Cache-enabled Device-to-Device Communications: Offloading Gain and Energy Cost
11 Cache-enabled Device-to-DeviceCommunications: Offloading Gain and EnergyCost
Binqiang Chen, Chenyang Yang and Andreas F. Molisch
Abstract
By caching files at users, content delivery traffic can be offloaded via device-to-device (D2D) linksif a helper user is willing to transmit the cached file to the user who requests the file. In practice, theuser device has limited battery capacity, and may terminate the D2D connection when its battery haslittle energy left. Thus, taking the battery consumption allowed by the helper users to support D2D intoaccount introduces a reduction in the possible amount of offloading. In this paper, we investigate therelationship between offloading gain of the system and energy cost of each helper user. To this end,we introduce a user-centric protocol to control the energy cost for a helper user to transmit the file.Then, we optimize the proactive caching policy to maximize the offloading opportunity, and optimizethe transmit power at each helper to maximize the offloading probability. Finally, we evaluate the overallamount of traffic offloaded to D2D links and evaluate the average energy consumption at each helper,with the optimized caching policy and transmit power. Simulations show that a significant amount oftraffic can be offloaded even when the energy cost is kept low.
Index Terms
Caching, D2D, Traffic offloading, Energy cost.
I. I
NTRODUCTION
Device-to-device (D2D) communications boosts the throughput of cellular networks by of-floading traffic [1–4], and thus is a promising way to achieve the goal of 5th generation (5G)mobile networks. Traditional D2D communication, which does not cache content locally, can
Binqiang Chen and Chenyang Yang are with the School of Electronics and Information Engineering, Beihang University,Beijing, China, Emails: [email protected], [email protected]. Andreas F. Molisch is with the Ming Hsieh Department ofElectrical Engineering, University of Southern California, Los Angeles, CA, USA. Email: [email protected].
June 10, 2016 DRAFT a r X i v : . [ c s . I T ] J un only offload peer-to-peer (P2P) traffic from cellular networks if source and destination are inproximity at the time they wish to communicate, such as gaming and relaying [1–5]. However,the lion’s share of cellular traffic is video dissemination, a kind of client/server (C/S) services,which will generate more than / of mobile data traffic by 2019 [6].Motivated by the fact that a large amount of content requests are asynchronous but redundant,i.e., the same content is requested at different times, caching at the wireless edge has become atrend for content delivery, which improves the throughput and energy efficiency of the networkand the quality of experience (QoE) of the users [7–13].Recent work [14, 15] has shown that caching at the user devices enables offloading alsoof C/S traffic, in particular video, to D2D connections. Without caching at the devices, theusers need to fetch their requested video via base station (BS) from a remote server. By pre-downloading popular files to users during the off-peak time, say at night, the file requestedby a user can be transmitted via D2D links by other users in proximity that have cached thefile. Such a proactive caching policy largely alleviates the burden to the BSs during the peaktime, yielding high offloading gain [14–18]. To improve the performance of cache-enabled D2Dcommunications, proactive caching policies were optimized in [17, 18], and a distributed reactivecaching mechanism was designed in [16].When D2D communications are used for supporting P2P services, the users acting as trans-mitters are by definition willing to send messages to the destination users. However, offloadingcontent delivery traffic by cache-enabled D2D communications needs the help of other userswho are not obligated to help. Due to the limited battery capacity, a natural question from ahelper user in such a network is: “why should I spend energy of my battery to provide you with faster video download? [14]” This makes the energy consumption of a helper user a bigconcern in cache-enabled D2D communications. In practice, a helper user may only be willingto use a fraction of its battery for transmitting files to other users, if properly rewarded by theoperator. It is thus important to quantify the offloading gain when the helper users’ allowedbattery consumption is taken into account, and to evaluate the average energy consumed by ahelper user to deliver the files to others.In previous research efforts for cache-enabled D2D communications [14–18], the energy of thebattery is implicitly assumed infinite and the energy costs at helper users are never considered.Consequently, (i): maximal transmit power is used by all D2D transmitters to deliver the files, June 10, 2016 DRAFT and (ii): once a D2D link is established, the file is assumed to be able to be delivered completelywithout considering whether there is still energy in the battery or whether a helper is willing tocontribute more energy.In this paper, we quantify the offloading gain of a cache-enabled D2D communication systemby taking maximal permissible battery consumption into account, and evaluate the energy costfor a user to transmit the file. With the allowed battery consumption, a helper user may onlytransmit part of a file to the user requesting the file. To control the energy spent by the helper userfor transmitting a file, we consider a user-centric caching and transmission strategy, where onlythe users within a collaboration distance r c of the requesting user can serve as helpers. Whenthe collaboration distance is large, the probability that the users can fetch their desired contentsvia D2D links is high, and thus more traffic can be offloaded. However, since the possible D2Dlink distance increases, the energy cost of a helper user also grows and then more files cannotbe conveyed completely via D2D links.Aimed to find the maximal offloading gain, we first introduce a user-centric probabilisticcaching policy, where the users proactively cache files according to a r c -dependent cachingdistribution. We optimize the policy to maximize the amount of traffic that can be possiblyoffloaded with a given collaboration distance and the user demands statistics. In [17], a cluster-centric caching policy was proposed, which was optimized to maximize the same objective withgiven cluster size and demands statistics, but is not optimal under the user-centric framework.Then, we optimize the transmit power at each helper to maximize the probability that a requestedfile can be found in adjacent users and transmitted completely via a D2D link, considering twoextreme cases in terms of interference level. Finally, we quantify the total offloaded amountof traffic by taking complete and partial transmission into account, evaluate the average energyconsumption for each D2D transmitter with optimized caching policy and transmit power, andcharacterize the relationship between offloading gain and energy cost.The contributions of this paper are summarized as follows: • We analyze the offloading gain when the user only allows partial energy in its battery to beconsumed. To the best of the authors’ knowledge, this is the first paper to characterize theoffloading gain given limited battery consumption in cache-enabled D2D communications. • We investigate the relationship between the offloading gain of the system and the energycosts of the helper user, and show the impact of the allowed battery consumption.
June 10, 2016 DRAFT
The rest of the paper is organized as follows. Section II presents the system model. SectionIII optimizes the caching policy. Section IV optimizes the transmit power, and evaluates theoffloading gain and energy cost. Section V shows simulations. Section VI concludes the paper.II. S
YSTEM M ODEL
Consider a cell where users’ locations follow a Poisson Point Process (PPP) with density λ .Each single-antenna user has local cache to store files, and can act as a helper to transmit butwith only a fraction of its battery capacity. For simplicity of notation, assume that each useronly stores one file in its local cache as in [15, 18], though generalization to storage of multiplefiles is straightforward.When a helper transmits a file in the local cache via D2D link to a user requesting the file,i.e., a D2D receiver (DR), the helper becomes a D2D transmitter (DT). To control the energyspent by a DT for transmitting to a DR, we introduce a user-centric protocol. A DT will send acached file to the DR only if their distance is smaller than a given value r c , called collaborationdistance . The users with distance r less than r c are called adjacent users . Assume that a fixedbandwidth is assigned to the D2D links to avoid the interference between D2D and cellular links[5], and all DTs transmit with same transmit power. The BS is aware of the files cached at theusers and coordinates the D2D communications. A. Content Popularity and Caching Placement
We consider a static content catalog consisting of N f files that all users in the cell may request,which are indexed in descending order of popularity, i.e., the 1st file is the most popular file.Each file has size of F bits, but the analysis can be easily extended to general cases by dividingeach file into chunks of equal size. The probability that the i th file is requested follows a Zipfdistribution p r ( i ) = i − β (cid:80) Nfk =1 k − β , (1)where (cid:80) N f i =1 p r ( i ) = 1 , and the parameter β reflects how skewed the popularity distribution is,with large β meaning that a few files are responsible for the majority of requests [19].Since deterministic caching policy designed for wired networks with fixed topology is notapplicable for a wireless scenario with user locations that are unknown a priori , we considera probabilistic caching policy. Specifically, each user caches a file according to a r c -dependent June 10, 2016 DRAFT c r
33 35 3 912 11110 adjacentuser
20 3 a user who caches the 3 rd file Fig. 1. Illustration for a user-centric cache-enabled D2D network caching distribution, which is the probability that the i th file is cached at users, i = 1 , · · · , N f .All users in the cell that have cached with the i th file constitute a user set, called the i th helperset , as shown in Fig. 1. B. User Allowed Battery Consumption and Content Delivery
The content popularity usually changes at a much slower speed than the traffic variation ofcellular networks (e.g., one week for movies [14]), which is often regarded as invariant overa period. Consequently, the files can be proactively downloaded by the BS during the off-peak time, without the need to be updated frequently. The energy consumed at users duringcontent placement is negligible since users will usually be connected to the AC power duringthe download time (say at night).Assume that each user requests one file from the catalog independently. If a user can find itsrequested file in the local caches of its adjacent users , a D2D link is established between theuser and its nearest adjacent user cached with the file to convey the file. Assume that each userdevice has the same battery capacity of Q (mAh), and only a fraction ρ of each DT’s batterycapacity can be consumed for transmitting a file to the DR. Denote the operating voltage of theuser device as V . When a DT has consumed ρQV energy to transmit a file to a DR, the DT June 10, 2016 DRAFT interrupts the D2D link, and the DR has to receive the remaining data of the file from the BS. Infact, another helper in the adjacent of the DT can take over the transmission. We do not considerthe hand over among DTs due to the following reason. The distances between the DR and othernot-busy helpers are always longer than the distance between the DR and its first-establishedDT, and hence the corresponding channel conditions are worse in high probability (e.g., when r c = 100 m and β = 1 , this probability is 97 %). As a result, the handover will introduce higherenergy cost for other DTs and more signaling overhead for the BS to coordinate. Therefore, twocases may occur for the established D2D links depending on their channel conditions. • Complete transmission:
A DR can receive a complete file via D2D link, which is calleda satisfied DR . • Partial transmission:
A DR only receives a fraction of the file from a DT, which needsto access to the BS to fetch the remaining file.If a user cannot find its requested file in the local caches of its adjacent users , the userfetches the file from the BS. If a user can find the desired file in its own local cache, such aself-serve can offload traffic without establishing D2D link. Since we focus on the energy costof a DT in cache-enabled D2D communications, we ignore self-serve (also called self-offloadingin literature) in the forthcoming analysis (similar to [15–18]), but we will evaluate its impactvia simulations in section V.We consider two metrics regarding offloading by the cache-enabled D2D communications. • Offloading probability:
This is the probability that a DR enjoys complete transmission viaD2D links, which reflects the percentage of the satisfied users. • Offloading ratio:
This is the ratio of the amount of data offloaded by both complete andpartial transmission via D2D links to the total amount of data in the cell, which reflects theoffloading gain of the system.To focus on the energy cost issue, we assume that the distance between DT and DR remains fixed ρ can reflect the user incentive in terms of battery consumption to serve as a helper. We assume that all users are initiallywith full battery and hence each user allows to employ the same amount of energy to help others. In practice, the user devicesmay have different battery capacities. Moreover, a helper may be requested more than once over several hours before rechargingits battery, especially when the file is very popular. When a DT serves the second request, the remaining energy in its batterymay be less than ρQV . For analysis simplicity, we assume that one user only sends one request, and hence each DT is onlyrequested once. The impact that a DT serves multiple requests will be shown via simulation later. June 10, 2016 DRAFT during transmission (again following most of previous works [14–18] ), although user mobilityis one of the key factors that affects the offloading gain of cache-enabled D2D communications.III. O
PTIMAL C ACHING POLICY
To optimize the probabilistic caching policy with known user demand statistics, we need tofind the optimal caching distribution. Because the contents are proactively placed at users beforethey initiate requests, we optimize the caching distribution to maximize offloading opportunity asin [14–17], defined as the probability that the desired file of a user can be found in adjacent users.Such an opportunity reflects how much traffic can be possibly offloaded by D2D communicationsfor a given collaboration distance under the assumption of infinite battery capacity.Denote the probability that the i th file is cached at a user as p c ( i ) . Then, the set { p c ( i ) } =[ p c (1) , p c (2) , ..., p c ( N f )] constitutes the caching distribution. The locations of the users whobelong to the i th helper set follow a PPP with density λ i = λp c ( i ) according to the thinningproperty of PPP [20]. Thus, the probability that a user requesting the i th file can find its desiredfile in the cache of any user within the collaboration distance r c is p f ( i ) = 1 − e − λ i πr c . Then,the offloading opportunity with given caching distribution and r c can be derived as p o = (cid:80) N f i =1 p r ( i ) p f ( i ) = (cid:80) N f i =1 p r ( i )(1 − e − λp c ( i ) πr c ) . (2)The optimal caching distribution that maximizes the offloading opportunity can be found fromthe following problem max p c ( i ) p o s.t. (cid:80) N f i =1 p c ( i ) = 1 , p c ( i ) ≥ , i = 1 , · · · , N f . (3)Because the objective function is the sum of N f exponential functions and the constraints arelinear, this problem is convex [21]. It is not hard to show from its Karush-Kuhn-Tucker (KKT)conditions that the optimal caching distribution should satisfy the following conditions p ∗ c ( i ) = (cid:104) λπr c ln( p r ( i )) − λπr c ln( − µπλr c ) (cid:105) + , (4)where ≤ i ≤ N f , (cid:80) N f i =1 p ∗ c ( i ) = 1 , p r ( i ) is the Zipf distribution in (1), and [ x ] + = max( x, . Proposition 1: If ( N f ) Nf N f ! < e λπr cβ , then the optimal caching distribution is p ∗ c ( i ) = N f (cid:16) βλπr c (cid:80) N f j =1 ln( ji ) (cid:17) . (5) June 10, 2016 DRAFT
Otherwise, the optimal caching distribution is p ∗ c ( i ) = i ∗ (cid:16) βλπr c (cid:80) i ∗ j =1 ln( ji ) (cid:17) , i ≤ i ∗ , , i ∗ < i ≤ N f , (6)where i ∗ is upper and lower bounded as λπr c β − ≤ i ∗ ≤ λπr c β + ln( (cid:112) πN f ) + 1 . Proof:
See Appendix A.The gap between the upper and lower bounds of i ∗ in (6) is ln( (cid:112) πN f ) + 2 , which is small.For example, when N f = 1000 , the gap equals to . . This suggests that i ∗ and hence theoptimal caching distribution p ∗ c ( i ) , i = 1 , · · · , N f can be obtained efficiently. p ∗ c ( i ) depends onthe collaboration distance r c , user density λ , as well as content statistics N f and β .When r c → ∞ , ( N f ) Nf N f ! < e λπr cβ holds, and according to (5) p ∗ c ( i ) = N f . In this case, theoptimal caching distribution is a uniform distribution, i.e., each user can randomly choose a fileto cache, because the number of adjacent users for any user trends to infinity.By using the conditions below (6) and setting i ∗ = N f , it is not hard to show that when r c ≤ (cid:113) ( N f +1) βπλ , ( N f ) Nf N f ! ≥ e λπr cβ . In this case, p ∗ c ( i ) is computed with (6), and the less popularfiles with indices larger than i ∗ are never cached at the users. Because the number of adjacentusers are limited when r c is small, only the files with high popularity are cached. When r c → , p ∗ c (1) = 1 and p ∗ c ( i ) = 0 , < i ≤ N f , i.e., only the most popular file is cached at each user.IV. O FFLOADING G AIN AND A VERAGE E NERGY C OSTS
In this section, we investigate the offloading gain of the system and the energy cost at eachDT. To this end, we first optimize the transmit power of each DT to maximize the offloadingprobability , which yields maximal user satisfaction rate and hence high offloading gain. Then,we evaluate the offloading ratio and the average energy consumed at each DT to transmit a filevia D2D links with the optimized transmit power and optimized caching policy.Considering that the interference among D2D links has large impact both on the offloadinggain and the energy cost, for mathematical tractability we analyze two extreme cases in termsof interference level: full reuse and time division multi-access (TDMA). With full reuse, allDTs in a cell simultaneously transmit over the time and frequency resources are assigned forD2D communications without any interference coordination. With TDMA, only one DT in thewhole cell transmits at a time, and the DTs are scheduled according to round robin (or random)
June 10, 2016 DRAFT scheduling with equal time slot duration. While further improvements could be achieved throughscheduling, it is known that optimal scheduling in D2D networks is NP-hard. On the other hand,cluster-based scheduling as in [15] is not aligned with the user-centric transmission strategy thatforms the basis for our model.
A. Case 1: Full Reuse
Once a D2D link is established, the DT can transmit its cached file to the DR that requeststhe file. In the full reuse case, each DR treats the interference among the D2D links as noisewhen decoding the desired signal. The signal to interference plus noise ratio (SINR) at the DRrequesting the i th file from its corresponding DT is γ ( i, r ) = P t hr − α (cid:80) j (cid:54) = i P t h j r − αj + σ = hr − α I i,r + σ , (7)where P t is the transmit power at each DT, h is the channel power gain that follows an exponentialdistribution with unit mean for Rayleigh fading, r is the distance between the DT and the DR, α is the path loss exponent, I i,r = (cid:80) j (cid:54) = i h j r − αj is the total interference from other DTs normalizedby P t , σ is the variance of white Gaussian noise, and σ = σ /P t . Then, the data rate is R ( i, r ) = W log (cid:16) hr − α I i,r + σ (cid:17) , where W is the bandwidth assigned to D2D links.To evaluate the energy cost of each DT, we consider both circuit power and transmit power.Then, the energy consumed to transmit the i th file via a D2D link with distance r is E ( i, r ) = FW log (cid:18) hr − αIi,r + σ (cid:19) (cid:16) η P t + P c (cid:17) , (8)where η is the power amplifier efficiency and P c is the circuit power at the DT.Because only a fraction ρ of the battery capacity is permitted to be used at each DT to helpa DR, a DT can transmit the i th file completely only if E ( i, r ) ≤ ρV Q .
1) Optimal Transmit Power:
Because the files not completely delivered via D2D links needto be fetched from the BS, which not only introduces extra signaling overhead but also maydegrade the user experience, we optimize the transmit power at a DT to maximize the usersatisfaction rate. In other word, we maximize the offloading probability for a given collaboration Note that this model neglects shadowing and incorporating shadowing would lead to a change of the exponential channel gaindistribution to an approximate lognormal distribution [22]. We neglect shadowing, in line with most works in D2D literature.
June 10, 2016 DRAFT0 distance r c , which is the probability that a requested file can be found in adjacent users andtransmitted completely via a D2D link. Proposition 2:
The offloading probability in the full reuse case is p ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) e − φ i (Γ ,r ) dr, (9)where Γ = e F ( Pt + ηPc ) ln 2 WρQV η − , f i ( r ) = 2 πrλ i e − λ i πr is the probability density function (pdf) ofthe D2D link distance, φ i ( x, y ) (cid:44) xy α σ + π ( λ I ξ − λ di ξ ) y x /α , λ I = (cid:80) N f i =1 λ di is the densityof all DTs and λ di = λ i (cid:18) − (cid:16) λp r ( i )3 . λ i (cid:17) − . θ i (cid:19) is the density of DTs cached with the i thfile, θ i = Γ(3 . , − Γ ( . , (3 . λ i + λp r ( i )) πr c ) Γ(3 . , − Γ(3 . , . λ i πr c ) , Γ( s, x ) = (cid:82) ∞ x t s − e − t dt is the upper incomplete gammafunction [23], ξ (cid:44) (cid:82) + ∞ t α/ dt , and ξ (cid:44) (cid:82) x − α t α/ dt . Proof:
See Appendix BThe expression in (9) depends on the values of λ , r c , ρ and P t , but not on the user’s locationand channel. To maximize the offloading probability for the cache-enabled D2D communicationswith given values of λ , r c and ρ , the transmit power at each DT can be optimized as max P t p ( P t , ρ ) s.t. < P t ≤ P max , (10)where P max is the maximal transmit power of a DT.Due to the complicated expression of p ( P t , ρ ) , in general the optimal solution P ∗ t can only befound by using similar method as in [15]. When r c is small, all D2D links experience a line ofsight (LOS) environment [17], i.e., α = 2 . In such a special case, both closed-form expressionsof p ( P t , ρ ) and P ∗ t can be obtained. Proposition 3:
When α = 2 , the offloading probability can be approximated as p ( P t , ρ ) ≈ (cid:80) N f i =1 p r ( i ) πλ i ϕ i ( P t ) (1 − e − ϕ i ( P t ) r c ) , (11)which first increases and then decreases with P t , where ϕ i ( P t ) is defined in (C.2). Proof:
See Appendix CThe approximation is accurate when the file catalog size N f is large. As shown in AppendixC, the closed-form solution of P ∗ t can be obtained by solving a cubic equation, which is notprovided herein for conciseness. June 10, 2016 DRAFT1
2) Offloading Gain:
To evaluate the offloading gain provided by cache-enabled D2D commu-nications, which is characterized by the offloading ratio, both complete transmission and partialtransmission should be taken into account.
Proposition 4:
The offloading ratio in the full reuse case is p a ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r )ln(1+Γ ) (cid:82) Γ e − φi ( t,r ) t dtdr, (12)and p ( P t , ρ ) ≤ p a ( P t , ρ ) ≤ p o , both equalities will hold if ρ → ∞ or if r α σ → and λ I → . Proof:
See Appendix DThe first condition ρ → ∞ means that all helpers have infinite battery capacity, which is thescenario where the user devices are charging when acting as the DTs. The second condition r α σ → and λ I → indicate that all interference are eliminated and the SNR is infinite,because r α σ = 1 / ( P t r − α σ ) is the inverse of the receive signal to noise ratio (SNR) at the DRaveraged over fading. In this case, although battery is limited, the data rate can be extremelyhigh to complete all transmission via D2D links. In either condition, the offloading probability,offloading ratio and offloading opportunity are equal.
3) Energy costs:
In what follows, we derive the energy cost of a DT for a given transmitpower and caching policy, with which we can evaluate the energy cost of a DT with the optimizedtransmit power and caching policy.
Proposition 5:
The average energy consumed at a DT for a complete transmission is ¯ E = ρV Q − ρV Q (cid:80) N f i =1 p r ( i ) p ( P t ,ρ ) (cid:82) r c f i ( r ) ln(1 + Γ ) (cid:82) e − φ i ( e t − ,r ) dtdr (13) Proof:
See Appendix EFor the satisfied DR, its DT consumes less energy than the allowed battery consumption. Forthe D2D link with partial transmission, the DT consumes energy ρV Q to convey part of thefile to the DR. Because a percentage p ( P t ,ρ ) p o of requested files can be completely conveyed viaD2D transmissions, the average energy consumed by a DT can be obtained as ¯ E a = p ( P t , ρ ) p o ¯ E + (cid:18) − p ( P t , ρ ) p o (cid:19) ρV Q ( a ) = ρV Q − ρV Q (cid:80) N f i =1 p r ( i ) p o (cid:82) r c f i ( r ) ln(1 + Γ ) (cid:82) e − φ i ( e t − ,r ) dtdr, (14)where (a) is obtained by substituting (2), (12) and (13).To reflect how much energy consumed at a DT by serving as a helper occupies the batterycapacity, we define the energy cost as ¯ e = ¯ E a V Q . June 10, 2016 DRAFT2
B. Case 2: TDMA
By using TDMA, the DT of a randomly scheduled D2D link transmits the requested file toits corresponding DR, while other DTs stay mute. The data rate of each DR is given by R ( r ) = WN a log (1 + P t hr − α σ ) , (15)where N a = p o λS is the average number of DRs in a cell and S is the area of the cell. Themuting DTs can turn off some circuits to save energy. We call the circuit power consumed by amuting DT as idle power , denoted as P c I , which ranges from a few to tens of mW [24]. Then,the energy consumed at a DT to transmit a file via the D2D link can be obtained as, E ( r ) = FN a R ( r ) (cid:16) η P t + P c (cid:17) + (cid:16) FR ( r ) − FN a R ( r ) (cid:17) P c I = FW log (1+ Pthr − ασ ) (cid:16) η P t + P Tc (cid:17) , (16)where P Tc (cid:44) P c + ( N a − P c I . Note that E ( r ) is not related to the file index i since the receivedsignals are only corrupted by noise, which is different from E ( r, i ) .
1) Optimal Transmit Power:
From the definition of the offloading probability and (16), it iseasy to obtain p ( P t , ρ ) by letting λ I ξ − λ di ξ = 0 in (9) as p ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) e − Γ r α σ dr, (17)where Γ = e F ( Pt + ηPTc ) ln 2 WρQV η − . With the growth of both the number of DRs N a and idle power P c I , the circuit power P Tc and hence Γ increase, which results in the reduction of p ( P t , ρ ) .To maximize the offloading probability for the cache-enabled D2D communications, thetransmit power at each DT can be optimized as follows max P t p ( P t , ρ ) s.t. < P t ≤ P max . (18)Again, the closed-form expression of p ( P t , ρ ) is hard to obtain in general. When α = 2 , byusing the similar way to derive (11), we can approximate the offloading probability as p ( P t , ρ ) ≈ (cid:80) N f i =1 p r ( i ) πλ i − e − ( σ πλi ) r c σ Γ + πλ i . (19)Despite that the offloading probability has complicated expression in general cases, the closed-form solution of the optimal transmit power for all values of α can be found as follows. June 10, 2016 DRAFT3
Proposition 6:
The optimal transmit power is P ∗ t = P max , P max < ηP Tc (cid:16)(cid:113) aηP Tc + − (cid:17) ηP Tc (cid:16)(cid:113) aηP Tc + − (cid:17) , otherwise , (20)where a = F ln 2 W ρQV η . Proof:
See Appendix F
2) Offloading Gain:
By considering both the complete and partial transmission, we can obtainthe offloading ratio by using a similar method as for the proof of Proposition 4 as p a ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r )ln(1+Γ ) (cid:82) Γ e − trασ t dtdr = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r )ln(1+Γ ) e r α σ (cid:82) Γ e − ( t +1) rασ t +1 d ( t + 1) dr = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r )ln(1+Γ ) e r α σ ( Ei ( − r α σ (Γ + 1)) − Ei ( − r α σ )) dr, where Ei ( x ) = (cid:82) x −∞ e t t dt is a frequently-used special function.
3) Energy costs:
By using the similar derivation as for Proposition 5, we can obtain theaverage energy consumed at a DT for a complete transmission with given transmit power andcaching policy as ¯ E = ρV Q − ρV Q (cid:80) N f i =1 p r ( i ) p ( P t ,ρ ) (cid:82) r c f i ( r ) ln(1 + Γ ) (cid:82) e − ( e t − r α σ dtdr. (21)Since a percentage p ( P t ,ρ ) p o of the requested files can be completely conveyed via D2D trans-missions, and the DT only consumes energy ρV Q to help the DR for a partial transmission, byconsidering both complete and partial transmission, the average energy consumption for a DTcan be obtained as ¯ E a = p ( P t , ρ ) p o ¯ E + (cid:18) − p ( P t , ρ ) p o (cid:19) ρV Q ( a ) = ρV Q − ρV Q (cid:80) N f i =1 p r ( i ) p o (cid:82) r c f i ( r ) ln(1 + Γ ) (cid:82) e − ( e t − r α σ dtdr, (22)where (a) is obtained by substituting (2), (17) and (21).Then, the energy cost for a DT to transmit a file is ¯ e = ¯ E a V Q .V. S IMULATIONS
In this section, we validate previous analytical results, and evaluate the offloading gain of thesystem and the energy cost at each DT via simulations.
June 10, 2016 DRAFT4 File Index O p t i m a l C a c h i ng P r obab ili t y r c =20m, =0.01, - =1r c =100m, =0.01, - =1r c =500m, =0.01, - =1r c =100m, =0.005, - =1r c =100m, =0.01, - =0.5 (a) Optimal caching distribution versus file index Collaboration Distance r c (m) O ff l oad i ng O ppo r t un i t y Optimal-5Optimal-2Optimal-1Popularity-5Popularity-2Popularity-1Uniform-5Uniform-2Uniform-1 (b) Offloading opportunity versus r c Fig. 2. Optimal caching distribution and offloading opportunity with different caching policies.
We consider a square cell with side length m. The users’ locations follow a PPP with λ = 0 . , so that in average there is one user in a m × m area. The path-loss model is . . ( r ) , where r is the distance of the D2D link [17]. W = 20 MHz and σ = − dBm, the maximal transmit power of each DT is P max = 200 mW ( dBm), the circuit powerfor an active DT is P c = 115 . mW, and the power amplifier efficiency is η = 0 . [4]. The typicalidle power for a muting DT with TDMA is P c I = 25 mW [24]. The operating voltage at a userdevice is V = 4 V and the battery capacity is Q = 1800 mAh (typical for current-generationsmartphones). The file catalog contains N f = 1000 files, where each file has a size of Mbytes(a typical video size on the Youtube website [14]). The parameter of the Zipf distribution β = 1 .This setup is used in the sequel unless otherwise specified.Besides the optimized caching policy in Proposition 1 (with legend “Optimal-x”), we alsoprovide the results for uniform caching policy (i.e., each user selects a file from the cataloguniformly, with legend “Uniform-x”) and popularity based caching policy (i.e., each user selectsa file from the catalog according to the content popularity, with legend “Popularity-x”) as thebaseline caching policies, where “x” is the number of files cached at each user. A. Optimal Caching Distribution and Offloading Opportunity
In Fig. 2(a), we show the optimal caching distribution for different collaboration distance r c , Zipf parameter β , and user density λ . With the increase of β and λ or decrease of r c , the June 10, 2016 DRAFT5
Transmit Power (mW) O ff l oad i ng P e r f o r m an c e TDMA-S:Offloading ratioFull Reuse-S:Offloading ratioTDMA-S:Offloading probabilityFull Reuse-S:Offloading probability (a) Offloading performance
Transmitter Power (mW) E ne r g y C o s t TDMA-SFull Reuse-STDMA-NFull Reuse-N (b) Energy costFig. 3. Validation of the analytical results and show the impact of transmit power P t . r c = 100 m and ρ = 0 . . S-Simulationresults and N-Numerical results. (cid:13) and (cid:3) in both Fig. 3(a) and Fig. 3(b) represent the numerical results. probability of caching popular files increases, which makes the distribution more “skewed”, andvice versa. When r c is large enough, say r c = 500 m, the caching distribution reduces to auniform distribution. When r c is very small, say r c = 20 m, the caching distribution makes theprobability for caching most popular files very high, which agrees with Proposition 1.In Fig. 2(b), we show the simulated offloading opportunity versus the collaboration distance,where each user allows to cache more files. When each user has cached one or two files, theoptimized caching policy has the potential to offload more traffic than the popularity basedcaching policy and even more than the uniform caching policy. When each user caches morefiles, the offloading opportunity is improved for all policies, as expected. For large value of r c , say m, the offloading opportunities of all caching policies can achieve nearly . . This indicatesthat when r c is large and a user is willing to cache more files, uniform caching policy can alsoachieve high traffic offloading, despite that it is not good for D2D throughput in general [15].Nonetheless, the offloading opportunity for caching one file and more files exhibits the sametrend, which indicates that caching more files offers essentially the same insight with cachingone file, which justifies our assumption in previous analysis. B. Validation of Analytical Results
In Fig. 3, we compare the numerical and simulation results for the offloading probability p ( P t , ρ ) , p ( P t , ρ ) and offloading ratio p a ( P t , ρ ) , p a ( P t , ρ ) respectively for the full reuse and June 10, 2016 DRAFT6
TDMA cases in Fig. 3(a) and energy cost ¯ e and ¯ e for the two cases in Fig. 3(b) versus P t .We can see that the numerical results almost overlap with simulation results, which validatesour analysis. Moreover, the trend for p ( P t , ρ ) and p ( P t , ρ ) changing with P t are the same as p a ( P t , ρ ) and p a ( P t , ρ ) , respectively. This suggests that the optimized transmit power to maximizethe offloading probability can also maximize the offloading gain. For the full reuse case , p a ( P t , ρ ) first increases to achieve the maximal value and then decreases, and ¯ e first decreases and thenincreases. This is due to the severe interference and the allowed battery consumption. ComparingFig. 3(a) and Fig. 3(b), we can observe that the optimal value of P t to maximize the offloadingprobability p a ( P t , ρ ) can nearly minimize the energy cost ¯ e . This is because to maximize theoffloading ratio the transmit power should be reduced in an interference environment and thenmore DTs consume less power than the allowed battery consumption. For the TDMA case in theconsidered setting, increasing P t can always improve p a ( P t , ρ ) . Moreover, ¯ e always decreases,because increasing P t can shorten the duration of transmission and hence can reduce the circuitpower consumption.In Fig. 4, we compare the numerical and simulation results for the offloading probability underspecial channel models. By using the same approach as for deriving the closed form solution forthe LOS channel, we can also derive the closed-form expressions for the offloading probabilityunder α = 4 , which are not shown for conciseness. We can see that the numerical results almost Transmit power (mW) O ff l oad i ng P r obab ili t y Full Reuse-S-r c =100mTDMA-S-r c =100mTDMA-S-r c =50mFull Reuse-S-r c =50m (a) Offloading probability, α = 2 Transmit power (mW) O ff l oad i ng P r obab ili t y TDMA-S-r c =100mFull Reuse-S-r c =100mFull Reuse-S-r c =50mTDMA-S-r c =50m (b) Offloading probability, α = 4 Fig. 4. Validation of the analytical results in special cases. ρ = 0 . . S-Simulation results. (cid:13) and (cid:3) represent the numericalresults. For α = 2 , the interference generated by the DTs far away from m is ignored in analytical results. June 10, 2016 DRAFT7 overlap with the simulation results, which indicates that the approximations in (11) and (19) areaccurate. When α = 2 and α = 4 , the offloading probability for the full reuse case first increasesand then decreases with P t , and the offloading probability for TDMA case always grows with P t in the considered setting, which are the same as Fig. 3. It implies that the optimal solutionof P t can be found by bisection searching efficiently in general channels, because ≤ α ≤ inpractical channels among D2D links [17]. C. Impacts of Key Parameters and Self Offloading
In what follows, we analyze the impact of the file size, idle power, content popularity, energyconsumption allowed by user device, as well as the self offloading on the offloading gain andenergy cost for full reuse and TDMA cases with numerical results. The optimized transmit powerand optimized caching policy are used, unless otherwise specified.In Fig. 5, we show the impact of file size. We can see that for TDMA, P ∗ t (cid:54) = P max onlywhen ρ = 0 . and F > . GBytes. In all other cases, transmitting with P max is optimal forTDMA. We can also observe that with the growth of F , the offloading probability for bothfull reuse and TDMA cases decreases. This implies that cache-enabled D2D communications ismore applicable for offloading traffic of delivering small size files.In Fig. 6, we show the impact of the idle power P c I . We can see that the offloading ratio ofTDMA is always larger than that of the full reuse. The increase of P c I directly leads to higher File Size (Mbytes) O ff l oad i ng P r obab ili t y TDMA-r c =100m, ; =0.3TDMA-r c =50m, ; =0.01TDMA-r c =100m, ; =0.01Full Reuse-r c =100m, ; =0.3Full Reuse-r c =50m, ; =0.01Full Reuse-r c =100m, ; =0.01 (a) Offloading probability File Size (Mbytes) -1 O p t i m a l T r an s m i t P o w e r P * t ( m W ) Full Reuse-r c =50m, ; =0.01TDMA-r c =50m, ; =0.01Full Reuse-r c =100m, ; =0.01TDMA-r c =100m, ; =0.01TDMA-r c =100m, ; =0.3Full Reuse-r c =100m, ; =0.3 P max (b) Optimal transmit powerFig. 5. Impact of file size F on the offloading probability and optimal transmit power. June 10, 2016 DRAFT8
Idle Power (mW) O ff l oad i ng R a t i o TDMA-r c =100m, ; =0.01TDMA-r c =50m, ; =0.01Full Reuse-r c =100m, ; =0.01Full Reuse-r c =50m, ; =0.01 (a) Offloading ratio Idle Power (mW) E ne r g y C o s t Full Reuse-r c =100m, ; =0.01Full Reuse-r c =50m, ; =0.01TDMA-r c =100m, ; =0.01TDMA-r c =50m, ; =0.01 (b) Energy costFig. 6. Impact of the idle power of a muting DT under TDMA energy cost at each DT with TDMA. When only 1% battery capacity can be used and P c I > mW, the energy cost at a DT with TDMA is larger than full reuse. When ρ = 30 % (not shownin the figure), the energy cost of full reuse is around 12%, which is much larger than the energycost of TDMA that changes from 0 to 2.5 % with the increase of P c I .In Fig. 7, we show the impact of the Zipf parameter β and self offloading. As expected, theoffloading ratio increases rapidly with β due to the high cache hit rate. The energy cost increaseswith β for TDMA, but first increase and finally decreases with β for full reuse. This can be Zipf Parameter- - O ff l oad i ng R a t i o TDMA-D2D-r c =100mTDMA-D2D-r c =50mFull Reuse-D2D-r c =100mFull Reuse-D2D-r c =50m (a) Offloading ratio Zip Parameter- - E ne r g y C o s t Full Reuse-D2D-r c =100mFull Reuse-D2D-r c =50mTDMA-D2D-r c =100mTDMA-D2D-r c =50m (b) Energy costFig. 7. Impact of content popularity and self offloading. (cid:13) and (cid:3) represent taking self offloading into account, ρ = 0 . . June 10, 2016 DRAFT9 explained as follows. On one hand, larger β leads to smaller D2D link distance, which reducesthe energy cost. On the other hand, larger β leads to more DTs, which generates more severeinterference for full reuse and longer muting time for TDMA, both of which increase the energycost. Since the reduction in D2D link distance is dominant for full reuse, the energy cost finallydecreases. Besides, the offloading ratio including both cache-assisted D2D communications andself offloading is larger than that only contributed by the cache-enabled D2D, and the energycost including both is less than that only considering D2D. However, the contribution of selfoffloading on the performance is marginal.In Fig. 8, we show the impact of the allowed battery consumption ρ . We can see that theoffloading ratio first increases rapidly and then slowly with ρ , whereas the energy cost increaseswith ρ but is always much less than the allowed battery consumption. This is because for D2Dlinks with better channel state, the DTs can transmit complete files to corresponding DRs withless than ρQV of energy. The results suggest that choosing a proper ρ is important for operatorsto balance benefits (e.g., the offloading gain) and costs (e.g., rewarding users for a larger valueof ρ ). We can also observe that the energy cost for the full reuse case grows more rapidly thanthe TDMA case, because there are more partial transmission links in the full reuse case thanthe TDMA case, whose energy consumed by each DT equals the allowed battery consumption. ; O ff l oad i ng R a t i o TDMA-r c =100mTDMA-r c =50mFull Reuse-r c =100mFull Reuse-r c =50m (a) Offloading ratio ; E ne r g y C o s t Full Reuse-r c =100mFull Reuse-r c =50mTDMA-r c =100mTDMA-r c =50m (b) Energy costFig. 8. Impact of the allowed fraction of battery consumption ρ . June 10, 2016 DRAFT0
0% 5% 10% 15% 20%
Energy Cost O ff l oad i ng R a t i o TDMA- ; =0.3TDMA- ; =0.3TDMA- ; =0.1TDMA- ; =0.1Full Reuse- ; =0.3Full Reuse- ; =0.3Full Reuse- ; =0.1Full Reuse- ; =0.1 Optimal CachingPopularityCaching (a) Changing r c from 10 m to 400 m Energy Cost O ff l oad i ng R a t i o TDMA-N r =1TDMA-N r =5TDMA-N r =10Full Reuse-N r =1Full Reuse-N r =5Full Reuse-N r =10 Tradeoff Non-tradeoff ; (b) Changing ρ from 0 to 1, r c = r ∗ c .Fig. 9. Relationship between offloading gain and energy cost, λ = 0 . . N r is the number of requests each user sends. D. Relationships Between Offloading Gain and Energy Cost
In the sequel, we show the relation between offloading ratio and energy cost with the optimaltransmit power and optimal caching policy.In Fig. 9(a), the offloading ratio is adjusted by changing the collaboration distance r c from m to m, where popularity based policy is also simulated for comparison. We can observean optimal r c to maximize the offloading ratio for a given ρ , which are 350 m, 300 m, 100m, 80 m, respectively for “TDMA, ρ = 0.3”, “TDMA, ρ = 0.1”, “Full Reuse, ρ = 0.3”, and“Full Reuse, ρ = 0.1” with the optimal caching policy. This is because the full reuse schemeis interference limited and the TDMA scheme is transmit power limited. With the growth of r c , the average D2D communication distance increases, and hence the energy cost increases,whereas the very limited battery consumption allowed for helping others makes the offloadingratio decrease. Compared with popularity based caching policy, the optimal caching policy canimprove the offloading ratio and reduce energy cost.In Fig. 9(b), the offloading ratio is adjusted by changing ρ from to , where the optimal r c maximizing the offloading gain and optimal caching policy is used. To show what happens if ahelper serves multiple requests, here each user sends N r requests sequentially according to theZipf distribution. As a result, the helper that cached the most popular files may be requestedmultiple times and serve as a DT for multiple users. When N r = 1, there exists a tradeoffbetween offloading gain and energy cost. When N r > , a large energy cost may not yield a June 10, 2016 DRAFT1 high offloading gain. This is because with larger ρ , a DT will consume more energy beforeinterrupting the transmission for a D2D link with bad channel condition, and will soon run outof battery for serving subsequent requests. Consequently, each DT can serve fewer requests,which leads to the reduction of the offloading gain. Nonetheless, it is interesting to observe thatthe energy cost to support high offloading ratio is low. Even when N r = 10, to offload around of traffic, the average energy consumption at each DT with TDMA only consumes around battery capacity. This suggests that cache-enabled D2D communications is cost-efficient foroffloading by optimizing the collaboration distance and selecting a proper transmission scheme.In the following, we provide a brief summary of the simulation results. • Caching policy : When the collaboration distance is small or only one file is cached, opti-mizing caching policy can improve offloading gain and reduce energy cost. • Transmission scheme : When the file size is not large, TDMA is superior to full reuse withtypical value of idle power. The optimization of transmit power to maximize the offloadingprobability also helps increase the offloading gain and reduce the energy cost. For theTDMA case, the DT can simply transmit with P max to maximize the offloading gain if thefile size is not too large. For the full reuse case, optimizing transmit power is important. • Parameter setting : There exists an optimal value of r c that maximizes the offloading gainfor a given ρ . When each DT only serves one request, both offloading gain and energy costincrease with ρ . When each DT could serve multiple requests, a large value of ρ not onlycauses large energy cost but also reduces the offloading gain. • Gain and costs : When the file size is not very large, a high offloading gain can be achievedby a low energy cost if the collaboration distance, transmission scheme and caching policyare judiciously designed and the value of ρ is properly selected.VI. C ONCLUSION
In this paper, we quantified the offloading gain of cache-enabled D2D communications aftertaking the user allowed battery consumption into account and evaluated the energy consumed ata helper user. We considered a user-centric caching and transmission protocol, where the energyconsumed for transmission can be controlled by a collaboration distance. We first optimized aproactive caching policy with given collaboration distance, with which the offloading opportunitycan be maximized. For either full reuse or TDMA (round-robin) scheduling, we then optimized
June 10, 2016 DRAFT2 the transmit power to deliver a file via D2D link, where the percentage of satisfied users ismaximized. With the optimized probabilistic caching policy and optimized transmit power,we evaluated the offloading gain of the system and the energy cost of a D2D transmitter,and investigated their relationship. Simulation results showed that high offloading gain canbe obtained in practice by cache-enabled D2D with low energy cost at each help user, if thecollaboration distance, transmission scheme and caching policy are optimized and the allowedbattery consumed by each D2D transmitter for conveying one file is properly set.A
PPENDIX AP ROOF OF P ROPOSITION x i (cid:44) ln( p r ( i )) λπr c and v (cid:44) λπr c ln( − µπλr c ) . Then, considering (cid:80) N f i =1 p c ( i ) = 1 and from (4)we have (cid:80) N f i =1 [ x i − v ] + = 1 . (A.1)Since problem (3) is convex, the solution of v found from this necessary condition is globallyoptimal, and with it the optimal caching distribution can be obtained.As shown in (4), p ∗ c ( i ) decreases when p r ( i ) decreases. As shown in (1), p r ( i ) is a decreasingfunction of file index i . This indicates that p ∗ c ( i ) is a decreasing function of i . Thus, there existsa unique file index i ∗ ≤ N f , with which p ∗ c ( i ) > if i ≤ i ∗ , p ∗ c ( i ) = 0 otherwise. As a result,finding the solution of v from (A.1) is equivalent to finding the index i ∗ from (cid:80) i ∗ i =1 ( x i − v ) = 1 .Once i ∗ is found, the solution of (A.1) can be obtained as v ∗ = (cid:80) i ∗ i =1 x i − i ∗ . (A.2) Case 1 : When i ∗ = N f , from (4) and p c ( i ) > we have p ∗ c ( N f ) = x N f − v > , which canbe rewritten as (cid:80) N f i =1 ( x i − x N f ) < after substituting v in (A.2), then (cid:80) N f i =1 ( x i − x N f ) = (cid:80) N f i =1 ln( p r ( i )) − ln( p r ( N f )) λπr c = βλπr c (cid:80) N f i =1 ln( N f /i ) = βλπr c ln( N Nff N f ! ) < , (A.3)which can be rewritten as ( N f ) Nf N f ! < e λπr cβ . By substituting v ∗ in (A.2) into (4), the optimalcaching distribution can be derived as p ∗ c ( i ) = βλπr c N f (cid:80) N f j =1 ln( ji ) + N f . (A.4) June 10, 2016 DRAFT3
Case 2 : When i ∗ < N f , p ∗ c ( i ∗ ) = x i ∗ − v > and x i ∗ +1 − v ≤ . By substituting v in (A.2)into these two inequalities, we have (cid:80) i ∗ i =1 ( x i − x i ∗ +1 ) ≥ and (cid:80) i ∗ i =1 ( x i − x i ∗ ) < , which canbe further derived by substituting p r ( i ) in (1) and x i as βλπr c ln( ( i ∗ +1) i ∗ i ∗ ! ) ≥ , βλπr c ln( ( i ∗ ) i ∗ i ∗ ! ) < . (A.5)Then, i ∗ satisfies ( i ∗ +1) i ∗ i ∗ ! ≥ e λπr cβ and ( i ∗ ) i ∗ i ∗ ! < e λπr cβ . With Stirling formula [25], √ πn ( ne ) n
Using Theorem 2 in [26], when α > , L I i,r ( r α Γ ) can be derived as L I i,r ( r α Γ ) = exp (cid:16) − π (cid:0) λ I − λ di (cid:1) (cid:82) ∞ Γ +( v/r ) α vdv − πλ di (cid:82) ∞ r Γ Γ +( v/r ) α vdv (cid:17) = exp (cid:16) − πλ I (cid:82) ∞ Γ +( v/r ) α vdv + 2 πλ di (cid:82) r Γ +( v/r ) α vdv (cid:17) = exp (cid:18) − πλ I r Γ /α (cid:82) ∞ u α/ du + πλ di r Γ /α (cid:82) Γ − α u α/ du (cid:19) = exp (cid:16) − πr Γ /α ( λ I ξ − λ di ξ ) (cid:17) , (B.6)where ξ = (cid:82) ∞ u α/ du and ξ = (cid:82) Γ − α u α/ du .By substituting (B.3) and (B.6) into (B.2), we prove Proposition 2.A PPENDIX CP ROOF OF P ROPOSITION α > , in case of α = 2 , we need to re-derive the Laplacetransform L I i,r ( r α Γ ) . By ignoring the interference generated by the DTs with distance largerthan r max as in [29], we can approximate L I i,r ( r α Γ ) as L I i,r ( r α Γ ) ≈ exp (cid:16) − π (cid:0) λ I − λ di (cid:1) (cid:82) r max Γ +( v/r ) vdv − πλ di (cid:82) r max r Γ Γ +( v/r ) vdv (cid:17) ( a ) ≈ exp (cid:0) − πλ I r Γ (cid:82) r max u du (cid:1) = exp ( − πr Γ λ I ln(1 + r max )) , (C.1)where (a) is obtained by using λ I (cid:29) λ di when the file catalog size N f is large.Then, the offloading probability can be obtained by substituting L I i,r ( r α Γ ) into (B.3) as p ( P t , ρ ) ≈ (cid:80) N f i =1 p r ( i ) (cid:82) r c πλ i re − πλ i r e − Γ r σ − πλ I ln(1+ r max ) r Γ dr = (cid:80) N f i =1 p r ( i ) πλ i ϕ i ( P t ) (1 − e − ϕ i ( P t ) r c ) , (C.2)where ϕ i ( P t ) = σ Γ + πλ I ξ s Γ + πλ i , and ξ s = ln(1 + r max ) .By denoting g i ( P t ) = − e − ϕi ( Pt ) r c ϕ i ( P t ) , and taking the derivative of g i ( P t ) with respect to P t , wecan obtain g (cid:48) i ( P t ) = ϕ (cid:48) i ( P t ) r c e − ϕi ( Pt ) r c ϕ i ( P t ) − ϕ (cid:48) i ( P t ) (cid:16) − e − ϕi ( Pt ) r c (cid:17) ( ϕ i ( P t )) = ( κ ( ϕ i ( P t ) r c ) − ϕ i ( P t )) (cid:124) (cid:123)(cid:122) (cid:125) ( I ) ϕ (cid:48) i ( P t ) (cid:124) (cid:123)(cid:122) (cid:125) ( II ) , (C.3)where κ ( t ) (cid:44) (1 + t ) e − t , t ≥ . It is not hard to show that κ (cid:48) ( t ) = − te − t ≤ , so κ ( t ) is adecreasing function of t and κ (0) = 1 is the maximal value of κ ( t ) . Therefore, κ ( t ) ≤ andthe equality holds when t = 0 . Because ϕ i ( P t ) r c > , κ ( ϕ i ( P t ) r c ) < always holds. Then,part (I) in (C.3) ( κ ( ϕ i ( P t ) r c ) − ) ( ϕ i ( P t )) < always holds. June 10, 2016 DRAFT6
By changing variable P t → x and denoting a = F ln 2 W ρQV η , we have ϕ i ( x ) = σ e a ( x + ηPc ) − x + πλ I ξ s ( e a ( x + ηP c ) −
1) + πλ i , whose first-order derivative can be obtained as ϕ (cid:48) i ( x ) = σ x (cid:0) e a ( x + ηP c ) ax − e a ( x + ηP c ) + 1 (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) u ( x ) + aπλ I ξ s e a ( x + ηP c ) (cid:124) (cid:123)(cid:122) (cid:125) u ( x ) . (C.4)The first-order derivative of u ( x ) can be derived as u (cid:48) ( x ) = x ( a e a ( x + ηP c ) x − ae a ( x + ηP c ) x + 2 e a ( x + ηP c ) − (cid:124) (cid:123)(cid:122) (cid:125) v ( x ) . (C.5)It is not hard to obtain that v (cid:48) ( x ) = a e a ( x + ηP c ) x ≥ . Therefore, v ( x ) is an increasing functionof x and v (0) = 0 . Because x > , we know that v ( x ) ≥ always holds, i.e., u (cid:48) ( x ) ≥ , andhence u ( x ) an increasing function.By using the same approach, we can show that u ( x ) = aπλ I ξ s e a ( x + ηP c ) in (C.4) is anincreasing function. Then, ϕ (cid:48) i ( x ) is an increasing function. Besides, when x → , lim x → ϕ (cid:48) i ( x ) =lim x → ( aπλ I ξ s e aηP c − ( e aηPc − σ x ) = −∞ .When x = P max , ϕ (cid:48) i ( P max ) = σ P ( e a ( P max + ηP c ) aP max − e a ( P max + ηP c ) +1)+ aπλ I ξ s e a ( P max + ηP c ) > σ P ( − e a ( P max + ηP c ) ) + aπλ I ξ s e a ( P max + ηP c ) = e a ( P max + ηP c ) ( aπλ I ξ s − σ P ) > , because σ P (cid:28) in practice. Further considering that part (I) in (C.3) is negative, g (cid:48) i ( P t ) is first greater than zeroand then less than zero, and hence g i ( P t ) first increases and then decreases with P t . Therefore,the global optimal transmit power P ∗ i to maximize g i ( P t ) can be obtained by solving ϕ (cid:48) i ( P t ) = 0 .It is worthy to note that although ϕ i ( x ) depends on i , its first-order derivative does not. Therefore,the optimal transmit power P ∗ t to maximize p ( P t , ρ ) in (C.2) is the same as that to maximize g i ( P t ) .Considering that today’s device battery capacity is usually large, the file size is not very large(typically less than 3 GBytes), and ρQV ηP max + ηP Tc is the maximal time that a DT can transmit with P max , we have a ( P max + ηP c ) = F ln 2( P max + ηP c ) W ρQV η = F ln 2 W P max + ηP c ρQV η (cid:28) . By using the approxi-mation e t ≈ t , when t (cid:28) , ϕ (cid:48) i ( x ) = σ x (cid:0) e a ( x + ηP c ) ax − e a ( x + ηP c ) + 1 (cid:1) + aπλ I ξ s e a ( x + ηP c ) ≈ σ x (( a ( x + ηP c ) + 1) ax − ( a ( x + ηP c ) + 1) + 1) + aπλ I ξ s ( a ( x + ηP c ) + 1) . Then, the optimal x ∗ satisfying ϕ (cid:48) i ( x ) = 0 can be obtained by solving the cubic equation a µx +( a ( σ + µηP c ) + µ ) ax + a ηP c σ x − aηP c σ = 0 , where µ = πλ I ξ s . From the equation,we can obtain the closed-form of P ∗ t .This proves Proposition 3. June 10, 2016 DRAFT7 A PPENDIX DP ROOF OF P ROPOSITION δ ( i, r ) as the ratio of the data conveyed via D2D links to the file size F , which canbe obtained as δ ( i, r ) = min (cid:18) R ( i, r ) ρV QF ( η P t + P c ) , (cid:19) = min (cid:18) log (1 + γ ( i, r )) W ρV QF ( η P t + P c ) , (cid:19) .From the definition, the offloading ratio can be obtained as p a ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) E h [ δ ( i, r )] dr = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) P [ δ ( i, r ) = 1] dr + (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) P [ δ ( i, r ) < E h [ δ ( r ) | δ ( i, r ) < dr ( a ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) P [ E ( i, r ) ≤ ρV Q ] dr + (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) P [ δ ( i, r ) < E h [ δ ( i, r ) | δ ( i, r ) < dr = p ( P t , ρ ) + (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) P [ δ ( i, r ) < E h [ δ ( i, r ) | δ ( i, r ) < dr, (D.1)where (a) comes from the fact that P [ δ ( i, r ) = 1] = P [ E ( i, r ) ≤ ρV Q ] .From the expression of δ ( i, r ) and R ( i, r ) , we have E h [ δ ( i, r ) | δ ( i, r ) <
1] = E h (cid:20) log (1 + γ ( i, r )) W ρV QF ( η P t + P c ) | log (1 + γ ( i, r )) W ρV QF ( η P t + P c ) < (cid:21) ( a ) = E h (cid:104) ln(1+ γ ( i,r ))ln(1+Γ ) | ln(1+ γ ( i,r ))ln(1+Γ ) < (cid:105) = ) E h [ln(1 + γ ( i, r )) | ln(1 + γ ( i, r )) < ln(1 + Γ )] , where (a) is obtained by substituting ln(1 + Γ ) = F ( P t + ηP c ) ln 2 W ρQV η , and Γ = e F ( Pt + ηPc ) ln 2 WρQV η − .For a positive random variable x with cdf F ( x ) and pdf f ( x ) , we have E [ x | x < X ] ( a ) = (cid:82) X x f ( x ) F ( X ) dx = X − F ( X ) (cid:82) X P [ x < t ] dt, (D.2)where (a) comes from the fact that the conditional pdf of x given x < X is f ( x ) F ( X ) .Then, we have E h [ln(1 + γ ( i, r )) | ln(1 + γ ( i, r )) < ln(1 + Γ )]= ln(1 + Γ ) − P [ln(1+ γ ( i,r )) < ln(1+Γ )] (cid:82) ln(1+Γ )0 P [ln(1 + γ ( i, r )) < t ] dt ( a ) = ln(1 + Γ ) − P [ h< Γ ( σ + I i,r ) r α ] (cid:82) ln(1+Γ )0 P [ h < ( e t − σ + I i,r ) r α ] dt ( b ) = ln(1 + Γ ) − − e − φi (Γ1 ,r ) (cid:82) ln(1+Γ )0 − e − φ i ( e t − ,r ) dt, (D.3)where (a) and (b) are respectively obtained analogous to deriving (B.2) and (B.3), and φ i ( x, y ) = xy α σ − π ( λ I ξ − λ di ξ ) y x /α . June 10, 2016 DRAFT8
Therefore, we have E h [ δ ( i, r ) | δ ( i, r ) <
1] = 1 − − e − φi (Γ1 ,r ) (cid:82) ln(1+Γ )0 1 − e − φi ( et − ,r ) ln(1+Γ ) dt. (D.4)On the other hand, we can obtain P [ δ ( i, r ) <
1] = P [ E ( i, r ) > ρV Q ] = 1 − P [ E ( i, r ) ≤ ρV Q ] ( a ) = 1 − e − φ i (Γ ,r ) , (D.5)where (a) is obtained according to (B.3). By substituting (D.5) and (D.4) into (D.1), we canobtain the expression of p a ( P t , ρ ) in Proposition 4.From (D.1), we can show that p ( P t , ρ ) ≤ p a ( P t , ρ ) . Considering δ ( i, r ) ≤ , we canobtain p a ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) E h [ δ ( i, r )] dr ≤ (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) dr = p o . Then, p ( P t , ρ ) ≤ p a ( P t , ρ ) ≤ p o . When φ i (Γ , r ) = 0 , according to Proposition 2, p ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) e dr = (cid:80) N f i =1 p r ( i )(1 − e πλ i r c ) dr = p o and both equalities hold, where theconditions that lead to φ i (Γ , r ) = 0 can be further derived as follows.From the expression of Γ , we have lim ρ →∞ Γ = lim ρ →∞ e F ( Pt + ηPc ) ln 2 WρQV η − . Hence, lim ρ →∞ φ i (Γ , r ) =lim ρ →∞ Γ r α σ + π ( λ I ξ − λ di ξ ) r Γ /a = 0 . Since ≤ λ di ≤ λ I , λ I → leads to λ di → . Furtherconsidering that SNR = P t r − α /σ = 1 /r α σ , we have lim SNR →∞ λ I → φ i (Γ , r ) = 0 . Therefore, theupper bound of p a ( P t , ρ ) can be achieved when ρ → ∞ , or when SNR → ∞ and λ I → .A PPENDIX EP ROOF OF P ROPOSITION ¯ E = (cid:80) N f i =1 p (cid:48) r ( i ) (cid:82) r c f (cid:48) i ( r ) E h [ E ( i, r ) | E ( i, r ) < ρV Q ] dr, (E.1)where p (cid:48) r ( i ) is the probability that the i th file is requested by a satisfied DR, f (cid:48) i ( r ) is the pdf ofthe D2D link distance for a DR that requests the i th file and is satisfied.We can obtain p (cid:48) r ( i ) as p (cid:48) r ( i ) = p r ( i ) (cid:82) rc f i ( r ) P [ E ( i,r ) ≤ ρV Q ] drp ( P t ,ρ ) ( a ) = p r ( i ) (cid:82) rc f i ( r ) e − φi (Γ1 ,r ) drp ( P t ,ρ ) , where p r ( i ) (cid:82) r c f i ( r ) P [ E ( i, r ) ≤ ρV Q ] dr is the probability that a DR requests the i th file and can besatisfied, p ( P t , ρ ) is the probability that a DR requesting any file can be satisfied, and (a) is ob-tained analogously to deriving (B.1). The cdf of the D2D distance for a satisfied DR that requeststhe i th file can be obtained as F (cid:48) i ( R ) = P [ r ≤ R ] = (cid:82) R f i ( r ) P [ E ( i,r ) ≤ ρV Q ] dr (cid:82) rc f i ( r ) P [ E ( i,r ) ≤ ρV Q ] dr = (cid:82) R f i ( r ) e − φi (Γ1 ,r ) dr (cid:82) rc f i ( t ) e − φi (Γ1 ,r ) dr ,where (cid:82) R f i ( r ) P [ E ( i, r ) ≤ ρV Q ] dr is the probability that a DR that desires the i th file can be June 10, 2016 DRAFT9 satisfied with a D2D transmission distance smaller than R , and (cid:82) r c f i ( r ) P [ E ( i, r ) ≤ ρV Q ] dr is the probability that a DR desiring the i th file can be satisfied. Then, the pdf f (cid:48) i ( r ) can beobtained as f (cid:48) i ( r ) = d F (cid:48) i ( r ) dr = f i ( r ) e − φi (Γ1 ,r ) (cid:82) rc f i ( r ) e − φi (Γ1 ,r ) dr .Considering that E ( i, r ) = ln(1+Γ ) ρV Q ln(1+ γ ( i,r )) , we have E h [ E ( i, r ) | E ( i, r ) < ρV Q ] = ln(1 + Γ ) ρV Q E h (cid:104) γ ( i,r )) | γ ( i,r )) < Γ (cid:48) (cid:105) , (E.2)where Γ (cid:48) = +1) .Moreover, the expectation in (E.2) can be derived as E h (cid:104) γ ( i,r )) | γ ( i,r )) < Γ (cid:48) (cid:105) ( a ) = Γ (cid:48) − P (cid:104) γ i,r )) < Γ (cid:48) (cid:105) (cid:82) Γ (cid:48) P (cid:104) γ ( i,r )) < t (cid:105) dt ( b ) = Γ (cid:48) − P [ h> Γ (cid:48) ( σ + I r,i ) r α ] (cid:82) Γ (cid:48) P (cid:104) h > ( e t − σ + I r,i ) r α (cid:105) dt ( c ) = Γ (cid:48) − e − Γ (cid:48) σ rα − π ( λIξ − λdi ξ r (cid:48)
1) 2 α (cid:82) Γ (cid:48) e − ( e t − σ r α − π ( λ I ξ − λ di ξ ) r ( e t − α dt ( d ) = Γ (cid:48) − e − φi (Γ (cid:48) ,r ) (cid:82) Γ (cid:48) e φ i ( e t − ,r ) dt, (E.3)where (a) is obtained according to (D.2), (b) is obtained by substituting the definition of γ ( i, r ) in (7), (c) is because h follows an exponential distribution with unit mean, and (d) is because φ i ( x, y ) = xy α σ − π ( λ I ξ − λ di ξ ) y x /α .By substituting p (cid:48) r ( i ) , f (cid:48) i ( r ) and (E.3) into (E.2) and (E.1) and after some further manipulations,Proposition 5 follows. A PPENDIX FP ROOF OF P ROPOSITION A = r α σ , a = F ln 2 W ρQV η and g ( P t ) = A Γ P t , the offloading probability can beexpressed as p ( P t , ρ ) = (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) e − g ( P t ) dr, (F.1)where Γ = e a ( x + ηP Tc ) − .To simplify the notation, we change the variable P t → x . We can obtain the first-orderderivative of g ( x ) as g (cid:48) ( x ) = Ax ( d Γ d x x − Γ ) . Then, the first-order derivative of p ( x, ρ ) can beobtained as p (cid:48) ( x, ρ ) = − (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) g (cid:48) ( x ) e − g ( x ) dr = − g ( x ) (cid:80) N f i =1 p r ( i ) (cid:82) r c f i ( r ) Ax e − g ( x ) dr, (F.2) June 10, 2016 DRAFT0 where g ( x ) = d Γ d x x − Γ . The first-order derivative of g ( x ) with respect to x can be obtainedas g (cid:48) ( x ) = d Γ d x x = a e a ( x + ηP Tc ) x ≥ .Therefore, g ( x ) is an increasing function, where g (0) = 1 − e aηP Tc < and g (+ ∞ ) → + ∞ . Then, from (F.2), we can see that p ( P t , ρ ) first increases and then decreases, and theoptimal x to maximize p ( x, ρ ) can be obtained by solving g ( x ) = 0 . Again, considering that a ( P max + ηP Tc ) (cid:28) and using the approximation e t ≈ t that is accurate when t (cid:28) , g ( x ) can be derived from (F.2) as g ( x ) = e a ( x + ηP Tc ) ( ax −
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