Cache-Aided Heterogeneous Networks: Coverage and Delay Analysis Mohamed A. Abd-Elmagid∗†, Ozgur Ercetin†, and Tamer ElBatt∗‡ ∗Wireless Intelligent Networks Center (WINC), Nile University, Giza, Egypt. † Faculty of Engineering and Natural Sciences, Sabanci University, Istanbul, Turkey. ‡Dept. of EECE, Faculty of Engineering, Cairo University, Giza, Egypt. Abstract—This paper characterizes the performance of a providesamoreflexibilitycomparedtocachingthesamecopy generic K-tier cache-aided heterogeneous network (CHN), in of contents. Differently from prior works, in this paper we 7 whichthebasestations(BSs)acrosstiersdifferintermsoftheir characterize the performance of a generic K-tier cache-aided 1 spatial densities, transmission powers, pathloss exponents, activ- heterogeneous network (CHN), in which each tier may adopt 0 ity probabilities conditioned on the serving link and placement 2 caching strategies. We consider that each user connects to the a different probabilistic placement caching strategy. BSwhichmaximizesitsaveragereceivedpowerandatthesame Our main contribution in this paper is to characterize the n time caches its file of interest. Modeling the locations of the performanceofK-tierCHNs.Westudyagenericmodelfora a BSs across different tiers as independent homogeneous Poisson J K-tierCHN,inwhichtheBSsacrosstiersaredistinguishedby Point processes (HPPPs), we derive closed-form expressions for 4 thecoverageprobabilityandlocaldelayexperiencedbyatypical theirspatialdensities,transmissionpowers,pathlossexponents 2 user in receiving each requested file. We show that our results and activity probabilities conditioned on the serving link. for coverage probability and delay are consistent with those Moreover, we consider the generic scenario in which the BSs ] previously obtained in the literature for a single tier system. across different tiers may employ different placement caching T strategies. Modeling the locations of BSs in each tier as an I . independent HPPP, we derive closed-form expressions for the s I. INTRODUCTION c coverage probability and local delay experienced by a typical [ Cache-aided small cell networks (CSNs) have recently user while trying to obtain each cached file in the network. 1 attracted considerable attention in the literature [1]–[5]. [1] Wefurtherobtainasimplifiedexpressionsforthespecialcase v proposed a new caching architecture, namely, Femtocaching, of having identical pathloss exponents across tiers, through 5 in which the small cell base stations (SBSs) are utilized as which we show the consistence of our obtained results with 3 distributed caching devices with either small or non-existing priorworksforthesingletierheterogeneousnetworkscenario. 7 backhaul capacity but with considerable storage space. At- 6 tributed to the nature of randomness in both the mobile user II. SYSTEMMODEL 0 . locations and the available contents inside caches, stochastic WestudyagenericmodelforaCHNconsistingofK inde- 1 0 geometry is considered to be a very relevant tool for the pendent network tiers. The BSs across tiers are distinguished 7 analysis of CSNs [2]–[5]. In [2], a probabilistic placement by their spatial densities, transmission powers, pathloss ex- 1 caching strategy is adopted at each SBS, in which each of the ponents, activity probabilities conditioned on the serving link v: popular files is cached with a certain probability. The prime and placement caching strategies. Particularly, the locations i goal was to characterize the optimal caching probabilities of BSs in the j-th tier are modeled as a HPPP φ with X j for different files to maximize the hit probability subject to spatial density λ . We consider a content library composed j ar the finite size caches constraints. Modeling the distributions of M different files and assume that each tier’s BSs may of mobile users and SBSs as HPPPs facilitates the ability adopt a different probabilistic placement caching strategy. Let to obtain closed-form expressions of various performance p = {p ,p ,...,p } denote the probabilistic placement j 1j 2j Mj metrics for different setups of CSNs [3]–[5], e.g., coverage caching strategy employed by the j-th tier’s BSs, where p mj probabilityandaverageachievablerate.[3]considereddevice- denotes the probability that a j-th tier’s BS caches file m to-device(D2D)cachingalongwithsmallcellcachingthrough such that (cid:80)M p = S files,∀j, where S is the cache m=1 mj j j the assumption that a subset of the mobile terminals are size associated with the j-th tier’s BSs. Conditioned on the equipped with finite cache sizes. The SBSs were assumed serving link, each interferer from the j-th tier is assumed to to cache the same copy of specific popular contents in [3], beindependentlyactivewithprobabilitya .Thisisduetothe j [4]. On the contrary, [5] considered that the content library is fact that the BSs may not be always active to save energy or partitionedintonon-overlappingsubsetsoffiles,andeachSBS to mitigate interference, etc. can only cache one of them with a certain probability, which According to Slivnyak’s theorem [6], an existing point in theprocessdoesnotchangethestatisticaldistributionofother This work was supported in part by the European Unions Horizon 2020 points of the PPP. Therefore, without loss of generality, we researchandinnovationprogrammeundertheMarieSkodowska-Curiegrant agreementNo690893. focus on the downlink analysis at a typical user located at the origin. Given that the typical user is associated with BS i in where A is the probability that the typical user is associated k the j-th tier located at Y ∈ φ , the received power at the withthek-thtierandC istheprobabilityofcoveragewhen ji j nk typical user is given by the user is associated with the k-th tier to obtain file n. Thus, C is given by Precv =P h (cid:107)Y (cid:107)−αj, (1) nk j j ji ji where Pj denotes the transmission power of the j-th tier, Cnk =Exnk[P(SIRk(xnk)>τ |xnk)] αj denotes the pathloss exponent of the j-th tier and hji is (cid:90) ∞ (cid:18) xαkτI (cid:19) the channel power gain from BS i of the j-th tier and the = P hk0 > nPk |xnk fXnk(xnk)dxnk 0 k tcyopeifcfiaclieunstesr.wWitehausnsiutmaveeirnadgeeppeonwdeenr,tiR.ea.,yhlejiigihsfaandienxgpocnheanntniaell (=a)(cid:90) ∞EI(cid:20)exp(cid:18)−xαnPkkτI(cid:19)(cid:21)fXnk(xnk)dxnk random variable with unit mean. 0 k We assume an open access network, i.e., the typical user (=b)(cid:90) ∞(cid:89)K L (cid:18)xαnkkτ(cid:19)f (x )dx , (5) Ij P Xnk nk nk is allowed to connect to any tier without any restrictions. We 0 j=1 k consider a cell association strategy1, where the typical user where I denotes the interference power of the j-th tier such j connectstotheBSwhichoffersthestrongestaveragereceived that I = (cid:80)K I , L denotes the Laplace transform of I , power and caches its requested file. For instance, the index of j=1 j Ij j f (x ) is the probability density function (PDF) of the the tier of the BS serving the user request file n is given by Xnk nk distance between the typical user and the serving BS in the k=arg max PjRn−jαj, (2) k-th tier. Note that (a) follows from hk0 ∼ exp(1) and (b) j∈K follows from the definition of Laplace transform along with where K = {1,2,···,K} and Rnj denotes the distance that fact that Ij, j = 1,2,···,K, are independent random betweentheuserandtheclosestBSinthej-thtierthatcaches variables. Before proceeding to derive the Laplace transform file n. of each tier, we provide the PDF of the distance between the Throughout this paper, we emphasize our analysis on a typical user and the serving BS in the k-th tier, f (x ), in Xnk nk scenario where the typical user tries to obtain file n cached the following Lemma. in the network. According to the thinning theorem of the PPP Lemma 1. The PDF of the distance between the typical user [6], the distribution of the k-th tier’s BSs that caches file n and its serving BS ∈φn, that caches file n, is given by can be viewed as a thinned HPPP φn with density p λ . k For instance, for a typical user associkated with a BS n∈kφknk f (x )= 2πpnkλkxnkexp(cid:34)−π(cid:88)K p λ P¯2/αjx2/α¯j(cid:35), located at a random distance x from the user, the signal to Xnk nk A nj j j nk nk k j=1 interference ratio (SIR) at the typical user is given by P α SIRk(xnk)= (cid:80)Kj=1(cid:80)i∈φjP\Bkkh0kt0jxiP−nkαjhkji (cid:107)Yji (cid:107)−αj, (3) wherePrPo¯jof=: NPokjteatnhdatα¯thje=evαenkjt,{i.Xe.n,kP¯k>=xnα¯kk}=is1e.quivalent to where B is the index of the serving BS and t is an the event {R >x } conditioned on the association of the k0 ji nk nk indicator random variable which represents the activity of BS typical user with a k-tier BS, thus we have i in the j-th tier. Thus, t takes value 1 with probability ji P(X >x )=P(R >x |Association index = k). a and 0 otherwise. Note that we assume the network is nk nk nk nk j interference limited, and hence the thermal noise power is The result follows from Lemma 3 in [7] by observing that the ignored compared to interference power. distribution of R can be obtained from its null probability nk as f (r )=2πp λ r exp(cid:0)−πp λ r2 (cid:1) [6]. III. COVERAGEPROBABILITYANALYSIS Rnk nk nk k nk nk k nk Now, our objective is to derive the Laplace transform of In this section we derive the coverage probability for the (cid:18)xαkτ(cid:19) the interference power L nk in order to evaluate the considered K-tier CHN. The coverage probability is defined Ij P k as the probability that a typical user is able to achieve some coverageprobability(5).WefirstderivetheLaplacetransform threshold SIR, denoted τ, when the user tries to obtain its file of the serving tier’s interference power, i.e., L , and then we Ik of interest from its associated BS. Recall that we focus on a derive the Laplace transform of the other tiers’ interference scenariowherethetypicalusertriestoobtainfilen.Sincethe powers.SincetheservingBSinthek-thtierlocatedatdistance typical user is associated with at most one tier, from the total x fromthetypicaluser,noneofthek-thtier’sBSswhichare nk probability law, the coverage probability of obtaining file n located inside the circle of radius x cache file n. Therefore, nk can be expressed as the k-th tier’s BSs located inside and outside the circle of radiusx constitutetwoHPPPswithdensitiesq λ andλ , K nk nk k k (cid:88) Cn = AkCnk, (4) respectively, and with activity probability ak, where qnk = 1 − p . L can be determined by evaluating the Laplace k=1 nk Ik transform of the k-th tier’s interference power from outside 1NotethatPjRn−jαj isalong-termaverageofPjrecvandfadingisaveraged and inside the circle of radius xnk separately. out,andhenceitdoesnotincludehji. Lemma 2. The Laplace transform of the k-th tier’s interfer- tier, ∀j (cid:54)=k, is at least x . Hence, the j-th tier’s BSs located j ence power from outside the circle of radius xnk is insideandoutsidethecircleofradiusxj constitutetwoHPPPs LoIukt(cid:18)xPαnkkkτ(cid:19)=exp(cid:2)−πλkakρ1(k)x2nk(cid:3), (6) wpriothbadbeilnitsyitiaejs,qwnhjeλrjeaqnndj =λj1, −resppnejc.tively, and with activity where ρ1(m)=τ2/αm(cid:82)τ∞−2/αm 1+u1αm/2 du. Ltfireoermm,tmhoeuatLs4ai.dpeGlatichveeentcrtaihrncasltfeothromef troyapfditichuaesljxu-stheirstiaesrs’osciinatteersfewreitnhcethpeokw-ethr j Proof: Following the same approach used in the proof of Theorem 1 in [8], the result can be obtained as follows Lout(cid:18)xαnkkτ(cid:19)=exp(cid:104)−πλ a P¯2/αjρ (j)x2/α¯j(cid:105), ∀j (cid:54)=k. Ij P j j j 1 nk   k LoIukt =Eφk,tki,hki (cid:89) exp(cid:0)−xαnkkτtkihki (cid:107)Yki (cid:107)−αk(cid:1) Proof: The result can be obtained in a similar way as in Lemma 2 by replacing the radius of the circle x with x i∈φk\B(0,xnk) nk j   and considering the transmission power of the j-th tier P . j (=a)Eφk,tkii∈φk\(cid:89)B(0,xnk)1+xαnkkτtki1(cid:107)Yki (cid:107)−αk traFnosflolorwminogf ththeej-stahmteiera’snainlytesrisferoefncLeepmomwaer4fr,omtheinLsiadpelathcee   circle of radius x is given by the following Lemma. j (=b)Eφki∈φk\(cid:89)B(0,xnk)1+xαnkkτa(cid:107)kYki (cid:107)−αk +(1−ak) Lweitmhmthae5k.-Cthontideirt,iotnheedLoanptlhaeceastsroacnisafotiromn ooffththeetyjp-itchaltiuesre’rs (7)   interference power from inside the circle of radius xj is (=c)exp−2πλkak(cid:90)x∞nk τ +(xτynk)αk ydy, LiInjs(cid:18)xPαnkkkτ(cid:19)=exp(cid:104)−πλjqnjajP¯j2/αjρ2(j)xn2/kα¯j(cid:105), ∀j (cid:54)=k. where B(0,x ) denotes the set of k-th tier’s BSs located UsingLemma4and5alongwithexploitingthefactthatthe nk inside a circle centered at the origin and with radius x , j-thtier’sinterferencepowerfromoutsideandinsidethecircle nk (a) follows from the independence of the random variables of radius xj are independent random variables, the Laplace h along with the fact that h ∼ exp(1), (b) follows from transformofthej-thtier’sinterferencepowercanbeexpressed ki ki the fact that the random variables t are independent and (c) as ki follows from the probability generating functional (PGFL) of (cid:18)xαkτ(cid:19) (cid:18)xαkτ(cid:19) (cid:18)xαkτ(cid:19) L nk =Lout nk ×Lins nk the PPP along with simple algebraic manipulations. By using Ij P Ij P Ij P the change of variables u = (y/xnk)2τ−2/αk, the result can k =exp(cid:104)−πλka P¯2/αjx2/α¯j(kρ (j)+q ρ (j))(cid:105). be directly obtained. j j j nk 1 nj 2 Following the same analysis of Lemma 2, the Laplace Theorem1. Thecoverageprobabilityofobtainingfilenwhen transform of the k-th tier’s interference power from inside the the typical user is associated with the k-th tier is circle of radius x is given by the following Lemma. The nk C proof of Lemma 3 is omitted for brevity. nk Lemma 3. The Laplace transform of the k-th tier’s interfer- =(cid:90) ∞ 2πpnAkλkxnkexp(cid:34)−π(cid:88)K λjP¯j2/αjx2n/kα¯j(pnj+aj(ρ1(j)+qnjρ2(j)))(cid:35) dxnk. ence power from inside the circle of radius x is 0 k j=1 nk Lins(cid:18)xαnkkτ(cid:19)=exp(cid:2)−πλ a q ρ (k)x2 (cid:3), (8) fileBans,eCd o,nisThdeeoterermmin1e,dthbeycopvluegragginegprCobabiinlittoy(o4f).obFtuaritnhienrg- Ik P k k nk 2 nk n nk k more, the coverage probability of obtaining any file m in the where ρ2(m)=τ2/αm(cid:82)0τ−2/αm 1+u1αm/2 du. cporonbteanbtililtiibersaroyfficlaennbaecrdoestserdmififneeredntbtyierrespwlaicthintghotsheeocfaficlheinmg Exploiting the fact that the k-th tier’s interference power inTheorem1.Next,forthespecialcaseofhavingαj =α, ∀j, from outside and inside the circle of radius x are indepen- we provide the coverage probability of obtaining file n. nk dent random variables, the Laplace transform of the serving Corollary 1. The coverage probability of obtaining file n tier’s interference power, LIk, can be expressed as when αj =α, ∀j, i.e., α¯j =1, ∀j, is L (cid:18)xαnkkτ(cid:19)=Lout(cid:18)xαnkkτ(cid:19)×Lins(cid:18)xαnkkτ(cid:19) C =(cid:88)K pnkλk . Ik Pk Ik Pk Ik Pk n (cid:80)K λ P¯2/α[p +a (ρ (j)+q ρ (j))] =exp(cid:2)−πλkakx2nk(ρ1(k)+qnkρ2(k))(cid:3). (9) k=1 j=1 j j nj j 1 nj 2 (10) Thetypicaluserisassociatedwiththek-thtiertoobtainfilen Equation (10) reflects the impact of the probabilistic avail- if R > P¯1/αjx1/α¯j, ∀j (cid:54)= k. Let x = P¯1/αjx1/α¯j. Given ability of file n at BSs across different tiers on the coverage nj j nk j j nk that the typical user with a BS at k-th tier, the distance from probability. Setting p =1, ∀j, leads back to the achievable nj the typical user to the closest BS that caches file n in the j-th coverage of conventional K-tier heterogeneous networks [7], where the requested files were assumed to be available at all where (a) follows from the change of variables u = BSs. On the other hand, by setting K = 1, we obtain the (y/xnk)2τ−2/αk and ρ3(m) is given by coverage probability of the single tier CHN (cid:90) ∞ 1 Cn = p +a (ρ p(1n)1+q ρ (1)), (11) ρ3(m)=τ2/αm τ−2/αm 1−am+uαm/2 du. (16) n1 1 1 n1 2 Similarly, considering the interference from inside the circle which is consistent with the obtained result for the single tier of radius x , we have nk scenario studied in [ [9], Theorem 1]. (cid:34) (cid:35) IV. LOCALDELAYANALYSIS Eφk Lins1 =exp(cid:2)πλkqnkakρ4(k)x2nk(cid:3), (17) Conditioned on the association of the typical user with the Ik|φk,xnk k-th tier to obtain file n, the local delay is defined as the where ρ (m) is given by 4 mean number of time slots required for the typical user to successfully receive file n from its serving BS. We assume ρ (m)=τ2/αm(cid:90) τ−2/αm 1 du. (18) that if the typical user fails to decode the transmitted file n in 4 0 1−am+uαm/2 acertaintimeslot,thefileisretransmittedinthenexttimeslot. Next, we derive the expectation inside the integral in (14) Theconditionalsuccessprobabilityinanarbitrarytimeslotis with respect to the PPP of the j-th tier, ∀j (cid:54)= k. The defined as PCnk|φ = P(SIRk(xnk)>τ |φ,xnk), where φ = corresponding expressions to (15) and (17) for the j-th tier ∪j∈Kφj.Thus,duetotheindependenceofconditionalsuccess case can be found is a similar way, and given respectively by events over time, the conditional local delay is a geometric rdaenladyomofvoabritaabinleingwifithlemnecaann1b/ePCexnpkr|eφss[e1d0]a.sHence, the local EφjLout1 =exp(cid:104)πλjajP¯j2/αjρ3(j)x2n/kα¯j(cid:105), ∀j (cid:54)=k, Dnk =Eφ,xnk(cid:20)P(SIR (x )1>τ |φ,x )(cid:21), (12) Ij|φj,xnk (19) k nk nk thusfromthetotalprobabilitylaw,thelocaldelayofobtaining   file n is given by EφjLins1 =exp(cid:104)πλjqnjajP¯j2/αjρ4(j)x2n/kα¯j(cid:105), ∀j (cid:54)=k. K Ij|φj,xnk (cid:88) (20) D = A D . (13) n k nk k=1 From (15), (17), (19) and (20) along with taking into account that P¯ =α¯ =1, we obtain that Byapplyingsimilaranalysisasin(5),D canbeexpressed k k nk as (cid:34) (cid:35)   Eφj L 1 =exp(cid:104)πλjajP¯j2/αjx2n/kα¯j(ρ3(j)+qnjρ4(j))(cid:105),∀j. Dnk =(cid:90) ∞Eφ(cid:89)K (cid:18)xαkτ1 (cid:19)fXnk(xnk)dxnk Ij|φj,xnk (21) 0 j=1LIj Pnk |φj,xnk Plugging (21) into (14) yields the local delay of obtaining file k   n when the typical user is associated with the k-th tier, as (=a)(cid:90)0∞j(cid:89)K=1EφjLIj(cid:18)xPαnkkτ1|φj,xnk(cid:19)fXnk(xnk)dxnk, teTyshptaiecboalrileshmuesder2b.iysTtahhseesoflcooiclalaotelwddinwegliatthyhetohofereokmb-.ttahintiienrgisfile n when the k (14) D (22) nk difwfehreenret (tiae)rsf.olIlnowosrdferromtothcheairnadcteepreiznedeDnce,ofwetheshPaPllPnsoowf =(cid:90) ∞ 2πpnkλkxnkexp(cid:34)−π(cid:88)K λ P¯2/αjx2/α¯j(p −a (ρ (j)+q ρ (j)))(cid:35) dx . nk A j j nk nj j 3 nj 4 nk calculatetheexpectationinsidetheintegralin(14)withrespect 0 k j=1 to each PPP. We first derive the expectation with respect to Based on Theorem 2, the local delay of obtaining file n, the PPP of the serving tier, which can be determined by con- D , is determined by plugging D (22) into (13). Now, for n nk sidering the conditional Laplace transform of the interference the special case of having α = α, ∀j, we provide the local j powerfromoutsideandinsidethecircleofradiusx .Hence, nk delay of obtaining file n in the following Corollary. from (7), we have Corollary 2. The local delay of obtaining file n when α = j   α, ∀j, i.e., α¯ =1, ∀j, is j (cid:34) (cid:35) Eφk LoIkut|φ1k,xnk =exp2πλkak(cid:90)x∞nk τ −τak+τy(cid:18)xy (cid:19)αkdy Dn =(cid:88)K (cid:80)K λ P¯2/α[p p−nkaλk(ρ (j)+q ρ (j))]. nk k=1 j=1 j j nj j 3 nj 4 (=a)exp(cid:2)πλ a ρ (k)x2 (cid:3), (15) (23) k k 3 nk 50 Coverage Probability000000000−.........011234567893 0 −25 −20 −15 −10τ (d−B5) 0 CCCCCCCC12121212 5−−−−−−−− IIIIDDDDSSSSSSSS,,,, ,,,,λλλλ 1λλλλ22220////2222λλλλ////λλλλ1111====1111====448844881 5 Local Delay (time slots)11223344−05050505051 5 −10 −5 τ (0dB) 5 DDDDDDDD12121212−−−−−−−−IIIIDDDDSSSSSSSS,,,,1 ,,,,λλλλ 0λλλλ2222////2222λλλλ////λλλλ1111 1111==== ==== 4488 448815 Coverage Probability of file 1 (C) − IS100000000........0123456789 a20.5 1 1 0a.51 0 (a)Coverageprobabilityvs.τ. (b)Localdelayvs.τ. (c)Coverageprobabilityvs.a1 anda2. Fig.1. NumericalResults. Setting K =1, leads back to the single tier case where the Fig. 1c shows the effect of changing the activity probabili- local delay is given by tiesontheachievablecoverageprobabilityoffile1inIScase. p We use λ2 = 4λ1 and τ = −5 dB. It is observed that the Dn = p −a (ρ (1n)1+q ρ (1)), (24) coverage probability is a monotonically decreasing function n1 1 3 n1 4 in both a and a . This is attributed to the fact that either 1 2 which is consistent with the obtained result for the single tier increasing a or a leads to a higher number of active BSs, 1 2 scenario studied in [ [9], Lemma 5]. and hence a higher received interference power at the serving BS which in turn leads to a lower coverage. V. NUMERICALRESULTS VI. CONCLUSION For clarity of exposition, we focus on a two-tier CHN Thispaperfocusesontheperformanceanalysisofageneric with a two-file distributed caching scenario, i.e., K = 2 and K-tiercache-aidedheterogeneousnetwork.Fortheconsidered M = 2. We further assume that S = S = 1, i.e., each 1 2 generic model, we drive closed-form expressions for both the BS in the network either caches file 1 or file 2 only. If not coverage probability and local delay of obtaining each file otherwise stated, we use the following parameters, P =0.1, 2 cached in the network. Our results are consistent with those P = 100P , λ = 1, a = a = 1 and α = α = 4. We 1 2 1 1 2 1 2 in the literature for the scenario of single tier cache-enabled consider two different placement caching strategies: 1) Iden- heterogeneous network. As part of our future work, we would tical placement caching strategy (IS), in which each file has like to characterize: 1) the performance of closed access cell the same caching probability across the BSs of the two tiers, association strategy, and 2) the optimal placement caching and 2) Different placement caching strategy (DS), in which strategies to maximize the hit probability. each file has a different caching probability across the BSs of the two tiers. Particularly, in IS, we use p11 =p12 =0.2 and REFERENCES p = p = 0.8, whereas in DS, we use p = p = 0.5, 21 22 11 21 [1] N. Golrezaei, A. F. Molisch, A. G. Dimakis, and G. Caire, “Fem- p12 =0.2 and p22 =0.8. tocaching and device-to-device collaboration: A new architecture for In Fig. 1a and 1b, we plot the coverage probability and wirelessvideodistribution,”IEEECommun.Magazine,2013. [2] B. Blaszczyszyn and A. 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This networks: A stochastic geometry perspective,” in Asilomar Conference onSignals,SystemsandComputers,2015. can be seen from the expressions of coverage and delay in [10] W. Nie, Z. Yi, F. Zheng, W. Zhang, and T. O’Farrell, “Hetnets with Corollary1and2,inwhichcoverageanddelaydonotdepend random dtx scheme: Local delay and energy efficiency,” IEEE Trans. on the spatial densities of tiers for the scenario of IS. VehicularTechnology,2015.