Mobility Increases the Data Offloading Ratio in D2D Caching Networks
MMobility Increases the Data Offloading Ratio in D2DCaching Networks
Rui Wang ∗ , Jun Zhang ∗ , S.H. Song ∗ and K. B. Letaief ∗† , Fellow, IEEE ∗ Dept. of ECE, The Hong Kong University of Science and Technology, † Hamad Bin Khalifa University, Doha, QatarEmail: ∗ { rwangae, eejzhang, eeshsong, eekhaled } @ust.hk, † [email protected] Abstract —Caching at mobile devices, accompanied by device-to-device (D2D) communications, is one promising technique toaccommodate the exponentially increasing mobile data traffic.While most previous works ignored user mobility, there are somerecent works taking it into account. However, the duration of usercontact times has been ignored, making it difficult to explicitlycharacterize the effect of mobility. In this paper, we adopt thealternating renewal process to model the duration of both thecontact and inter-contact times, and investigate how the cachingperformance is affected by mobility. The data offloading ratio , i.e.,the proportion of requested data that can be delivered via D2Dlinks, is taken as the performance metric. We first approximatethe distribution of the communication time for a given user by betadistribution through moment matching. With this approximation,an accurate expression of the data offloading ratio is derived.For the homogeneous case where the average contact and inter-contact times of different user pairs are identical, we prove thatthe data offloading ratio increases with the user moving speed,assuming that the transmission rate remains the same. Simulationresults are provided to show the accuracy of the approximateresult, and also validate the effect of user mobility.
I. I
NTRODUCTION
The mobile data traffic is growing at an exponential rate,among which mobile video accounts for more than a half [1].Caching popular contents at helper nodes or user devices is apromising approach to reduce the data traffic on the backhaullinks, as well as improving the user experience of video stream-ing applications [2], [3]. In comparison with the commonlyconsidered femto-caching system, caching at devices enjoys aunique advantage, i.e., the devices’ aggregate caching capacitygrows with the number of devices [2]. Moreover, devicecaching can promote device-to-device (D2D) communications,where nearby mobile devices may communicate directly ratherthan being forced to communicate through the base station(BS)[4].Recently, caching in D2D networks has attracted lots ofattentions. In [5], the scaling behavior of the number ofD2D collaborating links was identified. Three concentrationregimes, classified by the concentration of the file popularity,were investigated. The outage-throughput tradeoff and optimalscaling laws of both the throughput and outage probabilitywere studied in [6]. Two coded caching schemes, i.e., cen-tralized and decentralized, were proposed in [7], where thecontents are delivered via broadcasting.
This work was supported by the Hong Kong Research Grants Councilunder Grant No. 610113.
So far, an important characteristic of mobile users, i.e., theuser mobility, has been ignored in previous studies of D2Dcaching networks. There are some works starting to considerthe effect of user mobility. Effective methodologies to utilizethe user mobility information in caching design were discussedin [8]. In [9], the effect of mobility was evaluated in D2Dnetworks with coded caching, with the conclusion that mobilitycan improve the scaling law of throughput. This result wasbased on the assumption that the user locations are randomand independent in each time slot, which failed to take intoaccount the temporal correlation.The inter-contact model, which considers the temporalcorrelation of the user mobility, has been widely applied [10],where the timeline for an arbitrary pair of mobile users aredivided into contact times and inter-contact times . Specifically,the contact times denote the time intervals when the mobileusers are located within the transmission range. Correspond-ingly, the inter-contact times denote the time intervals betweencontact times [11]. This model has been used to developdevice caching schemes to exploit the user mobility patternin [12]. The throughput-delay scaling law was developed bycharacterizing the inter-contact pattern of the random walkmodel [13]. In these works, it was assumed that a fixed amountof data can be delivered within one contact time, while theduration of the contact times was not considered. However,as the user moving speed will affect the durations of boththe contact and inter-contact times, it is critical to account fortheir effects when investigating the impact of user mobility oncaching performance.In this paper, we shall analytically evaluate the effect ofmobility in D2D caching networks, by adopting an alternatingrenewal process to model the mobility pattern so that boththe contact and inter-contact times are accounted for. The data offloading ratio , which is defined as the proportion ofdata that can be obtained via D2D links, is adopted as theperformance metric. The main contribution is an approximateexpression for the data offloading ratio, for which the maindifficulty is to deal with multiple alternating renewal processes.We tackle it by first deriving the expectation and varianceof the communication time of a given user, and then use abeta random variable to approximate it by moment matching.Furthermore, we investigate the effect of mobility in a ho-mogeneous case, where the average contact and inter-contacttimes for all the user pairs are the same. In the low-to-mediummobility scenario, by assuming that the transmission rate isirrelevant to the user speed, it is proved that the data offloadingratio increases with the user speed for any caching strategy that a r X i v : . [ c s . I T ] F e b ontact timeinter-contact timetime … time … Fig. 1. The timeline for an arbitrary pair of mobile users. does not cache the same contents at all devices. Simulationresults validates the accuracy of the derived expression, as wellas the effect of the user mobility.II. S
YSTEM M ODEL AND P ERFORMANCE M ETRIC
In this section, we will first introduce the alternatingrenewal process to model the user mobility pattern, and discussthe caching and file delivery models. Then, the performancemetric, i.e., the data offloading ratio, will be defined.
A. User Mobility Model
The inter-contact model, which captures the temporal cor-relation of the user mobility [10], is used to model the usermobility pattern. Specifically, the timeline of each pair of usersis divided into contact times , i.e, the times when the usersare in the transmission range, and inter-contact times , i.e.,the times between consecutive contact times. Considering thatcontact times and inter-contact times appear alternatively inthe timeline of a pair of users, similar to [14], an alternatingrenewal process is applied to model the pairwise contactpattern, as defined below [15].
Definition 1.
Consider a stochastic process with state space { A, B } , and the successive durations for the system to be instates A and B are denoted as ξ k , k = 1 , , · · · and η k , k =1 , , · · · , respectively, which are i.i.d.. Specifically, the systemstarts at state A and remains for ξ , then switches to state B for η , then backs to state A for ξ , and so forth. Let ψ k = ξ k + η k .The counting process of ψ k is called as an alternating renewalprocess .As shown in Fig. 1, if the pair of users is in contact at t = 0 , ξ k and η k represent the contact times and inter-contacttimes, respectively; otherwise, ξ k and η k represent the inter-contact times and contact times, respectively. It was shown in[16] that exponential curves well fit the distribution of inter-contact times, while in [17], it was identified that exponentialdistribution is a good approximation for the distribution of thecontact times. Thus, same as [14], we assume that the contacttimes and inter-contact times follows independent exponentialdistributions. For simplicity, the timelines of different userpairs are assumed to be independent. Specifically, we consider N u users in a network, and the index set of the users isdenoted as S = { , , · · · , N u } . The contact times andinter-contact times of users i ∈ S and j ∈ S\{ i } followindependent exponential distributions with parameters λ Ci,j and λ Ii,j , respectively.
B. Caching and File Transmission Model
There is a library with N f files, whose index set is denotedas F = { , , · · · , N f } , each with size C . Each user device transmission rangetransmission range Mobile userCaching contentTransmission linkRequested fileafter a period of timeuser 3 user 2user 1user 3user 1 user 2 Received Content Fig. 2. A sample network with three mobile users. contact timeinter-contact time … time … users & users & …… User requests the file communication time Fig. 3. An illustration of the communication time. has a limited storage capacity, and each file can be completelycached or not cached at all at each user device. Specifically,the caching placement is denoted as x j,f = (cid:26) , if user j caches file f , , if user j does not cache file f , (1)where j ∈ S and f ∈ F . User i ∈ S , is assumed to requesta file f ∈ F with probability p ri,f , where (cid:80) f ∈F p ri,f = 1 . Whena user requests a file f , it will first check its own cache, andthen download the file from the users that are in contact andstore file f , with a fixed transmission rate, denoted as R . If theuser cannot get the whole file within a certain delay threshold,denoted as T d , it will download the remaining part from theBS. We assume that the delay threshold is larger than the timerequired to download each content, i.e., T d > CR . Fig. 2 showsa sample network, where user gets part of the requested fileduring the contact time with user , then gets the whole fileafter the contact time with user . C. Performance metric
The data offloading ratio , which is defined as the per-centage of requested content that can be obtained via D2Dlinks rather than downloading from the BS, is used as theperformance metric. Specifically, the data offloading ratio foruser i ∈ S is defined as P i = (cid:88) f ∈F p ri,f (cid:26) x i,f + (1 − x i,f ) E D i,f [min ( D i,f , C )] C (cid:27) , (2)where D i,f denotes the amount of requested data that can bedelivered via D2D links when user i requests file f . Sincea fixed transmission rate is assumed, D i,f can be written as i,f = RT ci,f , where T ci,f is the communication time for user i to download file f from other users caching file f withintime T d . We assume that user i can download file f while atleast one user caching file f is in contact, where the handovertime is ignored. Fig. 3 shows the communication time of user in Fig. 2. Then, the average data offloading ratio is P =1 N u (cid:88) i ∈S (cid:88) f ∈F p ri,f x i,f + (1 − x i,f ) E T ci,f (cid:104) min (cid:16) RT ci,f , C (cid:17)(cid:105) C . (3)In the following, we will evaluate the data offloading ratiogiven in (3) for any given caching strategy, and investigate theeffect of user mobility on caching performance.III. D ATA O FFLOADING R ATIO A NALYSIS
The main difficulty of evaluating the data offloading ratiois to find the distribution of the communication time. Asthis distribution is highly complicated, instead of derivingit directly, we will develop an accurate approximation. Inthis section, we will first approximate the distribution ofthe communication time by a beta distribution, and then, anapproximation of the data offloading ratio will be obtained.
A. Communication time analysis
To help analyze the communication time, we first definesome stochastic processes.
Definition 2.
Define H i,j , where i ∈ S and j ∈ S\{ i } , asthe continuous-time random process, i.e., H i,j = { H i,j ( t ) , t ∈ (0 , ∞ ) } with state space { , } , where H i,j ( t ) = 1 means thatusers i and j are in contact at the time instant t ; otherwise H i,j ( t ) = 0 . The durations of staying in states and followi.i.d. exponential distributions with parameter λ Ci,j and λ Ii,j ,respectively.
Definition 3.
Define H fi , where i ∈ S and f ∈ F , asthe continuous-time random process, i.e., H fi = { H fi ( t ) , t ∈ (0 , ∞ ) } with state space { , } , where H fi ( t ) = 1 means thatusers i can download file f from any other user caching file f at time instant t ; otherwise H fi ( t ) = 0 .At time t , since user i can download file f when at leastone user caching file f is in contact, we get H fi ( t ) = 1 − (cid:81) j ∈S\{ i } ,x j,f =1 [1 − H i,j ( t )] . Similar to [14], it is reasonable toassume that when a user requests a file, the alternating processbetween each pair of users has been running for a long time.Thus, denote T ri,f , i ∈ S and f ∈ F , as the time of user i requests file f , and the communication time T ci,f can be derivedas T ci,f = lim T ri,f →∞ (cid:82) T ri,f + T d T ri,f H fi ( t ) dt. In the following, we willderive the expectation and variance of the communication time.
Lemma 1.
When user i ∈ S requests file f ∈ F , which isnot stored at its own cache, the expectation and variance ofits communication time is E (cid:2) T ci,f (cid:3) = T d − (cid:89) j ∈S ,x j,f =1 λ Ci,j λ Ci,j + λ Ii,j . (4) and Var (cid:2) T ci,f (cid:3) =2 (cid:90) T d ( T d − u ) (cid:89) j ∈S ,x j,f =1 λ Ci,j ( λ Ii,j + λ Ci,j ) × (cid:104) λ Ci,j + λ Ii,j e − u ( λ Ci,j + λ Ii,j ) (cid:105) du − ( T d ) (cid:89) j ∈S ,x j,f =1 (cid:32) λ Ci,j λ Ci,j + λ Ii,j (cid:33) . (5) Proof:
See Appendix A.Since the communication time T ci,f is a bounded randomvariable, we propose to approximate its distribution by abeta distribution, which is widely used to model the randomvariables limited to finite ranges. Specifically, we consider T ci,f ≈ T d Y i,f , where Y i,f ∼ B ( α i,f , β i,f ) , i ∈ S and f ∈ F , if (cid:80) j ∈S\{ i } x j,f > , which means that user i maydownload file f from at least one user; otherwise, T ci,f = 0 .Let E [ T d Y i,f ] = E [ T ci,f ] and Var[ T d Y i,f ] = Var[ T ci,f ] , andthe parameters of the beta distribution to match the first twomoments can be derived as α i,f = E [ T ci,f ] ( T d − E [ T ci,f ])Var[ T ci,f ] T d − E [ T ci,f ] T d β i,f = T d − E [ T ci,f ] E [ T ci,f ] α i,f (6) B. Data offloading ratio approximation
Based on the above approximation, we get an approximateexpression of the data offloading ratio in Proposition 1. Sim-ulations will show that the approximation is quite accurate.
Proposition 1.
The data offloading ratio is approximated as P = 1 N u (cid:88) i ∈S (cid:88) f ∈F p ri,f [ x i,f + (1 − x i,f ) P i,f ] , (7) where P i,f is the data offloading ratio when user i requestsfile f , which is not in its own cache, approximated by P i,f ≈ − I CTdR ( α i,f , T d − E [ T ci,f ] E [ T ci,f ] α i,f )+ E [ T ci,f ] RC I
CTdR ( α i,f + 1 , T d − E [ T ci,f ] E [ T ci,f ] α i,f ) (cid:105)(cid:111) , (8) if (cid:80) j ∈S\{ i } x j,f > and elsewhere, where I r ( · , · ) is theincomplete beta function, and α i,f is given in (6).Proof: Following (3), (4), (5), and (6), the expression in(7) can be obtained. Due to the space limitation, the detail isomitted. IV. E
FFECT OF M OBILE U SER S PEED
In this section, we will consider a homogeneous case,where the contact and inter-contact parameters among all pairsof users are the same, i.e., λ C = λ Ci,j > and λ I = λ Ii,j > ,where i ∈ S and j ∈ S\{ i } . We will investigate how the The parameters of the beta distribution should be positive, and it can beproved that α i,f > and β i,f > , by e − u ( λ Ii,j + λ Ci,j ) ≤ . The detail isomitted due to the space limitation. ser moving speed affects the data offloading ratio for a givencaching strategy. If all users cache the same contents, the D2Dcommunications will not help the content delivery. Thus, inthe following, we assume that the contents cached at differentusers are not all the same. This investigation will be basedon the approximate expression in (7), and simulations will beprovided later to verify the results. A. Communication time analysis
Under the above assumptions, the expectation and varianceof the communication time can be simplified, as in the follow-ing corollary.
Corollary 1.
When λ C = λ Ci,j and λ I = λ Ii,j , where i ∈ S and j ∈ S\{ i } , the expectation and variance of a user requests file f , which is not stored at its own cache, are given by E [ T ci,f ] = T d (cid:20) − (cid:18) λ C λ C + λ I (cid:19) n f (cid:21) , (9) Var[ T ci,f ] = (cid:20) λ C ( λ C + λ I ) (cid:21) n f n f (cid:88) l =1 (cid:18) n f l (cid:19) ( λ C ) n f − l ( λ I ) l l ( λ C + λ I ) × (cid:34) T d − l ( λ C + λ I ) + e − l ( λ C + λ I ) T l ( λ C + λ I ) (cid:35) , (10) where i ∈ S and n f = (cid:80) j ∈S x j,f denotes the number of userscaching file f .Proof: See Appendix A.
B. Mobile user speed
We first characterize the relationship between the userspeed and the parameters λ C and λ I in Lemma 2. Lemma 2.
When all the user speeds change by s times, thecontact and inter-contact parameters will also change by s times, i.e., from λ C and λ I to sλ C and sλ I , respectively.Proof: The time for user i to move along a certain path L i can be given as a curve integral (cid:82) L i dzv i ( z ) , where v i ( z ) isthe speed of user i when passing by a point z on the path L i .When the speed of user i changes by s times, the time formoving along the path L i changes to (cid:82) L i dzsv i ( z ) = s (cid:82) L i dzv i ( z ) ,which scales by s times. During each contact or inter-contacttime, users i and j move along certain paths. When all the userspeeds change by s times, each contact or inter-contact timechanges by s times, and thus, the average ones change by s times. Since the contact and inter-contact times are assumed tobe exponential distributed with mean λ C and λ I , respectively,the parameters λ C and λ I scale by s times.Considering that a larger s means that users are movingfaster, in the following, we will investigate how changing s willaffect the data offloading ratio. For simplicity, we assume thatthe transmission rate is a constant, and will not change with theuser speed. This is reasonable in the low-to-medium mobilityregime. Firstly, the effect of user speed on the communicationtime is shown in Lemma 3 . Lemma 3.
When s increases, which is equivalent to increasingthe user speed, the expectation of the communication time u D a t a o ff l oad i ng r a t i o Theoretical results by (7)Simulation resultsC/R=300s,180s,60s
Fig. 4. Data offloading ratio with N f = 100 , T d = 300 s and γ r = 0 . . when a user i ∈ S requests file f ∈ F that is not in its owncache, i.e., E [ T ci,f ] , keeps the same, and the correspondingvariance, i.e., Var[ T ci,f ] , decreases, if the number of userscaching file f is larger than , i.e., n f > . Accordingly,the parameter α i,f of the beta distribution increases.Proof: See Appendix B.Then, we evaluate the relationship between α i,f and thedata offloading ratio when user i requests file f that is not inits own cache, i.e., P i,f in (8), in Lemma 4. Lemma 4.
When user i ∈ S requests file f ∈ F and cannotfind it in its own cache, the data offloading ratio, i.e., P i,f ,increases with α i,f .Proof: See Appendix C.Base on Lemmas 3 and 4, we can specify the effect of userspeed on the data offloading ratio in Proposition 2.
Proposition 2.
If the transmission rate does not change withthe user speed, and the average contact and inter-contact timesamong all the pairs are the same, the data offloading ratioincreases with the user moving speed.Proof:
See Appendix D.
Remark.
The result in Proposition 2 is valid for any cachingstrategy, only excluding the case that all the users have thesame cache contents.
V. S
IMULATION RESULTS
In the simulation, the content request probability follows aZipf distribution with parameter γ r , i.e., p f = f − γr (cid:80) i ∈F i − γr , f ∈ F [2]. Meanwhile, each user caches 5 contents, and a randomcaching strategy is applied [18], where the probabilities of thecontents cached at each user are proportional to the file requestprobabilities.Fig. 4 validates the accuracy of the approximation in (7).The inter-contact parameters λ Ii,j , i ∈ S , j ∈ S\{ i } are gen-erated according to a gamma distribution as Γ(4 . , / [19]. Similar as [14], we assume the average of the contactparameters are times larger than the inter-contact param-eters. Thus, the contact parameters are generated accordingto Γ(4 . × , / / . It is shown from Fig. 4 that thetheoretical results are very close to the simulation results, D a t a o ff l oad i ng r a t i o Theoretical results by (7)Simulation resultsC/R=240s,180s,120s, 60s
Fig. 5. Data offloading ratio with N u = 15 , λ C = 0 . s , λ I = 0 . s , N f = 100 , T d = 300 s and γ r = 0 . . which means the approximate expression (7) is quit accurate.Furthermore, the data offloading ratio increases with the num-ber of users, which is brought by the increasing aggregatecaching capacity and the content sharing via D2D links.In Fig. 5, the effect of s is demonstrated, where increasing s is equivalent to increasing the user speed. Firstly, the small gapbetween the theoretical and simulation results again verifies theaccuracy of the approximate expression in (7). It is also shownthat the data offloading ratio increases with s , which confirmsthe conclusion in Proposition 2. Moreover, from Fig. 5, theincreasing rate of the data offloading ratio is decreases withthe user moving speed.VI. CONCLUSIONS
In this paper, we investigated the effect of user mobilityon the caching performance in a D2D caching network. Thecommunication time of a given user was firstly approximatedby a beta distribution, through matching the first two moments.Then, an approximate expression of the data offloading ratiowas derived. For a homogeneous case, where the averagecontact and inter-contact times are the same for all the userpairs, we evaluated how the user moving speed affects thedata offloading ratio. Specifically, it was proved that the dataoffloading ratio increases with the user speed, assuming thatthe transmission rate is irrelevant to the user speed. Simulationresults validated the accuracy of the approximate expressionof the data offloading ratio, and demonstrated that the dataoffloading ratio increases with the user speed, while theincreasing rate decreases with the user speed.A
PPENDIX
A. Proof of Lemma 1 and Corollary 1
As the timeline of different user pairs are independent, theexpectation of the communication time when user i requestsfile f , which is not in its own cache, can be written as E [ T ci,f ] = lim T ri,f →∞ (cid:90) T ri,f + T d T ri,f − (cid:89) j ∈S ,x j,f =1 (1 − E H i,j ( t )) dt. (11)Since the timeline between each pair of users is modeled asan alternating renewal process, according to [15], we have lim t →∞ Pr[ H i,j ( t ) = 1] = λ Ii,j λ Ci,j + λ Ii,j . Thus, lim t →∞ E [ H i,j ( t )] = λ Ii,j λ Ci,j + λ Ii,j , and then, the expectation in (4) can be obtained. Let λ C = λ Ci,j and λ I = λ Ii,j , and we can get the expression in(9). The variance of the communication time is
Var[ T ci,f ] =2 lim T ri,f →∞ (cid:90) T ri,f + T d T ri,f (cid:90) τT ri,f Pr[ H fi ( t ) = 1 , H fi ( τ ) = 1] dtdτ − (cid:0) E [ T ci,f ] (cid:1) (12)According to [15], Pr[ H i,j ( τ ) = 0 | H i,j ( t ) = 0] = λ Ci,j λ Ci,j + λ Ii,j + λ Ii,j λ Ci,j + λ Ii,j e − ( λ Ci,j + λ Ii,j )( τ − t ) . Then, when T ri,f → ∞ , we can get Pr[ H fi ( τ ) = 1 , H fi ( t ) = 1] = 1 − (cid:89) j ∈S ,x j,f =1 λ Ci,j λ Ci,j + λ Ii,j + (cid:89) j ∈S ,x j,f =1 λ Ci,j ( λ Ii,j + λ Ci,j ) (cid:104) λ Ci,j + λ Ii,j e − ( λ Ci,j + λ Ii,j )( τ − t ) (cid:105) (13)Let u = τ − t and substitute (13) into (12), and we can get(5). Let λ C = λ Ci,j and λ I = λ Ii,j , and we can get (10) withthe binomial theorem.
B. Proof of Lemma 3
When the user speed changes by s times, the expectationof the communication time in (9) keeps the same, while thevariance changes to Var[ T ci,f ] = (cid:20) λ C ( λ C + λ I ) (cid:21) n f n f (cid:88) l =1 (cid:18) n f l (cid:19) ( λ C ) n f − l ( λ I ) l sl ( λ C + λ I ) × (cid:34) T d − sl ( λ C + λ I ) + e − sl ( λ C + λ I ) T d sl ( λ C + λ I ) (cid:35) , (14)To prove that Var[ T ci,f ] decreases with s , we will prove that ∂ Var[ T ci,f ] ∂s < . The partial derivation of Var[ T ci,f ] is ∂ Var[ T ci,f ] ∂s = (cid:20) λ C ( λ C + λ I ) (cid:21) n f n f (cid:88) l =1 (cid:18) n f l (cid:19) ( λ C ) n f − l ( λ I ) l s l ( λ C + λ I ) A ( x ) , (15)where A ( x ) = − x − xe − x − e − x − and x = sl ( λ C + λ I ) T d > . Since A (cid:48) ( x ) = − x ) e − x < − x ) x = 0 , A ( x ) is a decreasing function of x . Thus, A ( x ) < A (0) = 0 . According to (15), when n f > , wehave ∂ Var[ T ci,f ] ∂s < . The parameter α i,f given in (6) is adecreasing function of Var[ T ci,f ] , and thus increases with s . C. Proof of Lemma 4
To simplify the expression in (8), denote r (cid:44) CT d R ∈ (0 , , y (cid:44) T d − E [ T ci,f ] E [ T ci,f ] ≥ , and α (cid:44) α i,f . The expression in (8) canbe rewritten as a function of α , given as P i,f = 1 − (cid:82) r (1 − ur ) u α − (1 − u ) yα − duB ( α, yα ) . (16)et g ( α ) = 1 − P i,f , the derivation of g ( α ) is g (cid:48) ( α ) =1 B ( α, yα ) (cid:40) (cid:90) r (1 − ur ) u α − (1 − u ) yα − [ln u + y ln(1 − u )] du − (cid:90) r (1 − ur ) u α − (1 − u ) yα − duD ( y, α ) (cid:41) , (17)where D ( y, α ) = ψ ( α )+ yψ ( yα ) − (1+ y ) ψ [(1+ y ) α ] and ψ ( · ) is the digamma function. If r = 1 , g (cid:48) ( α ) = ∂ [ y/ (1+ y )] ∂α = 0 .Denote A ( r ) = B ( α,yα ) r g (cid:48) ( α ) , A (1) = 0 and lim r → + A ( r ) =lim r → + (cid:90) r ( r − u ) u α − (1 − u ) yα − [ln u + y ln(1 − u )] du (18)Since r ≥ u ≥ and y ≥ , ( r − u ) u α − (1 − u ) yα − ≥ and ln u + y ln(1 − u ) ≤ , thus, lim r → + A ( r ) ≤ . The derivationof A ( r ) is A (cid:48) ( r ) = (cid:90) r u α − (1 − u ) yα − [ln u + y ln(1 − u )] du − (cid:90) r u α − (1 − u ) yα − duD ( y, α ) . (19)Thus, A (cid:48) (1) = ∂B ( α,yα ) ∂α − ∂B ( α,yα ) ∂α = 0 and lim r → + A (cid:48) ( r ) ≤ .Then, we can get A (cid:48)(cid:48) ( r ) = r α − (1 − r ) yα − [ln r + y ln(1 − r ) − D ( y, α )] . Let A ( r ) = r − α (1 − r ) − yα A (cid:48)(cid:48) ( r ) , then,there is one zero point of A (cid:48) ( r ) = − (1+ y ) xx (1 − x ) in (0 , .Thus, there is one inflection point of A ( r ) . Considering that lim r → + A ( r ) = lim r → − A ( r ) = −∞ , the sign of A ( r ) maybe negative, or first negative, then positive, and then negative,while r increases in (0 , . If A ( r ) < , then A (cid:48)(cid:48) ( r ) < when r ∈ (0 , . However, we have lim r → + A (cid:48) ( r ) ≤ A (cid:48) (1) ,which means that A (cid:48) ( r ) can not be a decreasing function in (0 , . Thus, the sign of A ( r ) is first negative, then positive,and then negative, while r increases in (0 , . Since A (cid:48)(cid:48) ( r ) has the same sign with A ( r ) in (0 , , A (cid:48) ( r ) first decreases,then increases, and then decreases while r increases in (0 , .Considering that lim r → + A (cid:48) ( r ) ≤ and A (cid:48) (1) = 0 , the signof A (cid:48) ( r ) must be first negative, and then positive in (0 , .Therefore, while r increases in (0 , , A ( r ) first decreases,and then increases. Considering that lim r → + A ( r ) ≤ and A (1) = 0 , we have A ( r ) < in (0 , and A ( r ) = 0 when r = 1 . Since g (cid:48) ( α ) = rB ( α,yα ) A ( r ) , we get g (cid:48) ( α ) < in (0 , . Thus, g ( α ) decreases with α , and P i,f = 1 − g ( α ) increases with α . D. Proof of Proposition 2
The data offloading ratio in (7) increases with the increas-ing of P i,f if x i,f = 0 , i ∈ S , f ∈ F . Then, based on Lemmas3 and 4, we can get that the data offloading ratio when user i requests file f from other users, i.e., P i,f , decreases with theuser speed when n f > , otherwise P i,f = 0 . Accordingly,the data offloading ratio when user i requests file f , i.e., x i,f + (1 − x i,f ) P i,f , increases with the user speed when x i,f = 0 and n f > ; otherwise, it keeps the same, where i ∈ S , f ∈ F . Since we consider that not all the users cachethe same contents, there must exists i (cid:48) ∈ S , j (cid:48) ∈ S and f (cid:48) ∈ F ,where x i (cid:48) ,f (cid:48) = 0 and x j (cid:48) ,f (cid:48) = 1 , i.e, n f (cid:48) > . Thus, the dataoffloading ratio increases with the user speed.R EFERENCES[1] Cisco Systems Inc., “Cisco visual networking index: Global mobile datatraffic forecast update, 2015c2020,”
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