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

Transcription

1 O Capaity ad Delay of Multi-hael Wireless Networks with Ifrastruture Support Hog-Nig Dai, Raymod Chi-Wig Wog, Hao Wag arxiv:60403v [sni] 8 Apr 06 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 We show that a MC-IS-DA etwork has a eve lower delay of π θ C I ompared with a SC-IS etwork ad our MC-IS etwork For example, whe C I = ad θ = π, a MC-IS-DA a further redue the delay by 4 times lower that of a MC-IS etwork ad redue the delay by 88 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 [] [4] that i a radom ad ho etwork with odes, eah ode has a throughput apaity of ΘW/ log where W is the total etwork badwidth ad the average delay of this etwork is Θ /log Whe 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 Whe a ode trasmits, its eighborig odes i the same 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 [] However, the There are two kids of etwork plaemets: a a radom etwork, i whih odes are radomly plaed, ad the destiatio of a flow is also radomly hose ad b a arbitrary etwork, i whih the loatio of odes, ad traffi patters a be optimally otrolled We oly osider the radom etwork i this paper 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 We 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 [5] [0] 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 [] [9] that eah hael or up to Olog 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 Whe 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 We all this multi-hael wireless ad ho etwork as a MC-AH etwork Reet studies [] [6] ivestigated the performae improvemet by addig a umber of ifrastruture odes to a wireless etwork Speifially, as show i [], [5], deployig ifrastruture odes i the wireless etwork a sigifiatly improve the etwork apaity ad redue the average delay But, every ode both a ommo ode ad 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 i plae Besides, oly oe sigle hael is used i suh a etwork We 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 Eah ommo ode is equipped with a sigle etwork iterfae ard NIC Eah base statio is equipped with

2 TABLE I COMPARISON WITH OTHER EXISTING WIRELESS NETWORKS Pure Ad Ho Ad Ho with Ifrastruture Sigle Chael SC-AH etworks SC-IS etworks [] [4] [] [8] Multiple Chaels MC-AH etworks MC-IS etworks [5] [0] this paper 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 so we a igore the swithig delay of NICs 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 I fat, our proposed MC-IS etworks have provided a solutio to the ew appliatios, suh as Devie-to-Devie DD etworks [9], wireless sesor etworks WSNs, smart grid, smart home ad e-health systems [0], [] For example, the theoretial aalysis o the throughput ad the delay of our MC-IS etworks a be used to aalyze the performae of the overlaid DD etworks details a be foud i Setio VII-C 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 We 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 We propose a geeral theoretial framework to aalyze the apaity ad the average delay We 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 W, 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 iterfaes at a base statio, ompared with a sigle iterfae 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 We 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 We ame the exteded MC-IS etworks as MC-IS-DA etworks We show that a MC-IS-DA etwork a have a eve lower delay of ompared with both a MC-IS etwork π θ CI ad a SC-IS etwork, where θ is the beamwidth of a diretioal atea mouted at the base statio usually θ < π Cosider the ase of C I = ad θ = π that is feasible i Millimeter-Wave systems [] A MC-IS- DA a further redue the delay by 4 times lower tha that of a MC-IS etwork ad redue the delay by 88 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 We preset the models used i this paper i Setio III Setio IV the summarizes our mai results We 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 We 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 WORKS We 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

3 3 simultaeous trasmissios o a sigle iterfae ad iii the multi-hop ommuiatios [] [4] 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 etwork iterfaes istead of sigle etwork iterfae 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 [9] that eah hael up to Olog haels must be utilized by a dediated iterfae at a ode so that all the haels are fully utilized simultaeously ad thus the etwork apaity a be maximized Whe the oditio is ot fulfilled, the apaity degrades sigifiatly The third etwork related to our MC-IS etwork is a SC- IS etwork [] [8], [3] It is show i [], [5] 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 [4] [9], whih is the evolutio of multi-hael multi-iterfae wireless mesh etworks alled a MC-Mesh etwork [30], [3] 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 the same etwork i 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 A iterestig questio is whether usig diretioal ateas i MC-Mesh-IS etworks Fig X34 Base statio Commo ode Data flow X8 X4 B X4 X X0 X3 X5 X8 X3 X6 X6 B Data flow 4 X0 X5 X9 X Data flow X X4 X7 Data flow Network topology of a MC-IS etwork X5 B6 X35 X3 X33 X9 X3 X36 X8 X7 X X30 X3 Ad ho ommuiatios Ifrastruture ommuiatios Data flow 3 a brig the performae improvemet, whih might be oe of the future works 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: We 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 We 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] We 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 We exted our aalysis with osideratio of usig diretioal ateas i a MC-IS etwork Disussios o the mobility are also preseted i this paper see Setio VII for more details B3 III MODELS We adopt the asymptoti otatios [33] i this paper We 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 We assume that b a be expressed as a square of a ostat b 0 ie, b 0 B5 X7 B4 X X9 X X6

4 4 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 [], [5], [6], 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 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 With regard to Data flow, sie destiatio ode X 8 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 8 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 5 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 [], 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, whih is the earest BS to ode X 5 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 We the divide the total badwidth of W bits/se ito three parts: for ad ho ommuiatios, W I,U for uplik ifrastruture ommuiatios ad 3 W I,D for dowlik ifrastruture ommuiatios Sie W I,U is equal to W I,D, it is obvious that W = +W I,U +W I,D = +W I,U To simplify our aalysis, we use W I to deote either W I,U or W 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 Iterferee model: I this paper, we osider the iterferee model [], [9], [] [3], [5] Whe 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 distx k,x j + distx i,x j where distx i,x j deotes the distae betwee two odes X i ad X j Note that the physial iterferee model [] 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 [], [34] 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 : Per-ode throughput apaity of a MC-IS etwork with the throughput of λ is of order Θg bits/se if there are determiisti ostats h > 0 ad h < + suh that lim Pλ = hg is feasible = ad lim if Pλ = 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

5 5 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 [4] After averagig the delay of all the pakets trasmitted i 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], [4] 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 [35] 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 We 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 We 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 but oly some of them ad ot all For example, the apaity of SC-AH etworks ad SC-IS etworks is limited by Coetivity requiremet ad Iterferee requiremet as show i [] ad [] while the apaity of MC-AH etworks is limited by Coetivity requiremet, Iterferee requiremet ad Iterfae-bottleek requiremet [9] uder radom etwork plaemet 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, a be partitioed ito 3 ases: Case orrespodig to the ase whe = OF, Case orrespodig to the ase whe = ΩF ad = OF, ad Case 3 orrespodig to the ase whe = ΩF, where F = log ad F = loglogh log logh log Uder eah of the above ases, H a be partitioed ito two sub-ases Uder Case, H is partitioed ito sub-ases, amely Sub-ase ad Sub-ase Sub-ase is whe H = og ad Sub-ase is whe H = ΩG, where G = 3/log 3 Uder Case, H is partitioed ito subases, amely Sub-ase ad Sub-ase Sub-ase is whe H = og ad Sub-ase is whe H = ΩG, where G = 3C 6 A /log Uder Case 3, H is partitioed ito sub-ases, amely Sub-ase 3 ad Sub-ase 3 Subase 3 is whe H = og 3 ad Sub-ase 3 is whe H = ΩG 3, where G 3 = /log Fig 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 i whih Coetivity requiremet domiates Iterferee Coditio: orrespodig to Sub-ase i whih Iterferee requiremet domiates Destiatio-bottleek Coditio: orrespodig to Subase 3 i whih Destiatio-bottleek requiremet domiates Iterfae-bottleek Coditio: orrespodig to Subase, Sub-ase, or Sub-ase 3, i whih Iterfae-bottleek requiremet domiates B Summary of Results We 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 Whe Coetivity Coditio is satisfied, λ = Θ H log + Θ mi{ b, bm C I }W I, where λa = Θ H log ad λi = Θ mi{ b, bm C I }W I ; ii Whe Iterferee Coditio is satisfied, λ = W Θ A + Θmi{ b, bm C I }W I, where λ a = Θ C A H log C A H log ad λ i = Θ mi{ b, bm C I }W I ; iii Whe Destiatio-bottleek Coditio is satisfied, λ = loglogh Θ log + Θmi{ b, bm C I }W I, H log logh log

6 6 Case C = O F A C =Ω F A Case C =Ω F A ad C = O F A Case 3 H = og G H =Ω H = og H =Ω G H = og H =ΩG 3 3 Sub-ase Sub-ase Sub-ase Sub-ase Sub-ase 3 Sub-ase 3 Iterfae-bottleekCoetivity Iterfae-bottleekIterferee Iterfae-bottleekDestiatio-bottleek Fig All possible sub-ases osidered where λ a = Θ loglogh log H log logh log ad λ i = Θ mi{ b, bm C I }W I ; iv Whe Iterfae-bottleek Coditio is satisfied, λ = Θ H log WA +Θmi{ b, bm C I }W I, where λ a = Θ H log WA ad λ i = Θ mi{ b, bm C I }W I Theorem : 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} 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 [5] ad MC-AH etworks [9], 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 We 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 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 [], a MC-AH etwork [9], ad a SC-IS etwork [5] 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 [] 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, =, = W ad W I = 0 I that ofiguratio, the total badwidth is assiged for ad ho ommuiatios = W ad W I = 0, there is a sigle hael available = orrespodig to that of a SC-AH etwork [] B A MC-AH etwork is a speial ase of our MC-IS etwork: The theoretial bouds i the MC-AH etwork [9] 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 [9] I partiular, we have the followig ases: Case I: whe = Olog ad H = Θ /log Coetivity Coditio is satisfied, the per-ode throughput λ = ΘW/ log ad the average delay D = Θ /log, whih mathes the result of a MC- AH etwork [9]; Case II: whe = Ωlog ad = O loglogh log logh log, ad H = Θ /log Iterferee Coditio is satisfied, the per-ode throughput λ = ΘW/ ad the average delay D = Θ /log, whih mathes the result of a MC-AH etwork [9]; Case III: whe = Ω loglogh log ad H = logh log Θ /log Destiatio-bottleek Coditio is satisfied, the per-ode throughput λ = Θ loglogw log ad the average delay D = Θ /log, whih mathes the result of a MC-AH etwork [9] 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 [5] 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 3 Coetivity Coditio is satisfied, λ = Θ Wa H log + b W i ad D = Θ H3 log +, whih mathes the result of a SC-IS etwork [5]; Case II: whe = ad H = o 3/log 3 Iterfae-bottleek Coditio is satisfied, λ = ΘH log W a C a + mi{ b, bm C i }W I ad D = Θ H3 log +, whih mathes the result of a SC-IS

7 7 etwork [5] 4 Optimality of Results We 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 ; whe λ i domiates λ a Case : whe λ a domiates λ i ie W ad W I /W 0 We obtai the maximum per-ode throughput apaity as the followig sub-ases: i λ = Θ W H log with Coetivity W oditio; ii λ = Θ with Iterferee oditio; iii λ = Θ logw C H log loglogh with Destiatiobottleek oditio; iv λ = Θ CH log logh log H W log C with Iterfaebottleek 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 will show as follows that the above results are ot optimal ompared with Case Case : whe λ i domiates λ a ie W I W/ ad /W 0 I this ase, the maximum per-ode throughput apaity λ = Θ b W ad the average delay D = Θ mi{c I,m} It implies that whe whe λ i domiates λ a, 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 = ΘW, 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}, where is a ostat ad mi{c I,m} 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 = ΘW ad the optimal mi{c I,m} average delay D opt = Θ We summarize our key results ad ompare our results with other related etworks i Fig 3 I partiular, we ompare a MC-IS etwork with three existig etworks, amely a MC- AH etwork, a SC-IS etwork, ad a SC-AH etwork, i terms of the optimal per-ode throughput apaity λ ad the optimal average delay D i Fig 3 As show i Fig 3, a MC-IS etwork a ahieve the optimal per-ode throughput apaity λ opt = ΘW 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 I other words, there is o apaity degradatio i the optimal per-ode throughput of a MC-IS etwork Besides, ompared with other existig etworks, a MC- D log mi CI, m π CI θ W A log B C ' C W SC-AH MC-AH SC-IS MC-IS MC-IS-DA λ Fig 3 Capaity ad delay regios uder differet etworks The sales of the axes are i terms of the orders i IS etwork a ahieve the smallest delay Θ mi{c I,m} poit C i Fig 3 whe the optimal per-ode throughput apaity λ = ΘW is ahieved It is show i [3], [4] that i a SC-AH etwork ad a MC-AH etwork, the ireased apaity pays for the higher delay due to the multihop 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 iterfaes at eah base statio, whih a support multiple simultaeous trasmissios Speifially, as show i Fig 3, a MC-IS etwork poit C i Fig 3 has a delay redutio gai of mi{c I,m} over a SC-IS etwork poit B i Fig 3 For example, a MC-IS etwork with C I = m = eg, there are C I = o-overlappig haels i IEEE 80a [36], i whih we assig a dediated iterfae for eah hael, has a delay times lower tha a SC-IS etwork Besides, whe we 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, we a obtai a eve lower delay of as show i poit π C i Fig 3, θ where θ is the beamwidth CI of a diretioal atea mouted at the base statio usually θ < π Cosider the same ase of C I = ad θ = π that is feasible i most of mmwave systems [] A MC-IS-DA a further redue the delay by 4 times lower that of a MC-IS etwork ad redue the delay by 88 times lower tha that of a SC-IS etwork This is beause usig diretioal ateas a oetrate the trasmissios to the desired diretios ad a improve the spetrum reuse, potetially supportig more ourret trasmissios Details o this exteded work will be addressed i Setio VII V CAPACITY CONTRIBUTED BY AD HOC COMMUNICATIONS We first derive the upper bouds o the etwork apaity otributed by ad ho ommuiatios i Setio V-A Setio V-B presets ostrutive lower bouds o the etwork apaity otributed by ad ho ommuiatios, whih have the same order of the upper bouds, implyig that our results are quite tight We the give the aggregate throughput apaity i Setio V-C Note that our derivatios are sigifiatly

8 8 differet from those of the existig etworks, suh as SC- AH etworks, MC-AH etworks ad SC-IS etworks beause the apaity otributed by ad ho ommuiatios of MC-IS etworks is maily limited by all four aforemetioed requiremets simultaeously ot just subsets of these requiremets ad we eed to establish a ew theoretial framework to aalyze the results A Upper Bouds o Network Capaity Cotributed by Ad Ho Commuiatios We foud that the etwork apaity otributed by ad ho trasmissios i a MC-IS etwork, deoted by λ a, is maily affeted by Coetivity requiremet, Iterferee requiremet, 3 Destiatio-bottleek requiremet ad 4 Iterfae-bottleek requiremet We first derive the upper bouds o the per-ode throughput apaity uder Coetivity Coditio defied i Setio IV-A Before presetig Propositio, we have Lemma to boud the expetatio of the umber of hops, whih is deoted by h Lemma : The expetatio of the umber of hops h is bouded by ΘH Proof We first deote Ph = i by the probability of the evet that a paket traverses h = i hops Aordig to the H-max-hop routig sheme, Ph = 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, Ph = 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, Ph = i = i i πr πi r We the have h = Eh = Ph = + Ph = + +H Ph = H πr = πh r + 3πr πh r + +H H H πr πh r Sie H[H H ] i Eq are the series of hexagoal umbers, the Eq a be simplified as follows h = 6HH +4H H = 4H3 +3H H 6H It is obvious that h is a futio of H as show i Eq The limit of hh as H approahes is lim hh = ΘH, H whih a be diretly derived from the defiitio of the asymptoti otatio Θ ad Eq We the have Propositio that bouds the per-ode throughput apaity otributed by ad ho ommuiatios uder Coetivity oditio: Propositio : Whe Coetivity requiremet domiates, the per-ode throughput apaity otributed by ad ho ommuiatios is λ a = O H 3 log Proof We first alulate the probability that a ode uses the ad ho mode to trasmit, deoted by PAH, whih is the probability that the destiatio ode is loated withi H hops away from the soure ode Thus, we have PAH = πh r 3 Sie eah soure geeratesλ 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 PAH h λ a We ext prove that PAH h λ a is bouded by k r W A We deote the maximum umber of simultaeous trasmissios o a partiular hael by N max As proved i Lemma k 54 i [], N max is upper bouded by r, where k > 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 the total umber of trasmissios i the whole etwork is o more tha k r = = k r bits/se k Therefore, we have PAH h λ a r W A Combiig the above results with Lemma yields λ a k r πh 3 r kwa H 3 r, where k is a ostat Besides, to guaratee that the etwork is oeted with high probability whp log, we requirer > π [] Thus, we have λ a k3wa H 3 log, where k 3 is a ostat We the derive the upper bouds o the per-ode throughput apaity uder Iterferee Coditio Propositio : Whe Iterferee requiremet domiates, the per-ode throughput apaity otributed by ad ho ommuiatios is λ a = O A W C A H 3 log 3 Proof We preset a proof of the boud i Appedix A Before provig the upper bouds o the throughput apaity uder the destiatio-bottleek oditio, we eed to boud the umber of flows towards a ode uder the H-max-hop routig sheme Speifially, we have the followig result Lemma : The maximum umber of flows towards a ode uder the H-max-hop routig sheme is D H = Θ logh log loglogh 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 We have N t = i= N i Thus, the expeted umber of soure odes trasmittig i ad ho mode is: EN t = E N i = EN i i= i= We say that a evet e happes with a high probability if Pe whe

9 9 Sie fn i = = PAH = πh r ad r eeds to be Θ log/ to esure that the etwork is oeted, we have EN i = πh r + 0 πh r = πh r, ie, EN i = ΘπH log Therefore, EN t = πh log = πh log Reall the Cheroff bouds [37], we have for ay δ > 0, PN t > + δπh log < πh e δ +δ +δ log ; for ay 0 < δ <, PN t < δπh log < e πh log δ / I summary, for ay 0 < δ <, we a obtai P N t πh log > δπh log < e επh log, where ε > 0 Thus, whe, the total umber of soure odes trasmittig i ad ho mode is ΘH log whp I a radom etwork, eah soure ode a radomly hoose its destiatio The traffi for a soure-destiatio pair is deoted as a flow Thus, it is very likely that a ode will be the destiatio of multiple flows It is proved i [38] that the maximum umber of flows towards ay give ode i a radom etwork with N odes, deoted by DN, is upper bouded by Θ logn loglogn, whp Combiig the two results by replaig N = H log leads to the above result We the prove the upper bouds o the per-ode throughput apaity uder Destiatio-bottleek Coditio Propositio 3: Whe Destiatio-bottleek requiremet domiates, the per-ode throughput apaity otributed by ad 3 loglogh ho ommuiatios is λ a = O log H 3 log 3 logh log Proof Sie eah ode has oe iterfae that a support at most WA ad Sie eah ode has at most D H flows uder the H-max-hop routig sheme, the data rate of the miimum W rate flow is at most A C, where D AD H H is bouded by Θ logh log loglogh log by Lemma After alulatig all the data rates at eah ode times with the traversig distae, we have PAH λ a h r WA C AD H We the have λ a D H PAHhr πh 3 r 3 logh log loglogh log This is beause h = ΘH ad PAH = πh r are derived i Lemma ad Eq 3 i the proof of Propositio, respetively log Sie r = Θ as proved i [], we the have λ a 3 loglogh log H 3 log 3 logh log Fially, we prove the upper bouds o the per-ode throughput apaity uder Iterfae-bottleek Coditio Propositio 4: Whe Iterfae-bottleek requiremet domiates, the per-ode throughput apaity otributed by ad ho ommuiatios is λ a = O WA Proof I a MC-IS etwork, eah ode is equipped with oly oe iterfae, whih a support at most WA data rate Thus, Fig 4 a ell a Plae divided ito a umber of ells ad eah with area a λ a is also upper bouded by WA Note that this result holds for ay etwork settigs B Costrutive Lower Bouds o Network Capaity Cotributed by Ad Ho Commuiatios We 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 Θa odes, where a is the area of a ell Setio V-B We the desig a routig sheme to assig the umber of flows at eah ode evely Setio V-B Fially, we desig a Time Divisio Multiple Aess TDMA sheme to shedule the traffi at eah ode Setio V-B3 Cell Costrutio: We 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 [9], we set { } } 00log a = mi max, log 3, log 3 logh log C A 3 loglogh 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 Θa odes whp Proof Please refer to [9] We ext hek whether all the above values of a are properly hose suh that eah ell has Θa odes whp whe is large eough ie, Lemma 3 is satisfied 00log It is obvious that > 50log ad log 3 > 50log C A sie we oly osider i Coetivity Coditio ad Iterferee Coditio Besides, greater tha 50log with large sie log 3 logh log 3 loglogh log is also logh log loglogh log > ad log 3 > 50log 3 whe is large eough Besides, the umber of iterferig ells aroud a ell is bouded by a ostat, give by Lemma 4 as follows

10 0 Fig 5 oe seod 3 CA- CA mii-slot edge-olor slot TDMA trasmissio shedule 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 Routig Sheme: To assig the flows at eah ode evely, we desig a routig sheme osists of two steps: Assigig soures ad destiatios ad 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 Thus, after Step, there are at most +D H flows We deote the straight lie oetig a soure S to its destiatio D as a S-D lies I Step, 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 OH 3 a Proof The detailed proof is stated i Appedix C As show i Lemma 3, there are Θ a odes i eah ell Therefore, Step will assig to ay ode at most O H 3 a = OH 3 a flows Summarizig Step a ad Step, 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 = OH 3 a 3 Shedulig Trasmissios: We 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 ay two trasmissios o ay hael should ot iterfere with eah other We 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 The, we desribe the two time slots as follows i Edge-olor slot: First, we ostrut a routig graph i whih verties are the odes i the etwork ad a edge Time 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 = OH 3 a edges It is show i [9], [39] that this routig graph a be edge-olored with at most OH 3 a olors We the divide oe seod ito OH 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: We 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 We ostrut a iterferee graph i whih eah vertex is a ode i the etwork ad eah edge deotes the iterferee betwee two odes We 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 [9] Lemma 6: The iterferee graph a be vertex-olored with at most Oa olors Proof By Lemma 4, every ell has at most a ostat umber of iterferig ells Besides, eah ell has Θa odes by Lemma 3 Thus, eah ode has at most Oa 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 [9] [39] Hee, the iterferee graph a be vertex-olored with at most Oa olors We 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 iterfere with eah other We 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 + We 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 Whe Coetivity requiremet domiates, λ a is Ω WA bits/se; H 3 log Whe Iterferee requiremet domiates, λ a is W Ω A bits/se; H 3 C A log 3 3 Whe Destiatio-bottleek requiremet domiates,λ a 3 loglogh is Ω log bits/se; H 3 log 3 logh log 4 Whe 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 H 3 a k7 a

11 k7a bits a be trasported Sie k7a +, W we have λ a = Ω A k 7H 3 a +H 3 a bits/se Thus, W λ a = Ω MIN A O H 3 a, H 3 a bits/se 3 Reall { that a } is } set to mi max { 00log, log 3 C A, log3 logh log 3 loglogh log Substitutig the three values to λ a, we have the results, ad 3 Besides, eah iterfae a trasmit or reeive at the rate of WA bits/se Thus, λ a = Ω, whih is the result 4 C Aggregate Throughput Capaity The upper bouds proved i Propositios,, 3 math with the lower bouds proved i Propositio 5, implyig that our bouds are quite tight Besides, it is show i [5] that the total traffi of ad ho ommuiatios isπh r λ a Combiig Propositios,, 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 Whe Coetivity requiremet domiates, T A is Θ WA H log bits/se Whe Iterferee requiremet domiates, T A is W Θ A bits/se C A H log 3 Whe Destiatio-bottleek requiremet domiates, T A is Θ 3 loglogh log H log logh log bits/se 4 Whe Iterfae-bottleek requiremet domiates,t A is ΘH log WA bits/se VI CAPACITY CONTRIBUTED BY INFRASTRUCTURE COMMUNICATIONS I this setio, we aalyze the etwork apaity otributed by ifrastruture ommuiatios Speifially, we derive the upper bouds of the apaity i Setio VI-A ad give the ostrutive lower bouds of the apaity i Setio VI-B We give the aggregate apaity otributed by ifrastruture ommuiatios i Setio VI-C Note that our proposed MC- IS etworks have the multiple iterfaes at a base statio, ompared with a sigle iterfae 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 derivatios are also more ompliated tha those of SC-IS etworks Fially, Setio VI-D gives the proof of Theorem ad Theorem A Upper Bouds of Network Capaity Cotributed by Ifrastruture Commuiatios We 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: Whe C I m, T I = ObW I 3 MIN O f,g is equal to f if f = Og; otherwise, it is equal to g Whe C I > m, T I = Ob m C I W 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 W 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 = ObW I Case whe C I > m Whe the umber of iterfaes is smaller tha the umber of haels, 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 WI C I bits/se m Thus, eah base statio a support at most C I W I bits/se Coutig all the b base statios, we have T I = Ob m C I W 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 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 We 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 We group the b BS-ells ito a umber of lusters Eah luster has k 8 + BS-ells We 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 For example, all the lusters follow the same 9-TDMA trasmissio shedulig sheme, as show i Fig 6 Propositio 7: Uder the TDMA sheme Σ, the throughput apaity T I, is: Whe C I m, T I = ΩbW I Whe C I > m, T I = Ωb m C I W 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 TDMA sheme Σ, eah BS-ell is ative to trasmit every k 8 + time slots Whe a BS-ell is ative, there are at mostc I haels available to use

12 a BS-ell Fig 6 A example of the TDMA trasmissio shedule, i whih eah BS-ell i a luster beomes ative every 9 time slots Thus, the total badwidth ofw I of thosec I haels are fully used Thus, the per-ell throughput λ i is lower bouded by W I k Coutig all thebbase statios, we havet 8+ I = Ω bwi k 8+ Case whe C I > m: Similarly, eah BS-ell is ative to trasmit every k 8 + time slots i ase But, whe a BS-ell is ative, oly m haels available at a time ad eah hael a support at most WI C I data rate Thus, the per-ell throughput λ i is lower bouded Coutig all the b mw I C Ik 8+ base statios, we have T I = Ω bmwi C Ik 8+ C Aggregate Throughput Capaity After ombiig Propositio 6 ad Propositio 7, we have the followig theorem Theorem 4: The aggregate throughput apaity of the etwork otributed by ifrastruture ommuiatios is Whe C I m, T I = ΘbW I Whe C I > m, T I = Θb m C I W I It is show i Theorem 4 that the optimal throughput apaity otributed by ifrastruture ommuiatios T I = ΘbW 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 However, it is ot true that we always have C I = m sie the radio spetrum beomes sare [40] ad there may be fewer haels tha the iterfaes We will give a disussio o this issue i Setio VII D Proof of Theorem ad Theorem We fially give the proof of Theorem ad Theorem as follows: Proof of Theorem We 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 We the derive the average delay of a MC-IS etwork otributed by ad ho ommuiatios ad ifrastruture ommuiatios as follows Proof of Theorem We first derive the boud o the delay whe the pakets are trasmitted i the ifrastruture mode As show i [5], 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 We the derive the boud o the delay whe the pakets are trasmitted i ad ho mode If the pakets are trasmitted i the ad ho mode, the expetatio of the umber of hops h uder the H-max-hop routig strategy is bouded by ΘH as proved by Lemma Sie the time spet by a paket at eah relay is bouded by a ostat umber, the average delay is of the same order as the average umber of hops, ie, D = h = ΘH It is show i the proof of Lemma that the umber of trasmitters i the ad ho mode is πh log whp The the umber of trasmitters i the ifrastruture mode is πh log whp After applyig the aforemetioed aalysis, we have the average delay of all pak- πh log H+ πh log mi{c ets D = Θ I,m} πh log Θ mi{c I,m} Note that + is bouded by Θ Thus, D = Θ H 3 log 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 Note that our aalysis is o-trivial sie the existig aalytial models suh as MC-AH etworks, SC-IS etworks ad eve our MC-IS etworks aot be diretly used i the exteded MC-IS etworks beause the iterferee model is sigifiatly differet from those existig oes We 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 radio sigals i all diretios iludig some udesired diretios Reet studies suh as [4], [4] 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

13 3 Fig 7 Base statio Commo ode setors X4 X5 X3 B X X Ifrastruture diretioal liks Ad ho diretioal liks Network topology of a MC-IS-DA etwork i a BS-ell appliatio of diretioal ateas to ovetioal 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 mmwave ommuiatio systems, whih is a very promisig solutio for the ext geeratio ommuiatio systems 5G [43] It is feasible to deploy diretioal ateas at both base statios ad mobile devies i mmwave 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 mmwave ommuiatio systems [44] We 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 etwork Fig 7 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 that is typially used i previous works [0], [4], [45] I partiular, sidelobes ad baklobes are igored i this diretioal atea model This is beause the sidelobes/baklobes are so small that the impats of them a be igored whe the mai beamwidth is small eg, 30 i [46] Besides, smart ateas ofte have ull apability that a almost elimiate the sidelobes ad baklobes [47] 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 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 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 We first derive the apaity of a MC-IS-DA etwork I partiular, the apaity of a MC-IS-DA etwork otributed by ifrastruture ommuiatios is the same as that of a typial MC-IS etwork With regard to the apaity otributed by ad ho ommuiatios, we eed to exted our aalysis i Setio V to a MC-IS-DA etwork Speifially, a MC-IS- DA etwork has differet apaity regios o the per-ode throughput apaity λ a, ompared with a MC-IS etwork We show the mai results as follows Corollary : The per-ode throughput λ for a MC-IS-DA etwork has four regios as follows i Whe Coetivity Coditio is satisfied, λ = Θ 4π H log + Θ mi{ b, bm C I }W I, where λa = Θ 4π H log ad λi = Θ mi{ b, bm C I }W I ; π ii Whe Iterferee Coditio is satisfied, λ = Θ + Θmi{ b C A H log, bm C I }W I, where λ a = π Θ ad λ i = Θ mi{ b, bm C I }W I ; C A H log iii Whe Destiatio-bottleek Coditio is satisfied, λ = loglogh Θ log + Θmi{ b H log logh log, bm C I }W I, loglogh where λ a = Θ log ad H log logh log λ i = Θ mi{ b, bm C I }W I ; iv Whe Iterfae-bottleek Coditio is satisfied, λ = Θ H log WA +Θmi{ b, bm C I }W I, where λ a = Θ H log WA ad λ i = Θ mi{ b, bm C I }W I Proof The detailed proof is preseted i Appedix D 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 4π over a MC-IS etwork Whe Iterferee Coditio is satisfied, a MC-IS- DA etwork has a apaity gai π 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 Delay of a MC-IS-DA etwork Reall i Setio VI-C that we eed to have C I m so that the maximum throughput apaity otributed by ifrastruture ommuiatios a be ahieved We usually have C I = m so that there is o waste of iterfaes It implies 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