IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 12, DECEMBER

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER On Spatal Capacty of Wreless Ad Hoc Networks wth Threshold Based Schedulng Yue Lng Che, Student Member, IEEE, Ru Zhang, Member, IEEE, Y Gong, Senor Member, IEEE, and Lngje Duan, Member, IEEE Abstract Ths paper studes spatal capacty n a stochastc wreless ad hoc network. We propose a novel sgnal-tonterference-rato SIR threshold based schedulng scheme wth mult-stage probng and data transmsson, where each transmtter teratvely decdes to further probe or stay dle, dependng on whether the estmated SIR n the proceedng probng s no smaller than a predefned threshold. Though the locatons of the ntal transmtters can be modeled as a homogeneous Posson Pont Process PPP, the SIR based schedulng makes the PPP model no longer applcable n the subsequent probng and data transmsson phases. We frst focus on sngle-stage probng and fnd that when the SIR threshold s set suffcently small to assure an acceptable network nterference level, the proposed scheme can greatly outperform the reference scheme wthout any transmsson schedulng n terms of spatal capacty. We clearly characterze the spatal capacty wth exact/approxmate closed-form expressons, by proposng a new approxmate approach to deal wth the correlated SIR dstrbutons over non-ppps. Then, we successfully extend to mult-stage probng, by properly desgnng the multple SIR thresholds to assure gradual mprovement of the spatal capacty. Furthermore, we analyze the mpact of mult-stage probng overhead and present a probng-capacty tradeoff n schedulng desgn. Fnally, extensve numercal results are presented to demonstrate the schedulng performance. Index Terms Wreless ad hoc network, threshold based schedulng, spatal capacty, stochastc geometry. I. INTRODUCTION WIRELESS ad hoc networks have emerged as a promsng technology that can provde seamless communcaton between wreless users transmtter-recever pars wthout relyng on any pre-exstng nfrastructure. In such networks, the Manuscrpt receved December 0, 203; revsed June 20, 204; accepted October 6, 204. Date of publcaton October 4, 204; date of current verson December 8, 204. Ths work was presented n part at the IEEE Internatonal Symposum on Informaton Theory ISIT Istanbul, Turkey, July 7 2, 203. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was A. A. Abouzed. Correspondng author: Y. Gong. Y. L. Che s wth the School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore e-mal: chey0004@ ntu.edu.sg. R. Zhang s wth the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore, 7583, and also wth the Insttute for Infocomm Research, A STAR, Sngapore e-mal: elezhang@nus.edu.sg. Y. Gong s wth the Department of Electrcal and Electronc Engneerng, South Unversty of Scence and Technology of Chna, Shenzhen 58055, Chna emal: gong.y@sustc.edu.cn. L. Duan s wth the Engneerng Systems and Desgn Pllar, Sngapore Unversty of Technology and Desgn, Sngapore e-mal: lngje_duan@ sutd.edu.sg. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Object Identfer 0.09/TWC wreless users communcate wth each other n a dstrbuted manner. Due to the lack of centralzed coordnators to coordnate the transmssons among the users, the wreless ad hoc network s under compettve and nterference-domnant envronment n nature. Thereby, effcent transmsson schemes for transmtters to effectvely schedule/adapt ther transmssons are appealng for system performance mprovement, and thus have attracted wde research attentons n the past decade. Tradtonally, each transmtter s enabled to ndependently decde whether to transmt over a partcular channel based on ts own wllngness or channel strength [] [4], and the transmsson rate of each user can be maxmzed by fndng an optmal transmsson probablty or an optmal channel strength threshold, respectvely. Although easy to be mplemented, such ndependent transmsson schemes do not consder the resultng user nteractons n the wreless ad hoc networks due to the co-channel nterference, and thus do not acheve hgh system performance n general cases. Therefore, more complex transmsson schemes have been proposed to explot the user nteractons by explorng the nformaton of sgnal-tonterference-rato SIR. For example, by teratvely adaptng the transmt power level based on the estmated SIR, the Foschn-Mljanc algorthm [5] assures zero outage probablty and/or mnmum aggregate power consumpton for uplnk transmsson n a cellular network. In [6], Yates has studed power convergence condtons for such teratve power control algorthms. Moreover, there have been some recent studes e.g. [7] and [8] that extend the Foschn-Mljanc algorthm to the wreless ad hoc network through jont schedulng and power control transmsson schemes. In addton, by adaptng the transmsson probablty dependng on the receved SIR, [9] has studed varous random access schemes to mprove the system throughput and/or the user farness. However, [5] [9] ether requre each transmtter to know at least the wreless envronment nformaton of ts neghbors, or are of hgh mplementaton complexty, and thus are not approprate for practcal large-scale wreless ad hoc networks. On the other hand, due to the randomzed locaton of each transmtter and the effects of channel fadng, the network-level performance analyss s fundamentally mportant for the study of wreless ad hoc networks. It s noted that Gupta and Kumar n [0] studed scalng laws, whch quantfed the ncrease of the volume of capacty regon over the number of transmtters n ad hoc networks. Moreover, to determne the set of actve transmtters that can yeld maxmum aggregate Shannon capacty n the network, the authors n [] [3] addressed the capacty maxmzaton problem for an arbtrary wreless ad hoc IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 696 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 Fg.. Illustraton of the SIR-threshold based transmsson n P-Phase k: If the SIR n P-Phase k s no smaller than the threshold γ k 0 n P-Phase k, the transmtter decdes to transmt n P-Phase k; otherwse, t stays dle n the remanng tme of ths slot. network. However, [0] [3] dd not consder the mpact of spatal confguraton of the ad hoc network, whch s a crtcal factor that determnes the ad hoc network capacty [4]. It came to our attenton that as a powerful tool to capture the mpact of wreless users spatal randomness on the network performance, stochastc geometry [5] s able to provde more comprehensve characterzaton of the performance of wreless networks, and thus has attracted great attentons from both academy and ndustry [4], [6]. Among all the tools provded by stochastc geometry, homogeneous Posson pont process PPP [7] s the most wdely used one for network topology modelng and performance analyss. Under the assumpton that the transmtters are deployed accordng to a homogeneous PPP, the exact/approxmate capacty of a wreless ad hoc network under varous ndependent transmsson schemes, such as Aloha-based random transmsson [], channel-nverson based power control [2], and channel-threshold based schedulng [3], [4], can all be successfully characterzed by usng advanced tools from stochastc geometry. However, lmted work based on stochastc geometry has studed SIR-based transmsson schemes, where the user nteractons are nvolved. It s noted that [8] studed a probablty-based schedulng scheme, where each transmtter ndependently adjusts ts current transmsson probablty based on the receved SIR n the proceedng teraton. However, [8] only studed the convergence of the probablty-based schedulng, wthout addressng the network capacty wth spatalty dstrbuted users. To our best knowledge, there has been no exstng work on studyng the wreless ad hoc network capacty wth a SIR-based transmsson scheme. Hence, the mpact of SIR-based transmssons s lmtedly understood from the network-level pont of vew. A prncpal goal of ths study s to use stochastc geometry to fll the vod of wreless network capacty characterzaton by an effcent SIR-based transmsson scheme. To ths end, we propose a novel SIR-threshold based schedulng scheme for a sngle-hop slotted wreless ad hoc network. We consder a probe-and-transmt protocol, where mult-stage probngs are sequentally performed to gradually determne the transmtters that are allowed to transmt data n each slot. Specfcally, we assume there are n total N probng phases and one data transmsson phase n each slot, N<. We sequentally label the N probng phases as P-Phase 0, P-Phase,..., and P-Phase N, and label the data transmsson phase as D-Phase. As llustrated n Fg., f the feedback SIR from recever n P-Phase k, k N,snosmallerthan a pre-defned threshold, transmtter decdes to transmt n P-Phase k; otherwse, to mprove the system throughput as well as save ts own energy, transmtter stays dle n the remanng tme of the slot as n [9], so as to let other transmtters that have hgher SIR levels re-contend the current transmsson opportunty. Snce each transmtter only requres drect-channel SIR feedback from ts ntended recever for lmted tmes, the proposed scheme s approprate for dstrbuted mplementaton n practcal wreless ad hoc networks. In ths paper, we characterze the wreless ad hoc network capacty wth a metrc called spatal capacty, whch has been used n [20] and gves the average number of successful transmtters per unt area for any gven ntal transmtter densty. We am at closed-form spatal capacty characterzaton and maxmzaton by explorng the SIR-threshold based transmsson. The key contrbutons of ths paper are summarzed as follows. Novel SIR-threshold based schedulng scheme: In Secton II, we propose a novel SIR-threshold based transmsson scheme for a sngle-hop wreless ad hoc network, whch can be mplemented effcently n a dstrbuted manner. Though one can use a homogeneous PPP to model the stochastc locatons of the transmtters n the ntal probng phase, we fnd that due to the teratve SIRbased schedulng, the PPP model s no longer applcable to model the locatons of the retaned transmtters n all the subsequent probng or data transmsson phases. Furthermore, snce the SIR dstrbutons n all the probng and data transmsson phases are strongly correlated, t s challengng to analyze/characterze the spatal capacty of the proposed scheme. Sngle-stage probng for spacal capacty mprovement:in Secton III, we start up wth sngle-stage probng N = to clearly decde the SIR threshold for the proposed scheme and characterze the spatal capacty. We show that a small SIR threshold can effcently reduce the retaned transmtter number and thus the nterference level n the data transmsson phase, whle a large SIR threshold wll overly reduce the retaned transmtter number and does not help mprove the spatal capacty. We also propose a new approxmate approach to characterze the spatal capacty n closed-form, whch s useful for analyzng performance of wreless networks wth nteracted transmtters.

3 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 697 Mult-stage probng for spatal capacty mprovement: In Secton IV, we extend proposed schedulng scheme from the sngle-stage probng N =to mult-stage probng N > for greater spatal capacty mprovement. We show that once a sequence of ncreasng SIR thresholds are properly decded over probng phases, the spatal capacty s assured to gradually mprove. As mult-stage probng can ntroduce non-gnorable overhead n each tme slot, whch reduces the spatal capacty, we study an nterestng probng-capacty tradeoff over the probng-stage number N. Performance evaluatons for network desgn: In both Sectons III and IV, we also provde extensve numercal results to further evaluate the mpact of key parameters of the proposed scheme. In partcular, we present a denstycapacty tradeoff n Secton III-C, whch shows that a small ntal transmtter densty can help mprove the spatal capacty, whle a large one wll ntroduce hgh nterference level and thus reduce the spatal capacty. To hghlght the spatal capacty mprovement performance of the proposed scheme, we also compare the proposed scheme wth exstng dstrbuted schedulng schemes n Secton III-C2. Moreover, we consder a practcal scenaro wth SIR estmaton and feedback errors and show that the proposed desgn s robust to the SIR errors n Secton III-C3 by smulaton. In Secton IV, we study an example wth N =2and show the correspondng spatal capacty over both SIR thresholds n P-Phase and D-Phase. Interestngly, our numercal results show that the former SIR threshold plays a more crtcal role n determnng the spatal capacty than the latter one, snce the former SIR threshold decdes how many transmtters can have a second chance to contend the transmsson opportunty. It s noted that some of the exstng work has addressed the throughput/capacty analyss of a wreless communcaton system from the nformaton-theoretc pont of vew. For example, Tse and Hanly consdered a multpont-to-pont system and characterzed the throughput capacty regon and delaylmted capacty regon of the fadng multple-access channel n [2] and [22], respectvely, where the optmal power and/or rate allocaton that can acheve the boundary of the capacty regons was derved. Although appealng, both [2] and [22] have assumed multuser detecton at a centralzed recever and gnored the mpact of the random network topology drven by moble transmtters and recevers moblty, and thus cannot completely provde network-level system performance characterzaton wth dstrbuted sngle-user detecton.e., treatng the multuser nterference as nose recevers. Unlke Tse and Hanly s works n [2] and [22], we use stochastc geometry to model the large-scale random wreless ad hoc network topology, and novelly analyze the network-level performance of the teratve SIR-threshold based schedulng. In addton, t s also noted that some exstng work has adopted tools from stochastc geometry to study the non- PPP based wreless network. For example, by usng a PPP to approxmate the underlyng non-ppp based spatal dstrbuton of the transmtters locatons, [23] [27] have successfully characterzed the non-ppp based wreless network capacty. Unlke [23] [27], due to the teratve SIR-based schedulng of the proposed scheme, we need to address not only the non- PPP based spatal dstrbuton of the transmtters locatons, but also the resultng strongly-correlated SIR dstrbutons over all probng and data transmsson phases. To our best knowledge, such correlated SIR analyss/chracterzaton n non-ppp based wreless networks has not been addressed n the exstng work based on stochastc geometry. II. SYSTEM MODEL AND PERFORMANCE METRIC In ths secton, we descrbe the consdered transmsson schemes n ths paper. We then develop the network model based on stochastc geometry. At last, we defne the spatal capacty as our performance metrc. A. Transmsson Schemes We focus on the proposed scheme wth SIR-threshold based schedulng. For comparson, we also consder a reference scheme wthout any transmsson schedulng. For both transmsson schemes, we assume that all transmtters transmt n a synchronzed tme-slotted manner. We also assume that all transmtters transmt at the same power level, whch s normalzed to be unty for convenence. SIR-Threshold Based Scheme: Based on the probe-andtransmt protocol, n each tme slot, N probng phases wth N< are sequentally mplemented before the data transmsson phase. We assume N s a pre-gven parameter and ts effects wll be studed later n Secton IV-B. Moreover, as shown n Fg., we denote the duraton of a tme slot and a probng phase as T and τ, respectvely, wth τ T, such that Nτ < T, as n [9]. By normalzng over T,theeffectve data transmsson tme n a tme slot s obtaned as T Nτ T, whch reduces lnearly over N [28]. Furthermore, we assume f a transmtter transmts probng sgnals n a probng phase, ts ntended recever s able to measure the receved sgnal power over the total nterference power,.e., the SIR, and feeds t back to the transmtter at the end of the probng phase. The specfc algorthm desgn on SIR estmaton and feedback s out of the scope of ths paper and s not our focus. To obtan tractable analyss, we assume perfect SIR estmaton and feedback n ths paper, and thus the SIR value s exactly known at the transmtter; however, the mpact of fnte SIR estmaton and feedback errors on the network capacty s mportant to practcal desgn and thus wll also be evaluated by smulaton. Accordng to the feedback SIR level of ts own channel, each transmtter teratvely performs the threshold-based transmsson decson n each P-Phase or D-Phase, for whch the detals are gven as follows: In the ntal probng phase,.e., P-Phase 0, to ntalze the communcaton between each transmtter and recever In general, each transmtters can transmt at dfferent power levels by teratvely adjustng ts transmt power based on the feedback SIR nformaton, as n [5] or [6]. However, n ths paper, we manly focus on SIR-based transmsson schedulng and thus restrct transmt power adaptaton to be bnary for smplcty.

4 698 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 par, all transmtters ndependently transmt probng sgnals to ther ntended recevers. Each recever then estmates the channel ampltude and phase for possble coherent communcaton n the subsequent probng and data transmsson phases, and measures the receved SIR of the probng sgnal. Each transmtter receves the feedback SIR from ts ntended recever at the end of P-Phase 0. In each of the remanng probng phases from P-Phase to P-Phase N, by explotng the feedback SIR n the proceedng probng phase, each transmtter decdes whether to transmt n the current probng phase wth a predefned SIR threshold. Specfcally, suppose a transmtter transmts n P-Phase k, k N.Asshown n Fg., f the feedback SIR n P-Phase k s larger than or equal to the predefned SIR-threshold, denoted by γ k 0 for P-Phase k, the transmtter contnues ts transmsson n P-Phase k and thus receves the feedback SIR n P-Phase k; otherwse, to mprove the system throughput as well as save ts energy, the transmtter decdes not to transmt any more n the remanng tme of ths slot and wll seek another transmsson opportunty n the next slot, so as to let other transmtters that have hgher SIR levels to re-contend the current transmsson opportunty. In the D-Phase, smlar to the SIR-threshold based schedulng from P-Phase to P-Phase N, f a transmtter transmts n P-Phase N and ts feedback SIR n P-Phase N s larger than or equal to the predefned threshold, denoted by γ N 0 for the D-Phase, the transmtter sends data to ts ntended recever; otherwse, the transmtter remans slent n the rest tme of ths slot. The data transmsson s successful f the SIR at the recever s larger than or equal to the requred SIR level, denoted by β>0. 2 Reference Scheme: There s no transmsson schedulng n the reference scheme. In each tme slot, we assume all transmtters transmt data drectly to ther ntended recevers n an ndependent manner. Thus, the effectve data transmsson tme for the reference scheme s. 2 The data transmsson s successful f the SIR at the recever s larger than or equal to the requred SIR level β as the proposed scheme. Note that by mplementng an ntal probng phase before the data transmsson, the reference scheme can be mproved to be a proposed scheme wth sngle-stage probng. B. Network Model In the next, we develop the network model based on stochastc geometry. For both consdered transmsson schemes, we focus on sngle-hop communcaton n one partcular tme slot. For both schemes, we assume that all transmtters are ndependently and unformly dstrbuted n the unbounded twodmensonal plane R 2. We thus model the locatons of all the 2 It s worth pontng out that for the reference scheme, an ntal tranng s needed pror to data transmsson for the recever to estmate the channel for coherent communcaton, smlar to the ntal probng of the proposed scheme wth N =, but wthout the SIR feedback to the transmtter. Here, we have assumed that such tranng ncurs a neglgble tme overhead as compared to each slot duraton. transmtters by a homogeneous PPP wth densty λ. Due to the lack of central nfrastructure for coordnaton n the wreless ad hoc network, we assume the transmtters have no knowledge about ther surroundng wreless envronment, and thus ntend to transmt ndependently n a tme slot wth probablty θ 0,, as n [] [4]. Denote λ 0 = λθ as the densty of the ntal transmtters that have the ntenton to transmt n a partcular tme slot. Accordng to the Colorng theory [5], the process of the ntal transmtters for both schemes s a homogeneous PPP wth densty λ 0, whch s denoted by Φ 0. Wthout loss of generalty, we assume λ and θ and hence λ 0 are gven parameters, and wll dscuss the effects of λ 0 later n Secton III-C. We assume each transmtter has one ntended recever, whch s unformly dstrbuted on a crcle of radus d meters m centered at the transmtter. We denote the locatons of the -th transmtter and ts ntended recever as x, wth x Φ 0, and r not ncluded n Φ 0, respectvely. The path loss between the -th transmtter and the j-th recever s gven by l j = x r j α, where α>2 s the path-loss exponent. We use h j to denote the dstancendependent channel fadng coeffcent from transmtter to recever j. We assume flat Raylegh fadng, where all h j s are ndependent and exponentally dstrbuted random varables wth unt mean. We also assume that h j s do not change wthn one tme-slot. We denote the SIR at the -th recever as SIR 0, whch s gven by SIR 0 h d α = x j Φ 0,j h. jl j Note that for the reference scheme wthout transmsson schedulng, SIR 0 gves the receved SIR level at the -th recever for the data transmsson of transmtter. As a result, n the reference scheme, the data transmsson of transmtter s successful f SIR 0 β s satsfed. Unlke the reference scheme, n the proposed scheme, SIR 0 only gves the receved SIR level at the -th recever n the ntal probng phase P-Phase 0. We then denote the pont process formed by the retaned transmtters n P-Phase k wth k N, or the D-Phase wth k = N, asφ k.wealso denote SIR k as the receved SIR at the -th recever n Φ k. Clearly, we have Φ k = {x Φ k : SIR k γ k }, where the number of transmtters n Φ k s reduced as compared to that n Φ k. Thus, t s easy to verfy that SIR k SIR k for any gven γ k 0, Φ k Φ k. Moreover, smlar to SIR 0, for any Φ k, k {,...,N}, we can express SIR k as SIR k h d α = x j Φ k,j h, k {,...,N}. 2 jl j It s worth notng that due to the SIR-based schedulng, the transmtters are not retaned ndependently n Φ k. Thus, unlke SIR 0 n, whch s determned by the homogeneous PPP Φ 0,SIR k n 2 s determned by the non-ppp Φ k n general [5]. For the proposed scheme, the data transmsson of transmtter s successful f SIR k γ k, k {,...,N}, and β are all satsfed. SIR N

5 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 699 C. Spatal Capacty Due to the statonarty of the homogeneous PPP Φ 0, t s easy to verfy that Φ k, k {,...N}, s also statonary [4]. We thus consder a typcal par of transmtter and recever n ths paper. Wthout loss of generalty, we assume that the typcal recever s located at the orgn. The typcal par of transmtter and recever s named par 0,.e., =0. Denote the successful transmsson probablty of the typcal par n the data transmsson phase of the proposed scheme wth N probng phases or the reference scheme as P p,n 0 or P0 r, respectvely. We thus have P p,n 0 = P SIR 0 0 γ,...,sir N 0 γ N, SIR N 0 β. 3 P0 r = P SIR 0 0 β. 4 We adopt spatal capacty as our performance metrc, whch s defned as the spatal densty of successful transmssons, or more specfcally the average number of transmtters wth successful data transmsson per unt area. Consderng the effectve data transmsson tme n a tme slot, we thus defne the spatal capacty by the proposed scheme wth N probng phases and the reference scheme as C p,n and C r, respectvely, gven by C p,n Δ T Nτ = λ 0 P p,n 0, 5 T C r Δ = λ0 P0 r. 6 For the reference scheme, t s noted that P r,gvenn4, s the complementary cumulatve dstrbuton functon CCDF of SIR 0 0 taken at the value of β. We then have the followng proposton. Proposton 2.: The successful transmsson probablty n the reference scheme s P r 0 =exp πλ 0 d 2 β 2 α ρ, 7 where ρ = 0 dv. When α =4,wehaveρ = π +v α/2 2. The proof of Proposton 2. s smlar to that of [30, Theorem 2], whch s based on the probablty generatng functonal PGFL of the PPP, and thus s omtted here. Snce the network nterference level n the D-Phase ncreases over the ntal transmtter densty λ 0, we fnd that P0 r n 7 monotoncally decreases over λ 0 as expected. Moreover, from 6 and 7, we can obtan the expresson of C r as C r = λ 0 exp πλ 0 d 2 β 2 α ρ. 8 It s observed from 8 that unlke P0 r, the spatal capacty C r does not vary monotoncally over λ 0, snce C r can be benefted by ncreasng λ 0 f the resultng nterference s acceptable. Moreover, from 7 and 8, t s also expected that both P0 r and C r monotoncally decrease over the dstance d between each transmtter and recever par, due to the reduced sgnal power receved at the recever, and decrease over the requred SIR level β. Unlke the reference scheme, whch s determned by the homogeneous PPP Φ 0, the proposed scheme s jontly determned by Φ 0 and a sequence of non-ppps {Φ k }, k N, where the resultng SIR dstrbutons are correlated. Therefore, t s very dffcult to analyze/characterze the spatal capacty of the proposed scheme wth N probng phases. To start up, n the next secton, we focus on a smple case wth sngle-stage probng N =for some nsghtful results. III. SIR-THRESHOLD BASED SCHEME WITH SINGLE-STAGE PROBING In ths secton, we consder the proposed scheme wth snglestage probng,.e., N =. In ths case, there s only one round of SIR-based schedulng, whch s mplemented wth the threshold γ. For notatonal smplcty, for the case of N =, we omt the superscrpt N and use P p 0 and Cp to represent the successful transmsson probablty and the spatal capacty of the typcal transmtter, respectvely. Based on 3, the successful transmsson probablty for the case of N =s reduced to P p 0 = P SIR 0 0 γ, SIR = P SIR 0 0 γ P 0 β 9 SIR 0 β SIR 0 0 γ. 0 Moreover, when N =, the effectve data transmsson tme for the proposed scheme s T τ T. Snce τ T, we assume the sngle-stage probng overhead s neglgble; and thus, the effectve data transmsson tme becomes as the reference scheme. Consequently, based on 5, we can express the spatal capacty C p 0 as C p = λ0p p 0. Furthermore, by substtutng 0 to, we can express C p alternatvely as C p = λ 0 P SIR 0 0 γ P SIR 0 β SIR 0 0 γ = λ P SIR 0 β SIR 0 0 γ 2 where λ = λ 0 PSIR 0 0 γ s the densty of Φ n the D-Phase, wth λ λ 0. Based on Proposton 2., by replacng β wth γ, t s easy to fnd that λ = λ 0 exp πλ 0 d 2 γ 2 α ρ. 3 In the followng two subsectons, we compare the spatal capacty of the two consdered schemes, and characterze C p for the proposed scheme. A. Spatal Capacty Comparson and Closed-Form Characterzaton Wth γ =0and γ β In ths subsecton, we compare the spatal capacty of the proposed scheme wth that of the reference scheme. We then characterze the spatal capacty C p for the proposed scheme and obtan closed-form expressons for the cases of γ =0 and γ β. Frst, from 6 and, to compare C p and C r,thekeys to compare P p 0 and Pr 0. In the reference scheme, denote the

6 6920 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 total nterference power receved at the typcal recever as I 0 = x Φ 0, 0 h 0l 0. In the proposed scheme, the receved total nterference power at the typcal recever n P-Phase 0 s thus I 0, whle that n the D-Phase s gven by I = x Φ, 0 h 0l 0. For any γ 0, wehavei 0 I snce Φ Φ 0, and thus SIR SIR 0. As a result, by changng over the value of γ [0,, we obtan the followng proposton. Proposton 3.: Gven the requred SIR level β>0, for any γ [0,, wehave C p > C r, f 0 <γ <β C p = C r, conservatve transmsson regme f γ =0or γ = β neutral transmsson regme C p < C r, f γ >β aggressve transmsson regme. 4 Proof: Due to the space lmtaton, we omt the proof here, whch can be found n a longer verson of ths paper [3]. Remark 3.: Compared to the spatal capacty of the reference scheme, Proposton 3. shows that for the proposed scheme wth SIR-threshold based schedulng, due to the reduced nterference level n the D-Phase, the spatal capacty s mproved n the conservatve transmsson regme wth 0 < γ <β. However, n the case of the aggressve transmsson regme wth γ >β, where the transmtters that are able to transmt successfully n the D-Phase may also be removed from transmsson, the retaned transmtters n the D-Phase are overly reduced. Consequently, the spatal capacty s reduced n the aggressve transmsson regme. It s also noted that n the neutral transmsson regme wth γ =0 or γ = β, the spatal capacty s dentcal for the two schemes. At last, t s worth notng that Proposton 3. holds regardless of the specfc channel fadng dstrbuton and/or transmtter locaton dstrbuton. Next, we characterze the spatal capacty C p for the proposed scheme wth N =. We focus on dervng the successful transmsson probablty P p 0 n 4. Unlke Pr 0 n 4, whch s gven by the margnal CCDF of SIR 0 0 taken at value β, P p 0 s gven by the jont CCDF of SIR0 0 and SIR 0 taken at values γ,β. In the followng, we consder three cases γ =0, γ β, and 0 <γ <β, and fnd closed-form spatal capacty expressons for both cases of γ =0and γ β. Specfcally, for the smple case wth γ =0, we can nfer from Proposton 3. drectly that C p = C r, whch s gven n 8. For the case of γ β, snce SIR 0 > SIR 0 0, we have PSIR 0 β SIR 0 0 γ =. Accordng to 0, we thus obtan P p 0 = PSIR0 0 γ n ths case. By replacng β wth γ n Proposton 2., we further obtan that P p 0 = exp πλ 0 d 2 γ 2 α ρ. As a result, based on, we can express C p for the case of γ β as C p = λ 0 exp πλ 0 d 2 γ 2 α ρ. 5 Smlar to C r gven n 8, t s observed that C p for both cases of γ =0and γ β does not vary monotoncally over λ 0, but monotoncally decreases over the dstance d between each transmtter and recever par. Moreover, unlke C r and C p for γ =0, C p for γ β s not related to the requred SIR level β any more, snce all the retaned transmtters n the D-Phase meet the condton SIR β n ths case. However, for the case of 0 <γ <β, P p 0 cannot be smply expressed by a margnal CCDF of SIR 0 0 as n the above two cases. Moreover, from 4, due to the correlaton between SIR 0 0 and SIR 0 as well as the underlyng non-ppp Φ that, t s very dffcult, f not mpossble, to fnd an exact expresson of P p 0 and thus Cp n ths case. As a result, n the next subsecton, we focus on fndng a tght approxmate to C p wth a tractable expresson for the case of 0 <γ <β. determnes SIR 0 B. Approxmate Approaches for Spatal Capacty Characterzaton Wth 0 <γ <β Ths subsecton focuses on approxmatng the spatal capacty of the proposed scheme for the case of 0 <γ <β. We frst propose a new approxmate approach for C p and obtan an ntegral-based expresson. Next, to fnd a closed-form expresson for C p, we further approxmate the ntegral-based expresson obtaned by the proposed approach. At last, we apply the conventonal approxmate approach n the lterature and dscuss ts approxmate performance. The detals of the three approxmate approaches are gven as follows. Proposed Approxmaton: From 4, to fnd a good approxmate to P p 0 and thus Cp, the key s to fnd a good approxmate to the jont SIR dstrbutons n Φ 0 and Φ. Snce Φ Φ 0, we frst dvde the ntal PPP Φ 0 nto two dsjont non-ppps: one s Φ, and the other s ts complementary set Φ c =Φ 0 Φ, whch s the pont process formed by the nonretaned transmtters n the D-Phase. We denote the densty of Φ c as λ c = λ 0 λ. Clearly, Φ and Φ c are mutually dependent. Denote the receved SIR level at the typcal recever n Φ c as SIR,c 0 = h 00 d α / Φ h c 0 l 0. Snce Φ Φ c =Φ 0 and Φ Φ c =, wehave/sir 0 0 =/SIR 0 + SIR,c 0. As a result, 4 can be equally represented by usng the jont dstrbutons of SIR 0 and SIR,c 0. Next, we state an assumpton, based on whch we can use a homogeneous PPP to approxmate Φ and Φ c, respectvely, such that the exstng results on PPP nterference dstrbuton n the lterature can be appled to approxmate the jont dstrbutons of SIR 0 and SIR,c 0. Assumpton : In the proposed scheme wth N =,the transmtters are retaned ndependently n the D-Phase, wth probablty PSIR 0 0 γ. By applyng Assumpton, we denote the resultng pont processes formed by the retaned and non-retaned transmtters n the D-Phase as ˆΦ and ˆΦ c, respectvely. Clearly, both ˆΦ and ˆΦ c are homogeneous PPPs. Moreover, the densty of ˆΦ or ˆΦ c s the same as that of Φ or Φ c, respectvely. Snce the two homogeneous PPPs ˆΦ and ˆΦ c are dsjont, they are ndependent of each other [5]. Denote Î = ˆΦ h 0 l 0 and Î c = ˆΦ h c 0 l 0 as the receved nterference power at the typcal recever n ˆΦ and ˆΦ c, respectvely. We then use fî x and fîc x 2 to denote the probablty densty functons pdfs of

7 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 692 Î and Îc, respectvely. The followng lemma gves the general nterference pdf n a homogeneous PPP-based network wth Raylegh fadng channels, whch s a well-known result n the lterature e.g., [32]. Lemma 3.: For any homogeneous PPP of densty λ 0, f the channel fadng s Raylegh dstrbuted, the pdf of the receved nterference I at the typcal recever s gven by f I x = + Γ + δsnπδ λπ 2 δ πx! x δ, snπδ = 6 where δ =2/α. Moreover, when α =4, 6 can be further expressed n a smpler closed-form as f I x = λ 4 π x 3 2 exp π4 λ 2 6x. 7 As a result, based on Lemma 3., by substtutng λ = λ to 6 and 7, we can obtan fî x for the cases of general α and α =4, respectvely. Smlarly, wth λ = λ c, from 6 and 7 we can obtan fîc x 2 for general α and α =4, respectvely. Therefore, by approxmatng Φ and Φ c by ˆΦ and ˆΦ c, respectvely, we can easly approxmate the jont dstrbuton of SIR 0 and SIR,c 0 based on the nterference pdfs fî x and fîc x 2, and thereby obtan an ntegral-based approxmate to P p 0 n the followng proposton. Proposton 3.2: The successful transmsson probablty by the proposed scheme for the case of 0 <γ <β s approxmated as P p 0 0 e h 00 h 00 βd α 0 h 00 γ d fî x α x fîc x 2 dx 2 dx dh Proof: Please refer to [3]. Fnally, by multplyng λ 0 wth the rght-hand sde of 8, we obtan an ntegral-based approxmate to C p for the case of 0 <γ <βas h 00 e h βd α 00 h 00 γ d α x C p λ 0 fî x fîc x 2 dx 2 dx dh Note that the proposed approxmate approach consders the correlaton between SIR 0 0 and SIR 0, and only adopts PPPbased approxmaton to approxmate Φ and Φ c by ˆΦ and ˆΦ c, respectvely. Snce t has been shown n the lterature e.g., [23] [26] that such PPP-based approxmaton can provde tght approxmate to the correspondng non-ppp, the proposed approxmate approach s able to provde tght spatal capacty approxmate to C p for the case of 0 <γ <β. 2 Closed-Form Approxmaton for 9: Although the spatal capacty expresson obtaned n 9 s easy to ntegrate, t s not of closed-form. Thus, based on 8, we focus on fndng a closed-form approxmate to P p 0 and thus Cp. We frst ncrease the upper lmt of fîc x 2 n 8 from γ d α x to γ d α to obtan an upper bound for the rght-hand sde of 8. Then by properly lower-boundng the obtaned upper bound based on Chebyshev s nequalty [33], we obtan a closedform approxmate to P p 0, whch s shown n the followng proposton. Proposton 3.3: Based on the ntegral-based expresson gven n 8, a closed-form approxmate to P p 0 for the case of 0 <γ<βs obtaned as P p 0 exp πλ d 2 β 2 α ρ exp πλ c d 2 γ 2 α ρ. 20 Proof: Please refer to [3]. From, 3, and 20, we obtan a closed-form approxmate to spatal capacty of the proposed scheme for the case of 0 <γ <βas C p λ 0 exp πλ 0 d 2 γ 2 α ρ [ ] exp πλ 0 exp πλ 0 d 2 γ 2 α ρ d 2 β 2 α ρ [ ] exp πλ 0 exp πλ 0 d 2 γ 2 α ρ d 2 γ 2 α ρ. 2 3 Conventonal Approxmaton: It s noted that the conventonal approxmate approach n the lterature e.g., [23] [26], whch only focuses on dealng wth the non-ppp Φ, can often yeld a closed-form expresson. Thus, n the followng, we apply the conventonal approxmate approach and dscuss ts approxmate performance to C p. Frst, snce only the performance n Φ s concerned by the conventonal approxmate approach, t takes PSIR 0 β as the successful transmsson probablty of the typcal transmtter n the D-Phase. Next, the non-ppp Φ s approxmated by the homogeneous PPP ˆΦ under Assumpton. We denote the receved SIR at the typcal recever n ˆΦ as SIR ˆ 0 = h 00 d α / ˆΦ h 0 l 0. Thus, PSIR 0 β s approxmated by PSIR ˆ 0 β. At last, by adoptng the product of λ and PSIR ˆ 0 β as an approxmate to the spatal capacty C p,a closed-form approxmate to C p for the case of 0 <γ <β s obtaned as C p λ P SIR ˆ 0 β a = λ 0 exp πλ 0 d 2 γ 2 α ρ [ ] exp πλ 0 exp πλ 0 d 2 γ 2 α ρ d 2 β 2 α ρ where a follows by Proposton 2. and 3. Note that snce λ = λ 0 PSIR 0 0 γ, we can rewrte 22 as C p λ 0 PSIR 0 0 γ PSIR ˆ 0 β under the conventonal method. However, accordng to the defnton of C p for N =, whch s gven n 9 and, we have C p = λ 0 PSIR 0 0 γ, SIR 0 β, where the dstrbuton of SIR 0 s strongly dependent on that of SIR 0 0 as Φ Φ 0. As a result, the conventonal approxmate approach only focuses on the PPPbased approxmate to Φ, but gnores the dependence between Φ 0 and Φ. Therefore, 22 does not hold for representng, or reasonably approxmatng, the spatal capacty of the proposed scheme. In addton, by comparng 2 and 23, t s observed that for the case of 0 <γ <β, gven any λ 0 > 0 and d>0, the closed-form spatal capacty obtaned based on the proposed

8 6922 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 Fg. 2. Spatal capacty aganst γ wth λ 0 = and β =2.5. Fg. 3. Spatal capacty aganst λ 0. γ =0.6. β =2.5. approach s always outperformed that by the conventonal approach. C. Numercal Results Numercal results are presented n ths subsecton. Accordng to the method descrbed n [5], we generate a spatal Posson process, n whch the transmtters are placed unformly n a square of [0 m, 600 m] [0 m, 600 m]. To take care of the border effects, we focus on samplng the transmtters that locate n the nterm square of [200 m, 400 m] [200 m, 400 m]. We calculate the spatal capacty as the average of the network capacty over 2000 ndependent network realzatons, where for each network realzaton, the network capacty s evaluated as the rato of the number of successful transmtters n the samplng square to the square area of m 2. Unless otherwse specfed, n ths subsecton, we set α =4, β =2.5, and d =0 m. We also observe by smulaton that smlar performance can be obtaned by usng other parameters. In the followng, we frst valdate our analytcal results on the spatal capacty of the proposed scheme and the reference scheme wthout schedulng. To hghlght the spatal capacty mprovement performance of the proposed scheme, we then compare the spatal capacty acheved by the propose scheme wth that by two exstng dstrbuted schedulng schemes: one s the probablty-based schedulng n [8], and the other s the channel-threshold based schedulng n [3] and [4]. At last, we consder a more practcal scenaro wth SIR estmaton and feedback errors, and show the effects of the SIR errors on the spatal capacty of the proposed scheme. Valdaton of the Spatal Capacty Analyss: We valdate our spatal capacty analyss n Secton III-A and B for both proposed and reference schemes. Fg. 2 shows the spatal capacty versus the SIR threshold γ, for both the reference scheme wthout transmsson schedulng and the proposed scheme wth SIR-based schedulng. We set the ntal transmtter densty as λ 0 =0.0025/m 2 n both schemes. The analytcal spatal capacty of the reference scheme s gven n 8. By comparng the smulaton results for the proposed scheme wth the analytcal results for the reference scheme, we observe that C r s constant over γ as expected. We also observe that when γ <β, C p > C r ; 2 when γ =0 or γ = β, C p = C r ; and 3 when γ >β, C p < C r. Ths s n accordance wth our analytcal results n Proposton 3.. Moreover, for the proposed scheme, we adopt 8 and 5 as the analytcal spatal capacty for the cases of γ =0and γ β, respectvely, and observe that the analytcal results of the spatal capacty ft well to the smulaton counterparts. Furthermore, for the case of 0 <γ <β of the proposed scheme, where only approxmate expressons for the spatal capacty are avalable, we compare the approxmate performance of the three approxmate approaches gven n Secton III-B. It s observed that the ntegral-based expresson by the proposed approxmate approach, gven n 9, provdes a tght approxmate to C p for the case of 0 <γ <β. In addton, as a cost of expressng n closed-form, 2 s not as tght as 9, but 2 stll provdes a close approxmate to C p for the case of 0 <γ <β. At last, t s observed that the closed-form expresson gven n 23 by the conventonal approxmate approach cannot properly approxmate C p for the case of 0 <γ <βas expected. Fg. 3 shows the spatal capacty versus the ntal transmtter densty λ 0 when γ <β.wesetγ =0.6. For the proposed scheme, smlar to the case n Fg. 2, we observe tght and close approxmates are provded by 9 and 2, respectvely, based on the proposed approxmate approach, whle mproper approxmate s provded by 23 based on the conventonal approxmate approach. Moreover, t s observed that the spatal capacty of the proposed scheme s always larger than that of the reference scheme, gven n 8, for all values of λ 0, whch s as expected from Proposton 3. snce γ <βn ths example. Furthermore, for both the proposed and reference schemes, we observe an nterestng densty-capacty tradeoff: by ncreasng λ 0, the spatal capacty frst ncreases due to more avalable

9 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 6923 Fg. 4. Spatal capacty comparson wth exstng dstrbuted schedulng schemes. γ =0.4. β =2.5. transmtters, but as λ 0 exceeds a certan threshold, t starts to decrease, due to the more domnant nterference effect. Thus, to maxmze the spatal capacty, under the system scenaro set n Fg. 3, the optmal λ 0 should be set as 0.003/m 2. 2 Performance Comparson Wth Exstng Dstrbuted Schemes: We consder two exstng dstrbuted schedulng schemes for performance comparson. The frst scheme s the teratve probablty-based schedulng as n [8]. Denote the transmsson probablty for transmtter n P-Phase k, k N, and the D-Phase as φ k or φ N, respectvely. For any k {,...,N}, [8] sets φ k SIR =mn k β,. Intutvely, [8] provdes a smple and proper way to teratvely adjust the transmsson probablty φ k. The second scheme s the channel-threshold based schedulng wth sngle-stage probng as n [3] and [4], where the receved nterference power s not nvolved n the transmsson decson and each transmtter decdes to transmt n the D-Phase f ts drect channel strength n P-Phase 0 s no smaller than a predefned threshold γ,.e., h γ. For a far comparson, we consder snglestage probng wth N =for all the proposed SIR-threshold based scheme, the probablty-based schedulng n [8], and the channel-threshold based schedulng n [3] and [4]. Fg. 4 shows the spatal capactes acheved by the proposed scheme, the probablty-based schedulng, the channelthreshold based schedulng, and the reference scheme wthout schedulng. To clearly show the effects of nvolvng nterference n the transmsson decson for the proposed scheme, we set γ = γ =0.4 for the channel-threshold based schedulng. We obtan the spatal capacty of the channel-threshold based schedulng by applyng ts exact expresson gven n [4]. Due to the lack of an exact spatal capacty expresson for the probablty-based schedulng, we obtan ts spatal capacty by smulaton. We lst our observatons from Fg. 4 as follows: SIR based schemes v.s. channel-threshold based scheme: It s observed that by adaptng the transmsson decson to the SIR, the acheved spatal capactes by both the proposed scheme and the probablty-based schedulng are always hgher than that by the channel-threshold based schedulng, where the nterference nformaton s not exploted. Moreover, the spatal capacty of the channel-threshold based schedulng s smaller than that of the reference scheme when λ 0 s small, and becomes larger when λ 0 s suffcently large. Ths s n sharp contrast to the cases of the proposed scheme and the probablty-based schedulng, whch always guarantee capacty mprovement over the reference scheme wthout schedulng. SIR-threshold based schedulng v.s. probablty-based schedulng: It s nterestng to observe that although both the proposed scheme and the probablty-based schedulng adapt the transmsson decson to the SIR, the acheved spatal capacty by the former scheme s always hgher than that by the latter one n ths smulaton. Ths s because that the proposed scheme assures the mprovement of the successful transmsson probablty of each retaned transmtter n the D-Phase, whle the probablty-based schedulng only assures such mprovement wth some probablty. Moreover, t s observed that the optmal ntal transmtter densty that maxmzes the spatal capacty of the proposed scheme s λ 0 =0.0036, whch s larger than that for the probablty-based schedulng locatng at λ 0 = Note that for the proposed scheme, a lower SIR threshold γ allows more transmtters to retan n the D-Phase, so as to have a second chance to transmt. Thus, by comparng the smulaton results of the proposed scheme n Fg. 4 wth that n Fg. 3, t s observed that the acheved optmal spatal capacty over λ 0 wth γ =0.4 n Fg. 4 s larger than that wth γ =0.6 n Fg. 3. In addton, for all the consdered schemes n Fg. 4, we observe a densty-capacty tradeoff, whch s smlar to that n Fg Effects of the SIR Estmaton and Feedback Errors: We consder a more practcal scenaro, where SIR estmaton and feedback errors exst n the mplementaton of the proposed scheme, and show the effects of the SIR errors on the spatal capacty. Smlarly to [29], where the channel estmaton and feedback errors are assumed to be zero-mean Gaussan varables, respectvely, we assume the SIR estmaton and feedback errors follow zero-mean Gaussan dstrbutons wth varance σest 2 and σfed 2, respectvely. By further assumng that the two types of SIR errors are mutually ndependent, the sum of both SIR errors at transmtter, denoted by n, follows zero-mean Gaussan dstrbuton wth varance σ 2 = σest 2 + σfed 2. Thus, n the presence of SIR errors, the feedback SIR level at transmtter n P-Phase 0 s SIR 0 + n. Moreover, f the feedback SIR level SIR 0 + n γ for a gven SIR threshold γ n P-Phase, transmtter decdes to transmt n P-Phase ; otherwse, t decdes to be dle n the remanng tme of ths tme slot. Smlar to ts counterpart wthout SIR errors n Fgs. 3 and 4, the spatal capacty wth SIR errors s calculated as an average value over all the transmtters random locatons, the random fadng channels, as well as the random SIR errors.

10 6924 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 SIR level at each transmtter becomes very small; and even f SIR 0 γ for transmtter, SIR 0 s close to γ wth a large probablty. Thus, under the case wth zero-mean Gaussan dstrbuted error n, for the transmtters wth SIR 0 γ n the SIR error-free case, t s more lkely that these transmtters become SIR 0 + n <γ than SIR 0 + n γ n the SIR error-nvolved case. Smlarly, we can easly fnd that for the transmtters wth SIR 0 <γ n the SIR error-free case, t s also more lkely that these transmtters mantan SIR 0 + n < γ than SIR 0 + n γ n the SIR error-nvolved case. Thus, the number of transmtters wth SIR 0 + n <γ n the SIR error-nvolved case s larger than that wth SIR 0 <γ n the SIR error-free case n general. Hence, as compared to the case wthout SIR errors, more transmtters wll be refraned from transmttng n the D-Phase n the case wth SIR errors, whch mproves the successful transmsson probablty n the D-Phase due to the reduced nterference. As a result, t s nterestng to observe from Fg. 5 that when the ntal transmtter densty λ 0 ncreases to some sgnfcant pont, the spatal capacty wth SIR errors becomes slghtly hgher than that wthout SIR errors; and ther gap slowly ncreases over λ 0 after ths pont. Therefore, naccurate SIR may even help mprove the SIR-based schedulng performance n more nterference-lmted regme, whch makes the proposed desgn robust to SIR errors. IV. SIR-THRESHOLD BASED SCHEME WITH MULTI-STAGE PROBING Fg. 5. Effects of the SIR errors on the spatal capacty of the proposed scheme. γ =0.4. β =2.5. Fg. 5 numercally shows the spatal capactes of the proposed scheme n both cases wth and wthout SIR errors. We set σ 2 =0 2 and γ =0.4n ths example. It s observed from Fg. 5 that when λ 0 s small, due to the resultant small nterference n the network, each recever feeds back a suffcently to ts assocated transmtter, such that n has a small probablty to affect the transmtter s decson. Thus, we observe that when λ 0 s small, the spatal capacty wth SIR errors s tght to that wthout SIR errors. However, as hgh SIR level SIR 0 λ 0 ncreases, due to the decreased SIR 0 at each transmtter, the transmtters become more easly affected by the SIR errors n when decdng whether to transmt based on SIR 0 + n γ. It s noted that when λ 0 s suffcently large, the average In ths secton, we consder the proposed scheme wth multstage probng,.e., N>. In ths case, N probng phases are sequentally mplemented to gradually decde the transmtters that are allowed to transmt n the data transmsson phase. Accordng to 5, to fnd the spatal capacty C p,n wth N probng phases, we need to frst fnd the successful transmsson probablty P p,n 0 gven n 3. However, due to the mutually coupled user transmssons over dfferent probng phases, the successful transmsson probablty n P-Phase k, 0 < k N, s related to the SIR dstrbutons n all the proceedng probng phases from P-Phase 0 to P-Phase k. Moreover, due to the dfferent pont process formed by the retaned transmtters n each probng phase, the SIR correlatons of any two probng phases are dfferent. Thus, t s challengng to express the successful transmsson probablty and thus the spatal capacty for the case wth N> n general. As a result, nstead of focusng on expressng the spatal capacty C p,n, we focus on studyng how the key system desgn parameters, such as the SIR thresholds and the number of probng phases N, affect the spatal capacty of the proposed scheme wth N>. In partcular, unlke the case wth N =, where the sngle-stage overhead Nτ = τ T s neglgble, the mult-stage overhead Nτ wth N>may not be neglgble. In the followng, we frst study the mpact of multple SIR thresholds on the spatal capacty by extendng Proposton 3. for the case of N =to the case of N>. We then nvestgate the effects of the multstage probng overhead on the spatal capacty. A. Impact of SIR Thresholds From 3 and 5, the spatal capacty of the proposed scheme s determned by the values of SIR thresholds as well as the tme overhead Nτ for probng. To focus on the mpact of the SIR thresholds, n ths subsecton, we assume Nτ s neglgble and thus have C p,n =λ 0 P SIR 0 0 γ,...,sir N 0 γ N, SIR N 0 β 24 where the dstrbutons of SIR k 0 s, 0 k N, are mutually dependent and all the Φ k s, k N, are non-ppps n general. It s also noted that for any k N, wehaveφ k Φ k for γ k 0. Thus, the network nterference level n Φ k s reduced, as compared to that n Φ k. As a result, by extendng Proposton 3. for the case of N =, we obtan the followng proposton for the case of N>. Proposton 4.: Consder two proposed schemes wth arbtrary N and N probng phases, respectvely, N >. Suppose the two schemes adopt the same SIR threshold γ k 0 n each Φ k, k {,...,N }. Then gven β>0, by varyng the SIR threshold γ N [0, n the data transmsson phase for the proposed scheme wth N probng phases, we

11 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 6925 Fg. 6. Spatal capacty aganst γ and γ 2 for N =2. have the followng relatonshp between C p,n and C p,n based on 24: C p,n > C p,n, f γ N <γ N <β C p,n = C p,n, conservatve transmsson regme f 0 γ N γ N or γ N = β neutral transmsson regme C p,n < C p,n, f γ N >β aggressve transmsson regme. 25 Proof: Please refer to [3]. Remark 4.: Smlar to the case of Proposton 3., n Proposton 4., n the conservatve transmsson regme wth γ N <γ N <β, we obtan mproved spatal capacty; n the aggressve transmsson regme wth γ N >β, we obtan reduced spatal capacty; and n the neutral transmsson regon wth 0 γ N γ N or γ N = β, we obtan unchanged capacty. Moreover, based on the fact that the conservatve transmsson decson s benefcal for mprovng the spatal capacty of the proposed scheme, we obtan the followng corollary, whch gves a proper method to set the values of all the SIR-thresholds, such that the mprovement of spatal capacty over the number of probng phases s assured. Corollary 4.: For a proposed scheme wth N>probng phases wth neglgble overhead, f the desgned SIR thresholds are properly ncreased as 0 <γ < <γ N <β, the resultng spatal capacty C p,n ncreases wth the number of probng phases N. It s worth notng that based on 24, for a gven λ 0 > 0, C p,n s only determned by the successful transmsson probablty P p,n 0, gven n 3. Thus, both Proposton 4. and Corollary 4. also apply for P p,n 0. In the next, we provde a numercal example wth N =2 to further dscuss the mpact of SIR thresholds on the spatal capacty. In ths example, we set α =4, β =2, and d =0. Fg. 6 shows the correspondng spatal capacty over γ [0,β] and γ 2 [0,β]. It s observed from Fg. 6 that f 0 γ 2 γ, the spatal capacty acheved at γ,γ 2 remans unchanged over γ 2 ; and f γ 2 >γ, the spatal capacty acheved at γ,γ 2 Fg. 7. Spatal capacty aganst the number of probng phases. s always larger than that acheved at γ, 0. Apparently, ths s n accordance wth Proposton 4.. Moreover, among all the ponts over γ [0,β] and γ 2 [0,β], such trend s more obvously observed for small γ and small γ 2. In addton, t s also observed that the spatal capacty vares much faster over γ than over γ 2, and when γ s suffcently large, the resultng spatal capacty does not change much over γ 2.As a result, the SIR threshold γ plays a more crtcal role n determnng the spatal capacty than γ 2, snce γ determnes how many transmtters can have a second chance to contend the transmsson opportunty. Furthermore, t s observed that to acheve a hgher spatal capacty, t s preferred to start wth a smallγ > 0, and then setγ 2 <βwth an ncreasng stepsze,.e., γ 0 <γ 2 γ. As shown n ths example, the maxmum spatal capacty s acheved at γ =0.5 and γ 2 = 0.8; and the spatal capacty at γ =0.5 reduces very slowly over γ 2 [0.8,β]. B. Effects of Mult-Stage Overhead of Probng In ths subsecton, we assume the mult-stage overhead Nτ for probng s not neglgble and study the effects of Nτ on the spatal capacty. In ths case, t s easy to fnd from 5 that the effectve data transmsson tme s reduced over the probng-stage number N, whch reduces the spatal capacty. On the other hand, from Corollary 4., under the constrant that 0 <γ < <γ N <β, the successful transmsson probablty ncreases over N. As a result, from 5, there exsts a probng-capacty tradeoff over N under the condton that 0 < γ < <γ N <β. In the followng, we llustrate the probngcapacty tradeoff by a numercal example see Fg. 7. In ths example, we show the spatal capacty of the proposed scheme over the number of probng phases N. Wesetα =4, β =2, d =0, and the tme slot duraton T =second s. We consder two cases wth τ =0s and τ =0.04 s, respectvely, where the tme overhead for probng s zero for the former case and non-neglgble for the latter one. For both cases, as enlghtened by Fg. 6, we start wth γ =0.0 and gradually ncrease

12 6926 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 3, NO. 2, DECEMBER 204 γ k, 2 k N, based on γ k = γ k +0.0k, whch gves an ncreasng step-sze wth γ k γ k <γ k+ γ k. To ensure γ N <β, the maxmum allowable N s obtaned as 9. As shown n Fg. 7, t s observed that the spatal capacty ncreases over N for the case wth τ =0, whch valdates Corollary 4.. Moreover, for the case wth τ =0.04, the probng-capacty tradeoff s observed as expected: the spatal capacty frst ncreases over N, due to the mproved performance of successful transmsson probablty, but after N =0, the spatal capacty begns to decrease over N, due to the more domnant effects of the reduced data transmsson tme. V. C ONCLUSION In ths paper, we addressed the spatal capacty analyss and characterzaton n a wreless ad hoc network by an effcent SIR-threshold based scheme. For sngle-stage probng, we showed the condtons under whch the spatal capacty of the proposed scheme performs strctly better than that of the reference scheme wthout schedulng. We also characterzed the spatal capacty of the proposed scheme n closed-form. In partcular, we proposed a new approach to approxmate the spatal capacty, whch s useful for analyzng performance of wreless networks wth nteracted transmtters. For multstage probng, we extended the results for the case of snglestage probng, and gave the condton under whch the spatal capacty of the proposed scheme can be gradually mproved over the probng-stage number. We also studed the effects of mult-stage probng overhead and nvestgated the probngcapacty tradeoff. REFERENCES [] S. Weber, J. G. Andrews, and N. Jndal, An overvew of the transmsson capacty of wreless networks, IEEE Trans. Commun., vol. 58, no. 2, pp , Dec [2] N. Jndal, S. Weber, and J. G. Andrews, Fractonal power control for decentralzed wreless networks, IEEE Trans. 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Geranots, Adaptve closed-loop power control wth quantzed feedback and loop flterng, IEEE Trans. Wreless Commun., vol., no., pp , Jan [30] J. G. Andrews, F. Baccell, and R. K. Gant, A tractable approach to coverage and rate n cellular networks, IEEE Trans. Commun., vol. 59, no., pp , Nov. 20. [3] Y. L. Che, R. Zhang, Y. Gong, and L. Duan, On Spatal Capacty of Wreless Ad Hoc Networks Wth Threshold Based Schedulng. [Onlne]. Avalable: [32] M. Haengg and R. K. Gant, Inteference n large wreless networks, n Foundatons and Trends n Networkng. Delft, The Netherlands: Now Publsher, [33] C. P. Nculescu and J. Pečarć, The equvalence of Chebyshev s nequalty wth the Hermte-Hadamard nequalty, Math. Rep.,vol.262,no.2, pp , 200. Yue Lng Che S receved the B.E. and M.E. degrees n communcaton and nformaton engneerng from the Unversty of Electronc Scence and Technology of Chna, Chengdu, Chna, n 2006 and 2009, respectvely. She s currently workng towards the Ph.D. degree wth the School of Electrcal and Electronc Engneerng, Nanyang Technologcal Unversty, Sngapore. Her current research nterests nclude spatal modelng and analyss of random wreless networks, cogntve rado, and wreless energy transfer.

13 CHE et al.: ON SPATIAL CAPACITY OF WIRELESS AD HOC NETWORKS WITH THRESHOLD BASED SCHEDULING 6927 Ru Zhang S 00 M 07 receved the B.Eng. frstclass hons. and M.Eng. degrees from the Natonal Unversty of Sngapore n 2000 and 200, respectvely, and the Ph.D. degree from the Stanford Unversty, Stanford, CA, USA, n 2007, all n electrcal engneerng. Snce 2007, he has worked wth the Insttute for Infocomm Research, A-STAR, Sngapore, where he s now a Senor Research Scentst. In 200, he joned the Department of Electrcal and Computer Engneerng, Natonal Unversty of Sngapore as an Assstant Professor. Hs current research nterests nclude multuser MIMO, cogntve rado, cooperatve communcaton, energy effcent and energy harvestng wreless communcaton, wreless nformaton and power transfer, smart grd, and optmzaton theory. Dr. Zhang has authored/coauthored over 80 nternatonally refereed journal and conference papers. He was the recpent of the 6th IEEE ComSoc Asa- Pacfc Best Young Researcher Award n 200, and the Young Investgator Award of the Natonal Unversty of Sngapore n 20. He s now an elected member of IEEE Sgnal Processng Socety SPCOM and SAM Techncal Commttees, and an edtor for the IEEE TRANSACTIONS ON WIRELESS COM- MUNICATIONS and the IEEE TRANSACTIONS ON SIGNAL PROCESSING. Lngje Duan S 09 M 2 receved the Ph.D. degree from The Chnese Unversty of Hong Kong, Shatn, Hong Kong, n 202. He s an Assstant Professor of Engneerng Systems and Desgn Pllar, Sngapore Unversty of Technology and Desgn SUTD. Durng 20, he was a Vstng Scholar n the Department of EECS, Unversty of Calforna at Berkeley. Hs research nterests nclude network economcs, resource allocaton, and energy harvestng. He s a fnalst of the Hong Kong Young Scentst Award 204 and s now leadng the Network Economcs and Optmzaton Lab at SUTD. He s currently servng as the TPC Co-Char of the INFOCOM204 Workshop on GCCCN and as the ICCS204 Program Co-Char of the specal ssue on Economc Theory and Communcaton Networks. He s also a TPC member for many top-ter conferences. Y Gong S 99 M 03 SM 07 receved the Ph.D. degree n electrcal engneerng from the Hong Kong Unversty of Scence and Technology, Hong Kong, n He then joned the Hong Kong Appled Scence and Technology Research Insttute, Hong Kong, as a Member of Professonal Staff. He s now a Professor at the South Unversty of Scence and Technology of Chna, Shenzhen, Chna. Pror to hs current appontment, he was employed at Nanyang Technologcal Unversty, Sngapore, wth whch he stll has actve collaboratons. Snce 2006, he has served on the edtoral board of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS and the IEEE TRANSACTIONS ON VEHICULAR TECH- NOLOGY. Hs research nterests nclude cogntve rado, cooperatve communcatons, MIMO, OFDM, and physcal layer securty for wreless systems.