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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TWC.2017.2731760, IEEE Transactions on Wireless Communications 1

Content Placement for Wireless Cooperative Caching Helpers: A Tradeoff between Cooperative Gain and Content Diversity Gain Seong Ho Chae, Member, IEEE, Tony Q. S. Quek, Senior Member, IEEE, and Wan Choi, Senior Member, IEEE

Abstract—Depending on what and how caching helpers cache content in their finite storage, the caching helpers can offer either a content diversity gain by serving diverse content or a cooperative gain by jointly transmitting the same content. This paper identifies a tradeoff between the content diversity gain and the cooperative gain according to content placements and proposes a probabilistic content placement to optimally balance the tradeoff. Using stochastic geometry, we quantify this tradeoff by deriving the cache hit rate and the rate coverage probability. To efficiently control the tradeoff, we determine the near-optimal caching probabilities that maximize the average content delivery success probability with the cooperative caching helpers. Our analysis and numerical results reveal that our proposed content placement outperforms the conventional caching schemes, such as caching with uniform probabilities, caching the most popular contents, and caching the content maximizing the cache hit, in terms of the average content delivery success probability.

I. I NTRODUCTION The recent proliferation of new mobile devices such as smart-phones has led to unprecedented growth of traffic demand, and coping with the radically increasing traffic becomes one of the biggest challenges in the current wireless communication systems. Indeed, Cisco’s 2016 white paper forecasts that the monthly global mobile traffic demands will increase to 30.6 exabytes in 2020, which is nearly eightfold over 2015 [2]. Although there has been notable progress on resolving this issue and various techniques such as small cells, massive multi-input multi-output (MIMO), and mmWave have been developed [3], these techniques can incur high deployment costs due to many RF chains or backhaul installation. The recent price drop of storage devices creates new opportunities of utilizing storage memory as a wireless communication resource. Driven by the interesting observation that Manuscript received April 13, 2017; revised July 7, 2017; accepted July 13, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dusit Niyato. This work was supported by ICT R&D program of MSIP/IITP. [2015-0-00820, A research on a novel communication system using storage as wireless communication resource]. The work of T. Quek was supported by the MOE ARF Tier 2 under Grant MOE2014-T2-2-002. A part of this paper was presented at IEEE Global Commun. Conference (GLOBECOM), San Diego, CA, Dec. 2015 [1]. S. H. Chae is with the Agency for Defense Development (ADD), Daejeon, Korea (E-mail: [email protected]). W. Choi is with the School of Electrical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Korea (E-mail: [email protected]). T. Q. S. Quek is with Singapore University of Technology and Design, Singapore 487372 and also with the Department of Electronics Engineering, Kyung Hee University, Yongin-si, Gyeonggi-do, 17104, Korea (E-mail: [email protected]).

repeated downloading and streaming of some popular contents hold the majority of the data traffic, wireless edge caching has drawn much attention as a promising technique to alleviate the heavy network traffic [4]. The wireless local caching pre-fetches some popular content during off-peak hours in the storage installed at the network edges such as hand-held devices and small cells. As such, it reduces the backhaul burden and latency by avoiding download from a remote server via base stations (i.e., by locally sharing the content via the network edges). A content delivery success of wireless cooperative caching helpers depends on a cooperative gain as well as a content diversity gain. According to what the helpers have cached in their finite storage, they can offer either a content diversity gain by serving diverse contents or a cooperative gain by jointly transmitting the same content. As more helpers cache the same content, the signal-to-interference-plus-noise-ratio (SINR) can be improved by joint transmission of the same cached content, only if the cached content is requested, which is named as cooperative gain. On the contrary, if the helpers cache different content, various content requests can be served via caching, of which gain is termed as content diversity gain. That is, there exists a tradeoff between the cooperative gain and the content diversity gain according to content placements. This tradeoff was identified in [1], but the authors did not fully address the optimal content placement to balance such a tradeoff with a closed form solution, on the contrary, which is the main focus of this work. A. Related Works The wireless local caching has recently become a hot research topic. In [5]–[11], various caching strategies were proposed and their performance gains were investigated with simple cache-enabled network models. Centralized and decentralized coded caching strategies were proposed and their order optimality in terms of transmission rate was proved in errorfree networks [5], [6]. The optimal collaboration distance and throughput scaling laws were investigated in cache-enabled device-to-device (D2D) networks [7], [8]. The optimal content placement to minimize the average sum delay in downloading was studied for coded and uncoded cases in a caching helper network (e.g. femto-caching) [9], where the optimum file assignment for the uncoded case was shown to be NP-hard. The optimal throughput-outage tradeoff was characterized in terms of throughput scaling, and the order-optimal caching

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distribution was found in a wireless D2D network modeled as a simple grid [10]. A tradeoff between channel diversity and file diversity was identified and the content placement to minimize the average bit error rate (BER) was studied in [11]. However, these works ignore the random topology of wireless edge caching nodes and realistic wireless channel models, which can change the optimal content placement and the caching gain. In this vein, notable progress have been made on understanding wireless caching with more realistic network models with stochastic geometry [12]–[26]. Because finding the optimal content placement is difficult in complicated network models, various sub-optimal content placements which restrict candidate content to be stored were developed in a deterministic way to reduce search dimensions in [12]–[15]. The performance of different cache-enabled systems caching the most popular contents was analyzed in [12], [13]. A tradeoff between the cache memory size and the density of small cell base stations which store the most popular contents was investigated in [14]. A memory allocation for two deterministic caching schemes adopting different transmission techniques was studied in a cluster-centric cache-enabled small cell network [15]. In particular, the contents in the most popular content group are repeatedly cached and jointly transmitted for combining; on the other hand, the contents in the second most popular content group are partitioned and cached, and are also jointly transmitted for multiuser detection with successive interference cancellation at receivers. However, since deterministic content placement with restricted contents is obviously sub-optimal, the storage resources are not efficiently utilized to offer the full content diversity gain. Recently, probabilistic content placements have also been studied in cache-enabled D2D networks [16], [17], caching helper networks [18], and heterogeneous cache-enabled networks [19]. The probabilistic content placement caches the content independently and randomly with given probabilities, i.e., in a distributed manner, so it can be applicable even to complex networks with high flexibility. In the literature on optimal caching probabilities, the average file transmission success probability, the cache hit failure probability, and the average download time were investigated [16]–[19]. However, these works did not consider network interference and user interactions even though they are critical considerations for content placement and system performance. Taking into account network interference, the optimal geographic content placement to maximize the total cache hit probability was explored in cellular and heterogeneous cellular networks [20], [21]. In [22], the optimal caching distribution to maximize the density of successful receptions for single file transmission and simultaneous joint transmission of different files was numerically found in a stochastic wireless D2D network when the size of storage is unit. Sub-optimal random caching schemes with multi-casting [23] and channel selection diversity [24] were studied with multiple users. However, none of the above works address how probabilistic caching can efficiently control both the cooperative gain and content diversity gain.

B. Contributions The main contributions of this paper are summarized as follows. •

•

•

We propose the probabilistic content placement to balance the tradeoff between the cooperative gain and the content diversity gain in the presence of wireless cooperative caching helpers. Specifically, the helpers randomly and independently cache a subset of the contents in content library based on given caching probabilities, and those with the same content in the cooperative region jointly transmit their cached content (cache-based joint transmission). With stochastic geometry, we quantify the cooperative gain and the content diversity gain according to content placements by deriving the cache hit rate and the rate coverage probability. It is theoretically proved that the content diversity is maximized by caching with uniform probabilities, whereas the cooperative gain is maximized when all caching helpers cache the same content. To efficiently control the tradeoff between the cooperative gain and the content diversity gain, we find near-optimal caching probabilities based on a derived lower bound and an approximation of the average content delivery success probability. The proposed probabilistic content placement outperforms the conventional caching schemes, such as caching the most popular contents, caching with uniform probabilities, and caching the content maximizing the cache hit.

C. Organization The rest of the paper is organized as follows. In Section II, we describe the system model and performance metric considered in this paper. We analyze the average content delivery success probability in Section III and determine the near-optimal caching probabilities in Section IV. Numerical examples to validate our analysis are provided in Section V and the conclusion is drawn in Section VI. II. S YSTEM M ODEL AND P ERFORMANCE M ETRIC We consider a downlink wireless caching helper network, where the helpers are randomly deployed and jointly transmit their cached content to a served user, as depicted in Fig. 1. We assume that each user can be fully supported from cooperative caching helpers with the maximum available channel bandwidth, which corresponds to when the network load is relatively light and thus the effect of user density (i.e., load) is of subordinate importance. The caching helpers are modeled as a homogeneous Poisson point process (PPP) of density λ and their location sets are denoted by Φ. The caching helpers have a single antenna each and they do not have the channel state information (CSI) of the user whom they serve. The helpers are equipped with a cache memory which is capable of storing up to M different contents. The total number of contents is F and the set of their indices is denoted as F = {1, 2, · · · , F }, where F > M . We assume that the size of all content is



{1,2,4}

f1

{2,4,5}

{1,2,4}

f3

f5

{1,3,5}

f1

{2,4,5}

{1,3,4}

Rc f4

{2,3,4}

cooperative region Fig. 1. Non-coherent joint transmission in downlink caching helper network is shown when M = 3 and F = 5. The set {a, b, c} represents the indices of the content randomly cached at each helper.

normalized one1 and the content popularity profile fi follows a Zipf distribution as [1], [7], [8], [12], [15], [18]: 1/iγ fi = PF , for i ∈ F , γ j=1 1/j

(1)

where the exponent γ ≥ 0 reflects the skewness of the content popularity distribution. The larger the value of γ, the fewer popular contents hold a majority of the content requests. In addition, the lower indexed content has higher popularity, i.e., fi ≥ fj if i < j. Note that the popularity profile is not necessarily confined to the Zipf distribution and our analytic framework can be applied to any discrete popularity distributions. We assume that the content popularity profile is perfectly known with some estimation algorithms or platforms [27]–[29]. When the caching helpers adopt the probabilistic content placement, they independently cache content i with a probability pi . Due to the limited storage size, the following two constraints should be satisfied. 0 ≤ pi ≤ 1, ∀i ∈ F , F X

pi = M.

(2) (3)

i=1

For given caching probabilities {pi }F i=1 satisfying (2) and (3), each helper randomly builds up the list of M content to be stored by the probabilistic content caching method proposed in [20], [24]. The helpers caching content i can be modeled as an independent PPP with intensity λi (, pi λ) and S their location set can be represented by Φi , such that Φ = F i=1 Φi . We focus on the performance of a reference receiver located at the origin (0, 0) ∈ R2 , called a typical receiver. This is enabled by Slivnyak’s theorem that the statistics observed by a random point of a PPP Φ is the same as those observed by a point located at the origin in the process Φ ∪ (0, 0) [30]. The typical receiver is assumed to randomly request one of the F contents according to the content popularity profile fi . If the typical receiver requests content i, then the helpers caching content i in the cooperative region jointly transmit their cached 1 Although

this work does not consider unequal size of content, the analytic framework is still valid by partitioning content into small chunks of an equal size and treating them as individual content each.

content to the typical receiver with transmit power P (i.e., cache-based joint transmission). The cooperative region is defined as a certain geographical area around the typical receiver. We model this cooperative region as a circle of a radius Rc centered at the typical receiver for tractability and denote it as b(0, Rc ), which is depicted as the shaded area in Fig. 1. The transmitted signals from the caching helpers experience path losses with the path loss exponent α > 2 and the small scale fading channels are modeled as independent and identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. If there does not exist any helper caching content i in the cooperative region, the typical receiver receives the content from the nearest base station which is connected with the central download server. In this case, we assume that real-time streaming cannot be served without playback delay due to download delay via limited backhaul. When the helpers caching content i in the cooperative region jointly transmit, dropping the time index, the received signal at the typical receiver is given by X √ P hx |x|−α/2 s y= x∈Φi ∩b(0,Rc )

|

{z

}

desired signal

+

X

y∈Φ\{Φi ∩b(0,Rc )}

|

√ P hy |y|−α/2 sy +z,

{z

interference

(4)

}

where s indicates the transmitted symbol from the serving caching helpers; sy denotes the transmitted symbol from the helper located at y; z is a standard additive white Gaussian noise; hx is the complex Gaussian fading channel coefficient between the helper located at x and the typical receiver. As such, the received SINR of the typical receiver which requests content i is given by P P | x∈Φi ∩b(0,Rc ) |x|−α/2 hx |2 , (5) SINRi = σ 2 + Ii where σ 2 is the noise variance and Ii is the sum of interfering signal power given by X P |hy |2 |y|−α . (6) Ii = y∈Φ\{Φi ∩b(0,Rc )}

The desired signals are non-coherently added at the typical receiver due to absence of CSI at transmitter, as MBMS (multimedia broadcasting multicasting service) of LTE. The same content data transmitted from the caching helpers outside the cooperative region are regarded as interference due to their delayed arrivals2 . Note that the desired signal power and the interfering signal power are functions of Φi and thus they dynamically change according to the caching probabilities {pi }. As a performance metric, we consider the average content delivery success probability to quantify the success of 2 The received signals from the caching helpers in the cooperative region are assumed to be time synchronized with the help of cyclic prefix or equalization techniques, but the received signals from the caching helpers outside the cooperative region are not.



content delivery with cooperative caching helpers [1], [14]. If the achievable data rate from the caching helpers in the cooperative region exceeds a target bit rate of content i, the typical receiver can be served with video streaming service without any playback delay. Mathematically, the average content delivery success probability at the typical receiver is given by Ps =

F X i=1

fi · P [log2 (1+SINRi ) ≥ ρi , |Φi ∩ b(0, Rc )| 6= 0] , (7)

where |Φi ∩ b(0, Rc )| is the cardinality of set Φi ∩ b(0, Rc ) and ρi is the target bit rate of content i [bps/Hz]. III. AVERAGE C ONTENT D ELIVERY S UCCESS P ROBABILITY In this section, we identify the tradeoff between the cooperative gain and the content diversity gain according to content placements by deriving the cache hit rate, a lower bound, and an approximation of the rate coverage probability. In addition, we derive a lower bound and an approximation of the average content delivery success probability in order to find the caching probabilities. We assume that the network is interference-limited and thus the background thermal noise effect is negligible compared to the network interference. From the Baye’s rule, the average content delivery success probability with caching helpers in (7) can be written as

probabilities of other contents decrease due to the constraint PF i=1 pi = M . Therefore, we need to control the caching probabilities {pi } to maximize the cache hit rate, i.e., content diversity. Lemma 1. The cache hit rate is maximized by uniformly caching contents, i.e., {pi } = M/F and the maximized cache hit rate is an increasing function of the number of total contents, F . Proof: This lemma is readily proved by using Jensen’s inequality and non-void probability of PPP. From Lemma 1, we know that the content diversity is maximized by caching with uniform probabilities. However, uniformly caching diverse contents loses the chance of cooperation for a specific content because the cooperative gain improves as more caching helpers store the same content. To quantify such cooperative gain, we derive the rate coverage probability in the following lemma. Lemma 2. Conditioning on that at least one helper caching content i exists in the cooperative region, when the helpers caching content i in the cooperative region jointly transmit their cached content, the rate coverage probability for content i is given by Z Rc ∞ Z Rc Z Rc X ··· Si = ϕ(x, rn , τi )fRn (rn )dr1 · · · drn n=1 0 | x {z x } n≥1

Ps =

F X i=1

where

fi Ai Si ,

Ai = P [|Φi ∩ b(0, Rc )| 6= 0] ,

Si = P [log2 (1 + SIRi ) ≥ ρi | |Φi ∩ b(0, Rc )| 6= 0] ,

(8)

(9) (10)

and Ai indicates whether there exists at least one caching helper storing content i in the cooperative region. We call it cache hit probability of content i. Similarly, Si represents the probability whether the achievable data rate of content i from the serving caching helpers in the cooperative region is larger than the target bit rate. We call it rate coverage probability of content i. From the void probability of PPP [32], the cache hit probability of content i is given by (11) Ai = 1 − exp −πpi λRc2

and the cache hit rate for all contents becomes PF PF 2 i=1 1 − exp −πpi λRc i=1 Ai = . (12) F F The cache hit rate in (12) accounts for availability of the whole contents, so it measures the content diversity gain for given content placement. For a given {pi }, the cache hit rate increases as either λ or Rc increases. That is, for given content placement, the availability of each content increases as the average number of caching helpers in the cooperative region increases. For given Rc and λ, the cache hit rate of specific content i increases as pi increases, but the cache hit

×f|xi | (x)dx · pNi (n)+

Z

Rc

0

ψ(x, τi )f|xi | (x)dx · pNi (0),

(13)

where ϕ(x, rn , τi ) = e

−πλ

fRn (rn ) =

τi P −α x−α + n r k=1 k

n Y

k=1

2 Rc2 − x2

2/α n

Cα −pi Kα

R−α c τi P −α x−α + n r k=1 k

,

rk ,

2

2πpi λxe−πpi λx , Ai (πpi λ Rc2 − x2 )n −πpi λ(R2 −x2 ) c pNi (n) = e , n! 2 2/α −α α ψ(x, τi ) = e−πλx τi (Cα −pi Kα (Rc x τi )) .

f|xi | (x) =

2

Here, Ai = 1 − e−πpi λRc , τi = 2ρi − 1 is target SIR of content i such that log2 (1 + SIRi ) ≥ ρi , and Kα (a) = R a−2/α 1 du. 0 1+uα/2 Proof: Refer to Appendix A.

To quantify the cooperative gain, we derive the conditional rate coverage probability of content i in Lemma 2. However, its exact expression involves numerous integrals and thus makes our problem intractable. To circumvent this difficulty, we derive an approximation and a lower bound of the rate coverage probability of content i in the following lemma.



Lemma 3. When the helpers caching content i in the cooperative region jointly transmit their cached content, the rate coverage probability of the typical receiver for content i is approximated and lower bounded by

0.9 0.8 0.7

(14)

where Siapprox and Silower are given by (15) and (16), respectively, which are placed at the top of next page, where Cα = R a−2/α 2π 2π 1 ρi du. α csc α , τi = 2 − 1, and Kα (a) = 0 1+uα/2

Approx. R =200, λ=10

0.6

Si

Silower ≤ Si ≈ Siapprox ,

1

0.5

c

Monte-Carlo Lower bound -5 Approx. R =1000, λ=10

0.4 0.3

c


0.2

Proof: Refer to Appendix B.

c

0.1

Remark: As proved in Lemma 1, the cache hit rate (content diversity gain) is maximized by caching with uniform probabilities. In contrast, from Lemma 3, it is theoretically verified that the rate coverage probability of specific content (cooperative gain) increases with the number of caching helpers having the same content. Consequently, there is a tradeoff between the content diversity gain and the cooperative gain according to content placements. To properly balance such a tradeoff, we can control the caching probabilities {pi }. Figure 3 plots the rate coverage probability of content i for varying F when the caching helpers uniformly cache the content, i.e., when the content diversity is maximized. The detailed system parameters and values are presented in Section V. It is shown that the rate coverage probability of specific content i (cooperative gain for content i) decreases as F increases, but the content diversity increases. This verifies the tradeoff between the content diversity and the cooperative gain according to content placement.

0 -20

Monte-Carlo Lower bound

-15

-10

-5

0

5

10

15

20

τi (dB)

(a) A lower bound and an approximation of Si for various λ and Rc 1 0.9 0.8 0.7

Si

0.6 0.5 0.4 0.3 0.2

0 -20

Approx. α=3, λ=10 Monte-Carlo Lower bound

-4

Approx. α=3, λ=10 Monte-Carlo Lower bound

-3

Approx. α=2.5, λ=10 Monte-Carlo Lower bound

0.1

-15

-10

-5

-3

0

5

10

15

20

τi (dB)

(b) A lower bound and an approximation of Si for various λ and α Fig. 2. The derived lower bound and the approximation of rate coverage probability of caching helpers storing content i versus its target SIR (dB), τi , are plotted for various (a) λ (units/m2 ) and Rc (m) and (b) λ (units/m2 ) and α when M = 1 and pi = 1.

1 0.9 0.8 F=1,2,3,4 0.7 0.6

Si

Figs. 2(a) and 2(b) plot the lower bound and the approximation of the rate coverage probability of content i versus target SIR for various λ and Rc and for various λ and α, respectively. The derived approximation in (15) is considerably tight to the exact rate coverage probability obtained via simulations even for relatively large λ and Rc and small α. As proved in Appendix B, accuracy i hP of the approximation −α = for given |xi | depends on Var |x| x∈Φi ∩c(|xi |,Rc ) πpi λ 1 1 which is an increasing function with α−1 |xi |2α−2 − R2α−2 c respect to λ and Rc and a decreasing function of α > 2. However, Fig. 2(b) reveals that accuracy of the approximation is quite high even for a high density λ = 10−3 (equivalently, 1000(units/km2)) and small α = 2.5, which validates the proposed approximation. The lower bound in (16) is derived from the assumption that only the nearest helper caching content i transmits and the others caching content i in the cooperative region remain silent. As a result, the gap between the lower bound and the approximation accounts for the gain by joint transmission. From (14), it is noted that the rate coverage probability of content i is an increasing function of pi . That is, the cooperative gain of specific content i increases with the number of caching helpers that store content i for given λ and Rc . Intuitively, if more caching helpers participate in joint transmission for given network geometry, the desired signal power increases and interference decreases.

-5

c


0.5 0.4 0.3 F=1 F=2 F=3 F=4

0.2 0.1 0 -20

-15

-10

-5

0

5

10

15

20

τi (dB)

Fig. 3. Rate coverage probability of caching helpers caching content i, Si , for uniform caching versus its target SIR, τi = 2ρi − 1, when M = 1.



!2/α

τi Z Cα −pi Kα 2πpi λ Rc −πλ x−α + 2pα−2 i λπ x2−α −R2−α ( ) c xe Siapprox ≈ Ai 0 Z Rc 2/α 2 α −α 2 2πp λ i Silower = xe−πλτi x (Cα −pi Kα (τi x Rc ))−πpi λx dx Ai 0

To efficiently control the tradeoff, the optimal caching probabilities can be found to maximize the average content delivery success probability for a given memory constraint. However, due to absence of tractable form, we instead derive a lower bound and an approximation of the average content delivery success probability. Theorem 1. For given caching probabilities {pi } (i = 1, . . . , F ), when the caching helpers in the cooperative region jointly transmit the cached content, the average content delivery success probability is approximated and lower bounded as Pslower ≤ Ps ≈ Psapprox ,

(17)

where Pslower = Psapprox =

F X

i=1 F X i=1

fi Ai Silower ,

(18)

fi Ai Siapprox ,

(19)

where Ai is given in (11) and Siapprox and Silower are given in (15) and (16), respectively. Proof: From the definition of Ps in (8), the proof can be readily done with (11) and Lemma 3. IV. O PTIMIZED C ACHING P ROBABILITIES In this section, using the lower bound and the approximation of the average delivery success probability in Theorem 1, we determine the near-optimal caching probabilities according to specific conditions and shed light on how various system parameters affect on the caching performance.

R−α c τi 2pi λπ 2−α x2−α −Rc x−α + α−2

(

For the low target bit rate (∀ρi ≪ 1), Psapprox in (19) can be approximated as Z Rc F X 2 approx xe−πpi λx dx fi 2πpi λ Ps ≈ =

i=1 F X

= arg min {pi }

i=1

(20)

which is sufficiently tractable to find desirable caching placement. Using (20), we formulate an optimization problem to find near-optimal caching probabilities as follows. P1 : {ˆ p⋆i } = arg max (20) {pi }

(21)

dx

(15)

F X

2

fi e−πpi λRc

(22)

i=1

subject to (2), (3). 2

Defining gi (pi ) = e−πpi λRc , the second derivative of gi (pi ) 2 is d dg2ip(pi i ) > 0 and thus gi (pi ) is a convex function. Since a weighted sum of convex functions still satisfies convexity, problem P1 is a constrained convex optimization problem of which solution can be obtained with many known convex optimization algorithms such as sub-gradient or steepest descent algorithms [31]. Remark: If the target bit rate is low, the near-optimal caching solution is obtained by minimizing the cache miss probability for a given circle of radius of Rc . Intuitively, for low target bit rates, the rate coverage probability is sufficiently high if the typical user hits the cache in the cooperative region. Consequently, the optimal caching solution depends mainly on the cache hit event. Problem P1 is similar to the optimization problem in [16], so the algorithm proposed in [16] can also be utilized to find a solution. B. Large cooperative region If the radius of the cooperative region Rc is large, Kα (τi xα Rc−α ) in (18) can be approximated as Cα . Then, the lower bound of the average content delivery success probability is approximated as Pslower ≈ =

F X

i=1 F X

fi 2πpi λ fi

Z

∞

2/α 2

xe−πλτi

x (1−pi )Cα −πpi λx2

dx

0

pi 2/α τi Cα

2/α

+ (1 − τi

Cα )pi

,

(23)

which is sufficiently tractable to find desirable caching placement. With (23), an optimization problem to find a nearoptimal caching solution is formulated as P2 : {ˆ p⋆i } = arg max (23)

(24)

{pi }

subject to (2), (3).

0

2 fi 1 − e−πpi λRc

−πpi λx2

(16)

i=1

A. Low target bit rate

)

!!

pi Ai +(1−Ai )pi where Ai 2 of vi (pi ) is d dv2ip(pi i ) < 0

2/α

Cα > 0, the

Defining vi (pi ) =

= τi

second derivative

for 0 < Ai < 1 and

d2 vi (pi ) ≥ 0 for Ai ≥ 1. Therefore, vi (pi ) is a d2 pi if Ai ≥ 1 and a concave function if 0