3 2 loglog(h 2 log is also log(h 2 log loglog(h 2 log > ad log 3 2 whe is large eough 2 3 Besides, the umber of iterferig ells aroud a ell is bouded by a ostat, give by Lemma 4 as follows Lemma 4: Uder the iterferee model, the umber of iterferig ells of ay give ell is bouded by a ostat k 5, whih is idepedet of Proof The detailed proof is stated i Appedix B 2 Routig Sheme: To assig the flows at eah ode evely, we desig a routig sheme osists of two steps: ( Assigig soures ad destiatios ad (2 Assigig the remaiig flows i a balaed way I Step (, eah ode is the origiator of a flow ad eah ode is the destiatio of at most D H ( flows, where D H ( is defied i Lemma 2 Thus, after Step (, there are at most + D H ( flows e deote the straight lie oetig a soure S to its destiatio D as a S-D lies I Step (2, we eed to alulate the umber of S-D lies (flows passig through a ell so that we a assig them to eah ode evely Speifially, we have the followig result Lemma 5: The umber of S-D lies passig through a ell is bouded by O(H 3 (a( 2 Proof The detailed proof is stated i Appedix C As show i Lemma 3, there are Θ( a( odes i eah ( ell Therefore, Step (2 will assig to ay ode at most O H 3 (a( 2 = O(H 3 a( flows Summarizig Step ( a( ad Step (2, there are at most f( = O( + H 3 a( + D H ( flows at eah ode O the other had, H 3 a( domiates f( sie H > ad a( is asymptotially larger tha D H ( whe is large eough Thus, we have f( = O(H 3 a( 3 Shedulig Trasmissios: e ext desig a shedulig sheme to trasmit the traffi flows assiged i a routig sheme Ay trasmissios i this etwork must satisfy the two additioal ostraits simultaeously: eah iterfae oly allows oe trasmissio/reeptio at the same time, ad 2 ay two trasmissios o ay hael should ot iterfere with eah other e propose a TDMA sheme to shedule trasmissios that satisfy the above two ostraits Fig 5 depits a shedule of trasmissios o the etwork I this sheme, oe seod is divided ito a umber of edge-olor slots ad at most oe trasmissio/reeptio is sheduled at every ode durig eah edge-olor slot Hee, the first ostrait is satisfied Eah edge-olor slot a be further split ito smaller mii-slots I eah mii-slot, eah trasmissio satisfies the above two ostraits Details are desribed as follows (i Edge-olor slot: First, we ostrut a routig graph i whih verties are the odes i the etwork ad a edge deotes trasmissio/reeptio of a ode I this ostrutio, oe hop alog a flow is assoiated with oe edge i the routig graph I the routig graph, eah vertex is assiged with f( = O(H 3 a( edges It is show i [8], [37] that this routig graph a be edge-olored with at most O(H 3 a( olors e the divide oe seod ito O(H 3 a( edge-olor slots, eah of whih has a legth of Ω( H 3 a( seods ad is staied with a uique edge-olor Sie all edges oetig to a vertex use differet olors, eah ode has at most oe trasmissio/reeptio sheduled i ay edge-olor time slot (ii Mii-slot: e further divide eah edge-olor slot ito mii-slots The, we build a shedule that assigs a trasmissio to a ode i a mii-slot withi a edge-olor slot over a hael e ostrut a iterferee graph i whih eah vertex is a ode i the etwork ad eah edge deotes the iterferee betwee two odes e the show as follows that the iterferee graph a be vertex-olored with k 7 (a( olors, where k 7 is a ostat defied i [8] Lemma 6: The iterferee graph a be vertex-olored with at most O(a( olors Proof By Lemma 4, every ell has at most a ostat umber of iterferig ells Besides, eah ell has Θ(a( odes by Lemma 3 Thus, eah ode has at most O(a( edges i the iterferee graph It is show that a graph of degree at most k 0 a be vertex-olored with at most k 0 + olors [8] [37] Hee, the iterferee graph a be vertex-olored with at most O(a( olors e eed to shedule the iterferig odes either o differet haels, or at differet mii-slots o the same hael sie two odes assiged the same vertex-olor do ot iterfere with eah other, while two odes staied with differet olors may

10 0 iterfere with eah other e divide eah edge-olor slot ito k7a( mii-slots o every hael, ad assig the miislots o eah hael from to k7a( A ode assiged with a olor s, s k 7 a(, is allowed to trasmit i s mii-slot o hael (s mod + e ext prove the ostrutive lower bouds of the apaity Propositio 5: The ahievable per-ode throughput apaity λ a otributed by ad ho ommuiatios is as follows he Coetivity requiremet domiates, λ a is Ω( A bits/se; H 3 log 2 2 he ( Iterferee requiremet domiates, λ a is Ω A bits/se; H 3 C 2 A log he ( Destiatio-bottleek requiremet domiates,λ a 3 2 loglog(h is Ω 2 log bits/se; H 3 log 3 2 log(h 2 log 4 he Iterfae-bottleek requiremet domiates, λ a is Ω ( Proof Sie eah edge-olor slot with a legth of Ω ( k7a( H 3 a( seods is divided ito mii-slots over every hael, eah mii-slot has a legth of Ω( ( H 3 a( / k7a( seods Besides, eah hael a trasmit ( at the rate of bits/se, i eah mii-slot, λ a = Ω A H 3 k7 a( a( C A bits a be trasported Sie k7a( k7a( +, we ( have λ a = Ω A k 7H 3 a 2 (+H 3 a( bits/se Thus, λ a = ( Ω (MIN A O H 3 a 2 (, H 3 a( bits/se Reall that a( { { } } is mi max 00log, log3 2, log 2 3 log(h 2 log Substitutig the three values to λ a, we have the results, 2 ad 3 C 2 A 3 2 loglog(h 2 log Besides, eah iterfae a support the rate of A bits/se Thus, λ a = Ω (, whih is the result 4 C Aggregate Throughput Capaity It is show i [4] that the total traffi of ad ho ommuiatios is πh 2 r 2 (λ a Combiig Propositios, 2, 3, ad 5 with the total traffi leads to the followig theorem Theorem 3: The aggregate throughput apaity of the etwork otributed by ad ho ommuiatios is he Coetivity requiremet domiates, T A is Θ( A H log bits/se 2 he Iterferee requiremet domiates, T A is Θ( A bits/se C 2 A H log 2 3 he Destiatio-bottleek requiremet domiates, T A is Θ( 3 2 loglog(h 2 log H log 2 log(h 2 log bits/se 4 he Iterfae-bottleek requiremet domiates,t A is Θ(H 2 log A bits/se VI CAPACITY CONTRIBUTED BY INFRASTRUCTURE COMMUNICATIONS e first derive the upper bouds of the apaity i Setio VI-A ad give the ostrutive lower bouds of the apaity i Setio VI-B e give the aggregate apaity otributed by ifrastruture ommuiatios i Setio VI-C Fially, Setio VI-D gives the proof of Theorem ad Theorem 2 A Upper Bouds of Network Capaity Cotributed by Ifrastruture Commuiatios e derive the upper bouds of the throughput apaity otributed by ifrastruture ommuiatios as follows Propositio 6: Uder the H-max-hop routig sheme, the throughput apaity otributed by ifrastruture ommuiatios, deoted by T I, is: ( he C I m, T I = O(b I (2 he C I > m, T I = O(b m C I I Proof Sie eah paket trasmitted i the ifrastruture mode will use both the uplik ad the dowlik ommuiatios, we oly out oe for the throughput apaity Case ( whec I m It is obvious that themiterfaes at eah base statio a support at most I badwidth I other words, the C I haels are fully utilized by the m iterfaes Coutig all the b base statios, we have T I = O(b I Case (2 whe C I > m The umber of iterfaes is smaller tha the umber of haels, implyig that ot all the C I haels are fully used I fat, at most m haels a be used at a time Besides, eah hael a support at most I C I bits/se Thus, eah base statio a support at most m C I I bits/se Coutig all the b base statios, we have T I = O(b m C I I B Costrutive Lower Bouds of Network Capaity Cotributed by Ifrastruture Trasmissios The lower bouds are proved by ostrutig a routig sheme ad a trasmissio shedulig sheme o a regulartessellated BS etwork The derived orders of the lower bouds are the same as the orders of the upper bouds, implyig that the upper bouds are tight BS-Cell Costrutio by Regular Tessellatio: There are b base statios regularly plaed i the plae dividig the plae ito a umber of equal-sized BS-ells Note that the size of eah BS-ell may ot be eessarily equal to the size of a ell Besides, Lemma 4 still holds eve if the base statios are regularly plaed i the plae So, the umber of iterferig BS-ells is also bouded by a ostat, deoted by k 8, whih is also idepedet of b 2 Routig ad Shedulig Shemes: The routig sheme for the ifrastruture traffi is simple, ie, to forward the traffi to a base statio (uplik ad to forward the traffi from a base statio (dowlik e propose the followig TDMA shedulig sheme Σ to shedule the BS-ells to be ative i a roud-robi fashio ( Divide the plae ito b equal-sized BS-ells (2 e group the b BS-ells ito a umber of lusters Eah luster has (k 8 + BS-ells e the split the trasmissio time ito a umber of time frames Eah frame osists of (k 8 + time slots that orrespod to the umber of BS-ells i eah luster I eah time slot, oe BS-ell withi eah luster beomes ative to trasmit ad the BS-ells i eah luster take turs to be ative

11 Propositio 7: Uder the TDMA sheme Σ, the throughput apaity T I, is: ( he C I m, T I = Ω(b I (2 he C I > m, T I = Ω(b m C I I Proof Sie eah paket trasmitted i the ifrastruture mode will use both the uplik ad the dowlik, we oly out oe for throughput apaity Case ( whe C I m: Uder Σ, eah BS-ell is ative every (k 8 + time slots he a BS-ell is ative, there are at most C I haels available Thus, the total badwidth of I of those C I haels are fully used, implyig that the per-ell throughput λ i is lower bouded by I k 8+ Coutig all the b base statios, we have T I = Ω( bi k 8+ Case (2 whe C I > m: Similarly, eah BS-ell is ative to trasmit every (k 8 + time slots i ase (2 But, whe a BS-ell is ative, oly m haels available at a time ad eah hael a support at most I C I data rate Thus, the per-ell throughput λ i is lower bouded m I C I(k 8+ base statios, we have T I = Ω( bmi C I(k 8+ C Aggregate Throughput Capaity Coutig all the b Combiig Propositio 6 ad Propositio 7, we have Theorem 4: The aggregate throughput apaity of the etwork otributed by ifrastruture ommuiatios is ( he C I m, T I = Θ(b I (2 he C I > m, T I = Θ(b m C I I It is show i Theorem 4 that the optimal throughput apaity otributed by ifrastruture ommuiatios T I = Θ(b I is ahieved whe C I m Geerally, we have C I = m If C I m, some iterfaes are idle ad wasted It implies that to maximize T I, we shall assig a dediated iterfae per hael at eah base statio so that all the C I haels a be fully utilized D Proof of Theorem ad Theorem 2 e fially give the proof of Theorem as follows Proof of Theorem e first have the aggregate throughput apaity T = T A + T I, where T A is the aggregate apaity otributed by ad ho ommuiatios ad T I is the aggregate apaity otributed by ifrastruture ommuiatios give by give by Theorem 3 ad Theorem 4, respetively Sie there are at most odes i the etwork, we the divide T by ad fially have the results i Theorem This ompletes the proof e the derive the average delay of a MC-IS etwork otributed by ad ho ommuiatios ad ifrastruture ommuiatios as follows Proof of Theorem 2 e first derive the boud o the delay whe the pakets are trasmitted i the ifrastruture mode As show i [4], the average delay for the pakets trasmitted i the ifrastruture mode i a SC-IS etwork is bouded by Θ(, where is a ostat depedig o the trasmittig apability of the base statio Differet from a SC-IS etwork, where eah base statio is equipped with a sigle iterfae supportig at most oe trasmissio at a time, eah base statio i a MC-IS etwork a support mi{c I,m} simultaeous trasmissios at a time This is beause whec I m, a base statio with m iterfaes a support at most C I simultaeous trasmissios; whe C I > m, a base statio with m iterfaes a support at most m simultaeous trasmissios Thus, the average delay for the pakets trasmitted i the ifrastruture mode i a mi{c I,m} MC-IS etwork is bouded by Θ( e the derive the boud o the delay whe the pakets are trasmitted i ad ho mode The expetatio of h uder H-max-hop routig strategy is bouded by Θ(H as proved by Lemma Sie the time spet by a paket at eah relay is bouded by, the average delay is of the same order as the average umber of hops, ie, D = h = Θ(H It is show i the proof of Lemma 2 that the umber of trasmitters i the ad ho mode is πh 2 log whp The the umber of trasmitters i the ifrastruture mode is ( πh 2 log whp After applyig the above aalysis, we have the average delay ( πh 2 log H+( πh 2 log mi{c I,m} πh 2 log of all pakets D = Θ Note that Theorem 2 is bouded by Θ( Thus, we have VII DISCUSSIONS AND IMPLICATIONS I this setio, we first exted our aalysis to the searios of usig diretioal ateas i MC-IS etworks i Setio VII-A e the disuss the impats of mobility models i Setio VII-B Fially, we preset the impliatios of our MC- IS etworks i Setio VII-C A Usig Diretioal Ateas i MC-IS etworks Covetioal wireless etworks assume that eah ode is equipped with a omi-diretioal atea, whih radiates sigals i all diretios iludig some udesired diretios Reet studies suh as [38], [39] show that applyig diretioal ateas istead of omi-diretioal ateas to wireless etworks a greatly improve the etwork apaity The performae improvemet maily owes to the redutio i the iterferee from udesired diretios sie diretioal ateas oetrate radio sigals o the desired diretios Although diretioal ateas have umerous advatages, the bulky size ad the impats of diretioality also restrit the appliatio of diretioal ateas to wireless etworks However, with the evolutio of wireless ommuiatio tehologies, these hallegig issues will fially be solved I fat, a diretioal atea has beome a eessity i order to ompesate for the tremedous sigal atteuatio i millimeter-wave (mmave ommuiatio systems [40] It is feasible to deploy diretioal ateas at both base statios ad mobile devies i mmave ommuiatio systems sie their size will be quite ompat due to the fat that the atea size is iversely proportioal to the radio frequey (the frequey bad is ragig from 30GHz to 300GHz i mmave ommuiatio systems [4] e exted our aalysis o a MC-IS etwork with omidiretioal ateas (i the previous part of this paper to that with diretioal ateas I partiular, we ame a MC-IS etwork equipped with diretioal ateas as a MC-IS-DA

12 2 Fig 6 Base statio φ Commo ode setors X4 X5 X3 B X X2 Ifrastruture diretioal liks Ad ho diretioal liks Network topology of a MC-IS-DA etwork i a BS-ell etwork Fig 6 shows a example of MC-IS-DA etworks, i whih eah base statio is equipped with multiple diretioal ateas ad eah ommo ode is equipped with a sigle diretioal atea Similar to a MC-IS etwork, there are two types of ommuiatios i a MC-IS-DA etwork: ad ho ommuiatios betwee ommo odes ad ifrastruture ommuiatios betwee a ommo ode ad a base statio Differetly, both ad ho ommuiatios ad ifrastruture ommuiatios i a MC-IS-DA etwork osist of diretioal ommuiatio liks oly I this paper, we osider a flat-top atea model [9], [38], [42], i whih sidelobes ad baklobes are igored Our atea model assumes that a diretioal atea gai is withi a speifi agle, ie, the beamwidth of the atea, whih is ragig from 0 to π The gai outside the beamwidth is assumed to be zero I our MC-IS etwork, eah ommo ode is mouted with a sigle iterfae, whih is equipped with a diretioal atea with beamwidth φ Eah base statio is mouted with m iterfaes, eah of whih is equipped with a diretioal atea with beamwidth θ, where eah diretioal atea at eah base statio is idetial Note that the beamwidth φ of a atea at a ommo ode is ot eessarily equal to the beamwidth θ of that at a base statio Capaity of a MC-IS-DA etwork The apaity of a MC-IS-DA etwork otributed by ifrastruture ommuiatios is the same as that of a MC- IS etwork However, a MC-IS-DA etwork has differet apaity regios o the per-ode throughput apaity λ a from a MC-IS etwork Corollary : The per-ode throughput λ for a MC-IS-DA etwork has four regios as follows i he Coetivity Coditio is satisfied, λ = Θ ( 4π 2 φ 2 ( H log + Θ mi{ b, bm C I } I, where λa = Θ ( 4π 2 φ 2 H log ad λi = Θ ( mi{ b, bm C I } I ; ( 2π ii he Iterferee Coditio is satisfied, λ = Θ φ + Θ(mi{ b C 2 A H log, bm C I } I, where λ a = 2 ( 2π Θ φ ad λ i = Θ ( mi{ b, bm C I } I ; C 2 A H log 2 iii he ( Destiatio-bottleek Coditio is satisfied, λ = 2 loglog(h Θ 2 log + Θ(mi{ b H log 2 log(h 2 log, bm C I } I, ( 2 loglog(h where λ a = Θ 2 log ad H log 2 log(h 2 log λ i = Θ ( mi{ b, bm C I } I ; iv he ( Iterfae-bottleek Coditio is satisfied, λ = Θ H 2log A +Θ(mi{ b, bm C I } I, where λ a = ( Θ H 2log A ad λ i = Θ ( mi{ b, bm C I } I Proof The detailed proof is preseted i [43] As show i Corollary, a MC-IS-DA etwork has four apaity regios similar to a MC-IS etwork However, ompared with a MC-IS etwork, a MC-IS-DA etwork has the higher throughput apaity tha a MC-IS etwork whe Coetivity requiremet ad Iterferee requiremet domiate I partiular, whe Coetivity Coditio is satisfied, a MC-IS-DA etwork has a apaity gai 4π2 φ over a MC-IS 2 etwork he Iterferee Coditio is satisfied, a MC-IS- DA etwork has a apaity gai 2π φ over a MC-IS etwork This result implies that usig diretioal ateas i a MC- IS etwork a sigifiatly improve the apaity otributed by ad ho ommuiatios The apaity improvemet may owe to the improved etwork oetivity ad the redued iterferee Oe thig to ote that the apaity of MC-IS-DA etwork otributed by ifrastruture ommuiatios λ i is the same as that of a MC-IS etwork, implyig that usig diretioal ateas at base statios will ot improve the apaity However, our followig aalysis will prove that usig diretioal ateas at base statios a sigifiatly redue the delay otributed by ifrastruture ommuiatios 2 Delay of a MC-IS-DA etwork Reall i Setio VI-C that C I m so that the maximum throughput apaity otributed by ifrastruture ommuiatios a be ahieved e usually have C I = m so that there is o waste of iterfaes, implyig that we shall assig a dediated iterfae per hael at eah base statio so that all the C I haels a be fully utilized However, as the radio spetrum is beomig more ogested ad sare [44], it is extravagat ad impratial to let C I = m Thus, we exted our aalysis to the ase with C I < m e first equally divide m ateas ito κ groups, eah of ( H 3 log, 2π θ CI whih has m κ ateas (m is assumed to be divisible by κ though this aalysis a be easily exteded to the ase that m is ot divisible by κ ithi eah group, the m κ ateas are poited to the same diretio so that their beams over eah other, as show i Fig 6 e ame eah group of ateas as a setor It is obvious that eah setor will over θ There is o overlappig betwee ay two adjaet setors Therefore, there is o oflit betwee ay trasmissios from two adjaet setors The oflit oly happes betwee the ateas withi the same setor To avoid oflits, we a assig C I haels to the oflitig trasmissios withi the same setor I a MC-IS-DA etwork, eah base statio with multiple diretioal ateas a support more simultaeous trasmissios tha that of a typial MC-IS etwork Ituitively, a MC-IS-DA etwork a have a better performae tha a typial MC-IS etwork I partiular, we have the followig result Corollary 2: The average delay of all pakets ( i a exteded MC-IS etwork is D = Θ + Θ ( ( where D a = Θ H 3 log ad D i = Θ 2π θ Proof The detailed proof is preseted i [43] CI It is show i Corollary 2 that usig diretioal ateas at base statios i a MC-IS etwork a further redue the average delay otributed by ifrastruture ommuiatios D i i the ase C I < m sie obviously 2π θ C I > C I Besides, Corollary 2 also shows that the arrower atea

13 3 beamwidthθ is, the lower average delay D i is This result also implies that usig diretioal ateas i a MC-IS etwork a sigifiatly improve the spetrum reuse For example, suppose that we oly have oly oe hael available, ie C I =, whih a oly be used by oe omi-diretioal atea i a MC-IS etwork However, i a MC-IS-DA etwork where eah base statio is equipped with 2 diretioal ateas eah with beamwidth π 6 (ie, 30, this sigle hael a be simultaeously used by 2 ateas B Impats of Mobility Multi-hop ad short-raged ad ho ommuiatios ievitably result i the low throughput ad the high delay due to the iterferee amog multiple ourret trasmissios ad the time spet o multi-hop relays As show i [45], to allow a mobile ode to serve as the relay betwee the soure ad the destiatio a greatly redue the iterferee ad osequetly lead to the higher throughput tha the etwork without mobile relays I MC-IS etworks, we a also employ mobile odes to serve as the relays similar to [45] Note that the mobility a oly be applied to ommo odes istead of base statios sie all the base statios are oeted through a wired etwork ad they are usually fixed he there is the similar assumptio o the mobile model (ie radom walk to [45], we shall be able to derive the higher throughput apaity otributed by ad ho ommuiatios, whih shall be bouded by Θ( as suggested i [45] I additio to radom walk model, more realisti mobility models, suh as radom way-poit model [46] ad Browia motio model [47] a also be used i our MC-IS etworks It is ot the fous of our paper to osider mobility i our MC-IS etworks due to the followig reasos: ( most of existig mobility models a be diretly used i ad ho ommuiatios i our MC-IS etworks, whih basially have the similar features to ovetioal ad ho etworks; (2 itroduig mobile relay odes to the etwork also brigs the higher delay o matter whih mobility model is used, as idiated i [3], [47] This is beause it always takes a log time for relay odes to move from the soure to the destiatio C Impliatios of our results The peetratio of wireless ommuiatios with mobile itelliget tehologies is sigifiatly hagig our daily lives It arises a diversity of salable smart ommuiatio systems, eg, wireless sesor etworks (SNs, smart grid ad smart home [9], [20] The smart ommuiatio systems require smart devies (smart-phoes, smart appliaes, sesors, robots, surveillae devies oeted together Due to the heterogeeity of devies ad appliatios, heterogeeous traffis are geerated Take the smart grid as a example It may require the arrower badwidth to trasmit power osumptio iformatio from smart meters to the operatio eter tha that to trasmit surveillae videos The heterogeeity of the etwork performae requiremets of various appliatios leads to the ew researh halleges i this area [48], eg, how to improve the throughput apaity by offloadig the traffi at base statios Our MC-IS etworks provide a solutio to the above raised halleges he there are a large umber of low-volume traffis, eg, trasmittig moitored temperature iformatio from sesors to siks i a SN, we eed to let ad ho ommuiatios domiate, ieλ a domiates λ i, as implied from our results O the other had, whe there are high-volume traffis, suh as trasmittig images or surveillae videos obtaied from autoomous ameras to the otrollig eter of a smart grid, we eed to let ifrastruture ommuiatios domiate, ie λ i domiates λ a he there are some hybrid traffis of high-volume data ad low-volume data, we eed to assig ad ho ommuiatios ad ifrastruture ommuiatios proportioally There is a iterestig questio: how to assig the traffis to either ifrastruture ommuiatios or ad ho ommuiatios aordig to differet badwidth requiremets of various appliatios Devie-to-Devie (D2D ommuiatios have reetly attrated great attetios sie this tehology a offload the etwork traffi, improve the spetrum reuse ad irease the throughput apaity [8], [49] However, there are a umber of halleges i D2D etworks, suh as the iterferee maagemet, relay maagemet ad the spetrum alloatio D2D etworks have the ommo features of our MC-IS etworks - there are two kids of ommuiatios i a D2D etwork: (i D2D ommuiatios betwee devies (similar to ad ho ommuiatios i our MC-IS etworks ad (ii ellular ommuiatios betwee devies ad base statios (similar to ifrastruture ommuiatios i our MC-IS etworks Thus, our theoretial aalysis o MC-IS etworks a be used to aalyze the performae of D2D etworks For example, we a alloate haels for multi-hop D2D ommuiatios ad alloate C I haels for ellular ommuiatios i D2D etworks The throughput ad the delay of D2D etworks shall have the same bouds as our MC-IS etworks Meawhile, our proposed H-max-hop routig sheme a be applied to D2D etworks to solve the relay (routig issues with multi-hop D2D ommuiatios [50], [5] sie it is more pratial tha ovetioal ad ho routig shemes, whih ofte traverse the whole etwork while our H-max-hop routig sheme a loalize the ommuiatios withi H hops VIII CONCLUSION I this paper, we propose a ovel MC-IS etwork e derive the upper bouds ad lower bouds o the apaity of a MC- IS etwork Besides, we fid that a MC-IS etwork has a higher optimal apaity ad the lower average delay tha a MC-AH etwork ad a SC-AH etwork I additio, we show that a MC-IS etwork has the same optimal apaity as a SC-IS etwork while maitaiig a lower average delay tha a SC-IS etwork Moreover, sie eah ommo ode i a MC-IS etwork is equipped with a sigle iterfae oly, we do ot eed to make too may hages to ovetioal ad ho etworks while obtaiig high performae e exted our aalysis o a MC-IS etwork equipped with omi-diretioal ateas oly to a MC-IS etwork equipped with diretioal ateas oly, whih are amed as a MC-IS-DA etwork e show that a MC-IS-DA etwork has a eve lower delay of ompared with a SC-IS etwork ad our MC-IS 2π θ etwork CI

14 4 APPENDIX A Proof of Propositio 2 Let the average distae betwee a soure ad a destiatio be l, whih is roughly bouded by h r( I the etwork with odes ad uder the H-max-hop routig sheme, there are at most P(AH, where P(AH is the probability that a ode trasmits i ad ho mode ithi ay time period, we osider a bit b, b λp(ah e assume that bit b traverses h(b hops o the path from the soure to the destiatio, where the h-th hop traverses a distae of r(b, h It is obvious that the distae traversed by a bit from the soure to the destiatio is o less tha the legth of the lie joitig the soure ad the destiatio Thus, after summarizig the traversig distae of all bits, we have λ a l P(AH λap(ah h(b b= h= r(b,h Let T h be the total umber of hops traversed by all bits i a seod ad we have T h = λ ap(ah b= h(b Sie eah ode has oe iterfae whih a trasmit at most A, the total umber of bits that a be trasmitted by all odes over all iterfaes are at most A 2, ie, T h A 2 O the other had, uder the iterferee model, we have dist(x X 2 2 (dist(x 3 X 4 +dist(x X 2, where X ad X 3 deote the trasmitters ad X 2 ad X 4 deote the reeivers This i-equality implies that eah hop osumes a disk of radiums 2 times the legth of the hop Therefore, we have λ ap(ah h(b b= h= π 2 4 (r(b,h2, whih a be rewritte as λ ap(ah b= h(b h= T h (r(b,h 2 4 π 2 T h (3 Sie RHS of this i-equality is ovex, we have ( λap(ah h(b 2 λ ap(ah h(b r(b,h (r(b,h 2 T h T h b= h= b= h= (4 Joiig Eq (3 ad Eq (4, we have λap(ah h(b b= h= r(b,h 4T h π 2 Sie T h A 2, we have λ ap(ah h(b b= h= r(b,h 2 π 2 Besides, sie λ a l P(AH λap(ah b= h(b 2 A π 2 A hr(πh 2 (r( 2 h= r(b,h, we have λ a A 2 π 2 πh 3 (r( Sie r( > 3 fial prove the result APPENDIX B 2 π 2 l P(AH = log π, we Proof of Lemma 4 Cosider ay ell i Fig 4 The distae betwee ay trasmitter ad reeiver withi the ell a ot be more tha r max = 2a( Uder the iterferee model, a trasmissio a be suessful if o ode withi distae d s = (+ r max of the reeiver trasmits at the same time Therefore, all the iterferig ells must be otaied withi a disk D The umber of ells otaied i disk D is thus bouded by k 5 = ( 2d s 2 a( = ( 2(+ rmax 2 a( = 4( + 2, whih is a ostat, idepedet of APPENDIX C Proof of Lemma 5 a( Cosider a ell S otaied i a disk of radiusr 0 = 2 Suppose S i lies at distae x from the eter of the disk The agleαsubteded ats i by the disk is o more tha k7 x a( 2 It the destiatio ode D i is ot loated withi the setor of agle α, the lie l i aot iterset the disk otaiig the ell S Thus, the probability that L i itersets the disk is o a( 2 more tha k8h2 (r( 2 x Sie eah soure ode S i is uiformly distributed i the plae of uit area, the probability desity that S i is at a distae x from the eter of the disk is bouded by 2πx Besides, R 0 x H r( I additio, to esure the suessful trasmissio, the trasmissio rage r( 4R 0 = 8(a( As a result, we have ( P L i itersets S ad the trasmissio alog L i is usig badwidth H r( H 2 R o x ((a(3 2 2πxdx k 6 H 3 (a( 2 REFERENCES [] N Lu ad X S She, Salig laws for throughput apaity ad delay i wireless etworks - a survey, IEEE Commuiatios Surveys Tutorials, vol 6, o 2, pp , 204 [2] P Gupta ad P R Kumar, The apaity of wireless etworks, IEEE Trasatios o Iformatio Theory, vol 46, o 2, pp , 2000 [3] A E Gamal, J Mamme, B Prabhakar, ad D Shah, Optimal throughput-delay salig i wireless etworks-part I: The fluid model, IEEE Trasatios o Iformatio Theory, vol 52, o 6, pp , 2006 [4] A Raiwala ad T Chiueh, Arhiteture ad algorithms for a IEEE 802-based multi-hael wireless mesh etwork, i Pro of IEEE INFOCOM, 2005 [5] J So ad N H Vaidya, Multi-hael MAC for ad ho etworks: Hadlig multi-hael hidde termials usig a sigle traseiver, i Pro of ACM MobiHo, 2004 [6] P Bahl, R Chadra, ad J Duaga, SSCH: Slotted seeded hael hoppig for apaity improvemet i ieee 802 ad-ho wireless etworks, i Pro of ACM Mobiom, 2004 [7] R Draves, J Padhye, ad B Zill, Routig i multi-radio, multi-hop wireless mesh etworks, i Pro of ACM Mobiom, 2004 [8] P Kyasaur ad N H Vaidya, Capaity of multihael wireless etworks: Impat of umber of haels ad iterfaes, i Pro of ACM MobiCom, 2005 [9] H-N Dai, K- Ng, R C- og, ad M-Y u, O the Capaity of Multi-Chael ireless Networks Usig Diretioal Ateas, i Pro of IEEE INFOCOM, 2008 [0] B Liu, Z Liu, ad D Towsley, O the Capaity of Hybrid ireless Networks, i Pro of IEEE INFOCOM, 2003 [] U C Kozat ad L Tassiulas, Throughput apaity of radom ad ho etworks with ifrastruture support, i Pro of ACM MobiCom, 2003 [2] A Zemliaov ad G D Veiaa, Capaity of ad ho wireless etworks with ifrastruture support, IEEE Joural o Seleted Areas i Commuiatios, vol 23, pp , 2005 [3] B Liu, P Thira, ad D Towsley, Capaity of a ireless Ad Ho Network ith Ifrastruture, i Pro of ACM MobiHo, 2007 [4] P Li, C Zhag, ad Y Fag, Capaity ad delay of hybrid wireless broadbad aess etworks, IEEE Joural o Seleted Areas i Commuiatios (JSAC, vol 27, pp 7 25, 2009 [5] G Zhag, Y Xu, X ag, ad M Guizai, Capaity of Hybrid ireless Networks with Diretioal Atea ad Delay Costrait, IEEE Trasatios o Commuiatios, pp , 200 [6] P Li ad Y Fag, The apaity of heterogeeous wireless etworks, i Pro of IEEE INFOCOM, 200 [7] C Xie, J Li, X ag, X Tia, Y Zhag, ad X ag, Multiast apaity of wireless ad ho etworks with ifrastruture support, i Pro of IEEE ICC, 202 [8] A Asadi, Q ag, ad V Mauso, A survey o devie-to-devie ommuiatio i ellular etworks, IEEE Commuiatios Surveys Tutorials, vol 6, o 4, pp 80 89, Fourthquarter 204

15 5 [9] Y Zhag, R Yu, M Nekovee, Y Liu, S Xie, ad S Gjessig, Cogitive mahie-to-mahie ommuiatios: visios ad potetials for the smart grid, Network, IEEE, vol 26, o 3, pp 6 3, May 202 [20] Y Ya, Y Qia, H Sharif, ad D Tipper, A survey o smart grid ommuiatio ifrastrutures: Motivatios, requiremets ad halleges, Commuiatios Surveys Tutorials, IEEE, vol 5, o, pp 5 20, 203 [2] G R MaCartey, M K Samimi, ad T S Rappaport, Exploitig diretioality for millimeter-wave wireless system improvemet, i IEEE ICC, 205 [22] D M Shila, Y Cheg, ad T Ajali, Throughput ad delay aalysis of hybrid wireless etworks with multi-hop upliks, i INFOCOM, 20 [23] P Zhou, X ag, ad R Rao, Asymptoti apaity of ifrastruture wireless mesh etworks, IEEE Trasatios o Mobile Computig, vol 7, o 8, pp 0 024, Aug 2008 [24] M Masoori ad M Mahdavi, Asymptoti Throughput Capaity Aalysis of Multi-Chael, Multi-Iterfae ireless Mesh Networks, ireless Persoal Commuiatios, vol 68, o, pp , 203 [25] Fu ad D Agrawal, Capaity of hybrid wireless mesh etworks with radom aps, IEEE Trasatios o Mobile Computig, vol 2, o, pp 36 50, Ja 203 [26] M Masoori ad M Mahdavi, Throughput apaity aalysis of large multi-hael, multi-iterfae radom wireless mesh etworks, Performae Evaluatio, vol 70, o, pp 45 76, 203 [27] S Chieoha ad E Hossai, Chael Assigmet for Throughput Optimizatio i Multihael Multiradio ireless Mesh Networks Usig Network Codig, IEEE Trasatios o Mobile Computig, vol 2, o, pp 8 35, 203 [28] M Masoori, M Mahdavi, ad H A Khorasgai, Performae aalysis of large multi-iterfae wireless mesh etworks with multi-differet badwidth hael, IEEE Trasatios o Mobile Computig, vol 5, o 5, pp , 205 [29] I F Akyildiz, X ag, ad ag, ireless mesh etworks: a survey, Computer Networks, vol 47, o 4, pp , 2005 [30] M Kodialam ad T Nadagopal, Charaterizig the apaity regio i multi-radio multi-hael wireless mesh etworks, i Pro of ACM MobiCom, 2005 [3] H-N Dai, R C- og, ad Q Zhao, Multi-hael ireless Networks with Ifrastruture Support: Capaity ad Delay, i Pro of IEEE Iteratioal Coferee o Commuiatios (ICC, 204 [32] T H Corme, C E Leiserso, R L Rivest, ad C Stei, Itrodutio to Algorithms, seod editio, 3rd ed MIT Press, 2009 [33] T S Rappaport, ireless ommuiatios : priiples ad pratie, 2d ed Upper Saddle River, NJ: Pretie Hall PTR, 2002 [34] A E Gamal, J Mamme, B Prabhakar, ad D Shah, Throughput- Delay Trade-offs i ireless Networks, Part II: Costat-Size Pakets, IEEE Trasatios o Iformatio Theory, vol 52, o, pp 5 56, 2006 [35] R Motwai ad P Raghava, Radomized Algorithms Cambridge Uiversity Press, 995 [36] M Raab ad A Steger, Balls ito bis - A simple ad tight aalysis, i Pro of the Seod Iteratioal orkshop o Radomizatio ad Approximatio Tehiques i Computer Siee (RANDOM, 998 [37] D B est, Itrodutio to graph theory, 2d ed Upper Saddle River, NJ: Pretie Hall PTR, 200 [38] S Yi, Y Pei, ad S Kalyaarama, O the apaity improvemet of ad ho wireless etworks usig diretioal ateas, i MobiHo, 2003 [39] P Li, C Zhag, ad Y Fag, The Capaity of ireless Ad Ho Networks Usig Diretioal Ateas, IEEE Trasatios o Mobile Computig, vol 0, o 0, pp , 20 [40] T Rappaport, S Su, R Mayzus, H Zhao, Y Azar, K ag, G N og, J Shulz, M Samimi, ad F Gutierrez, Millimeter wave mobile ommuiatios for 5g ellular: It will work! IEEE Aess, vol, pp , 203 [4] J Qiao, X She, J Mark, Q She, Y He, ad L Lei, Eablig devieto-devie ommuiatios i millimeter-wave 5g ellular etworks, IEEE Commuiatios Magazie, vol 53, o, pp , Jauary 205 [42] T Nitshe, C Cordeiro, A Flores, E Kightly, E Perahia, ad J C idmer, IEEE 802ad: diretioal 60 GHz ommuiatio for multi- Gigabit-per-seod i-fi, IEEE Commuiatios Magazie, vol 52, o 2, pp 32 4, Deember 204 [43] H-N Dai, R C- og, ad H ag, O apaity ad delay of multi-hael wireless etworks with ifrastruture support, Teh Rep, April 206 [Olie] Available: [44] I F Akyildiz, -Y Lee, M C Vura, ad S Mohaty, Next geeratio/dyami spetrum aess/ogitive radio wireless etworks: A survey, Comput Netw, vol 50, o 3, pp , Sept 2006 [45] M Grossglauser ad D Tse, Mobility ireases the apaity of ad ho wireless etworks, IEEE/ACM Trasatios o Networkig, vol 0, 2002 [46] G Sharma ad R Mazumdar, Salig laws for apaity ad delay i wireless ad ho etworks with radom mobility, i Pro of IEEE ICCeffiiey of multi-hop devie-to-devie, 2004 [47] X Li, G Sharma, R Mazumdar, ad N Shroff, Degeerate delayapaity tradeoffs i ad-ho etworks with browia mobility, IEEE Trasatios o Iformatio Theory, vol 52, o 6, pp , Jue 2006 [48] E Khorov, A Lyakhov, A Krotov, ad A Gushi, A survey o IEEE 802ah: A eablig etworkig tehology for smart ities, Computer Commuiatios, vol 58, pp 53 69, 205 [49] M N Tehrai, M Uysal, ad H Yaikomeroglu, Devie-to-devie ommuiatio i 5g ellular etworks: halleges, solutios, ad future diretios, IEEE Commuiatios Magazie, vol 52, o 5, pp 86 92, May 204 [50] S- Jeo, S-N Hog, M Ji, ad G Caire, Cahig i wireless multihop devie-to-devie etworks, i IEEE ICC, 205 [5] L ei, R Q Hu, Y Qia, ad G u, Eergy-effiiey ad spetrumeffiiey of multi-hop devie-to-devie ommuiatios uderlayig ellular etworks, IEEE Trasatios o Vehiular Tehology, vol 65, o, pp , 206 Hog-Nig Dai is a Assoiate Professor i Faulty of Iformatio Tehology at Maau Uiversity of Siee ad Tehology He obtaied the PhD degree i Computer Siee ad Egieerig from Departmet of Computer Siee ad Egieerig at the Chiese Uiversity of Hog Kog i 2008 He also reeived the B S ad MPhil degrees i Computer Siee ad Egieerig from South Chia Uiversity of Tehology He also holds visitig positios at Departmet of Computer Siee ad Egieerig, The Hog Kog Uiversity of Siee ad Tehology ad Shool of Eletrial Egieerig ad Teleommuiatios, the Uiversity of New South ales, respetively His researh iterests ilude wireless etworks, mobile omputig, ad distributed systems Raymod Chi-ig og reeived the BS, MPhil ad PhD degrees i Computer Siee ad Egieerig i the Chiese Uiversity of Hog Kog (CUHK He is ow a Assoiate Professor of Computer Siee ad Egieerig of the Hog Kog Uiversity of Siee ad Tehology His researh iterests ilude database, data miig ad seurity Hao ag is a assoiate professor ad the head of Big Data Lab at the Faulty of Egieerig ad Natural Siees i Norwegia Uiversity of Siee ad Tehology i Aalesud, Norway He has worked as a researher i IBM Caada, MMaster, ad St Frais Xavier Uiversity before he moved to Norway He reeived a PhD degree i 2006 ad a BEg degree i 2000, both i omputer siee from South Chia Uiversity of Tehology His researh iterests ilude big data aalytis ad idustrial iteret of thigs, high performae omputig, rigorous software egieerig methods for safety-ritial systems, ad seurity ad privay i ommuiatios