Caching in Wireless Multihop Device-to-Device Networks - IEEE Xplore

IEEE ICC 2015 - Ad-hoc and Sensor Networking Symposium

Caching in Wireless Multihop Device-to-Device Networks Sang-Woon Jeon∗ , Song-Nam Hong† , Mingyue Ji‡ , and Giuseppe Caire§ ∗

Andong National University, South Korea, email:[email protected] Ericsson Research Lab., CA, email:[email protected] ‡ University of Southern California, CA, email:[email protected] § Technical University of Berlin, Germany, email:[email protected] †

Abstract—We consider a wireless device-to-device (D2D) network in which the nodes are uniformly distributed at random over the network area and can cache information from a library of possible messages (files). Each node requests a file in the library independently at random, according to a given popularity distribution, and downloads from other nodes having the requested file in their local cache via multihop transmission. Under the classical “protocol model” of wireless ad hoc networks, we characterize the optimal throughput scaling law by presenting a feasible scheme formed by a decentralized caching policy for the parameter regimes of interest and a local multihop transmission protocol. The scaling law optimality of the proposed strategy is shown by deriving a new throughput upper bound. Surprisingly, we show that decentralized uniform random caching yields optimal scaling in most of the system interesting regimes. We also observe that caching improves the throughput scaling law of classical ad hoc networks, and that multihop improves the previously derived scaling law of caching wireless networks under one-hop transmission. Index Terms—caching, multihop D2D networks, scaling laws.

I. I NTRODUCTION Wireless traffic has grown dramatically in recent years, under the pressure of on-demand video streaming [1]. It has been recently recognized that “caching at the wireless edge”, i.e., caching the content library directly in the wireless nodes, has the potential of solving this problem by providing per-node throughput that scales much better than conventional unicast transmission, in a variety of scenarios (see the short literature review provided below). As an example, consider a university campus where n ≈ 10000 users (distributed over a surface of ≈ 1km2 ) stream movies from a library of ≈ 100 files, such as Netflix or Amazon Prime weekly top-of-the chart titles. For such scenario, each user demand can be satisfied by local communication from a cache, without cluttering a cellular base station with thousands of unicast sessions, or without requiring a large number of small cell access points with highthroughput by costly backhaul. Intuitively, caching is efficient because the user demands are highly redundant, although, in this scenario, users do not request the same content at the same time (this type of redundancy is referred to in [2], [3] as asyncrhonous content reuse). Recently, there have been extensive researches on wireless caching networks [2]–[10]. In [3], a one-hop device-to-device (D2D) caching network is considered, where n user nodes stores M files from a library of m ≥ M files, and where

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delivery is restricted to be one-hop, i.e., either the file is found in the cache, or it is downloaded from a neighbor through a D2D wireless link. Under the classical “protocol model” of ad hoc networks pioneered in [11], it is shown in [3] that the per-node throughput behaves as Θ(M/m). This scaling (independent with the number of nodes) can be explained as an effect of the dense spatial spectrum reuse allowed by caching, for which the requested content is found within a short communication radius, and therefore a large number of simultaneous D2D links can be active on the same time slot. This is obtained at the expenses of a small outage probability, where outage events are essentially due to the case where a user does not find the requested content in its communication radius. A different caching network model is studied in [5], where a single transmitter (i.e., a base station with all files in the library) serves n users through a common noiseless communication link of fixed capacity (bottleneck link). The scheme proposed in [5] partitions each file into packets and each user store a carefully designed subset of packets from each file. This provides “side information” at each user such that, for any arbitrary set of user demands, the base station can compute a multicast network-coded messages that is transmitted on the common link such that each user can decode its own requested file from the multicast message and its cached information. The per-node throughput scaling is again given by Θ(M/m). In this case, the caching gain is explained in terms of “multicasting gain”, i.e., in the ability of turning unicast traffic into (network-coded) multicast traffic, such that one transmission satisfies multiple users. These remarkable results show that, for both these quite different one-hop network models, caching yields the same fundamental throughput scaling law.1 In this paper, we study a natural extension of the onehop D2D network, which considers multihop transmission. A related work is presented in [12], where a multihop transmission scheme for wireless caching networks has been studied under the protocol model. The key difference between this paper and [12] is as follows. First, the objective of [12] is to minimize the average number of flows passing through each node, which is the reciprocal of average per1 For completeness, we mention that several extensions and variations of these basic models have been studied in [4], [6]–[10], investigating aspects such as multilevel caching and heterogeneous network structures.

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node throughput under only certain network model; on the other hand, we study and provide the optimal scaling law of the average per-node throughput directly. Second, [12] proposed a centralized and deterministic caching placement according to the demand distribution; in contrast, we proposed a completely decentralized, independently at random caching placement according to a uniform distribution on the whole file library, which is “universal” since it is independent of the specific demand distribution and is robust when a small portion of the users move in and out of the network. Third, in [12], the distance (or number of hops) between the source– destination (SD) pairs can traverse the whole network; while in this paper, we propose a more practical scheme called local multihop protocol, where the number of hops between any SD pairs are independent with the number of users and decreases when the storage capacity per-node increases. The proposed caching placementpand delivery scheme yield pernode throughput scaling Θ( M/m). This result shows that multihop yields a much better p singlep throughput scaling than hop networks. In fact, Θ( M/m)/Θ(M/m) = Θ( m/M ), with m/M 1 in any reasonably practical situation. Furthermore, when the demands follows a Zipf distribution with Zipf exponent γ ∈ (0, 1), we show that the proposed policy is order-optimal (in terms of scaling law) if m < M n.2 Hence, in this case, tight centralized coordination of the user caches and an accurate knowledge of the popularity distribution are not required in order to achieve order-optimal throughput. II. P ROBLEM F ORMULATION A. Caching in Wireless D2D Networks We consider a wireless D2D network consisting of n nodes distributed uniformly at random over a unit square area [0, 1]2 . Let d(u, v) denote the distance between nodes u and v. It is assumed that communication between nodes follows the protocol model of [11]: the transmission from node u to node v is successful if and only if: i) d(u, v) ≤ r, and ii) no other active transmitter must be in a circle of radius (1 + ∆)r from the receiver node v. Here, r, ∆ > 0 are given protocol parameters. Also, each node sends its packets at some constant rate W bits/sec/Hz, where W is a non-increasing function of the transmission range r. As in the current information theoretic literature on caching networks (see Section I), a caching scheme is formed by two phases: caching placement and delivery. The problem consists of placing information in the caches such that the delivery is efficient for any set of user demands (i.e., requested files). Since the demands are not known a priori,3 the cache placement must made without a priori knowledge of the set of demands although, in general, it can depend on the demands statistics. 2 Throughout

the paper, an ‘order-optimal’ scheme means that it achieves the optimal throughput scaling law within the multiplicative gap of n for any > 0. 3 They can be either arbitrary (as in [5], [6]) or random (as in [2], [3] and in this paper.

During the placement phase, each node u stores M files in its local memory Mu from a library F of size m files, where M ≤ m. During the delivery phase, each node u requests a file fu ∈ F, independently with probability pr (fu ), and the network operates in order to satisfy all the requests. The probability mass function pr (·) is referred to in the following as the popularity distribution. B. Achievable Throughput and System Scaling Regime We consider m and l expressed as functions of n as m = nα and M = nβ ,

(1)

where α > 0 and β ∈ [0, α).4 Let Tn and Sn denote the per-node average throughput (expressed in files per unit time, averaged over the popularity distribution) and the sum average throughput supported by the network during the delivery phase. We have: Definition 1: The throughput Tn is said to be achievable if there exist a cache placement and a delivery transmission protocol such that all nodes can receive their requested files with average rate at least Tn with high probability (whp).5 Accordingly, the achievable sum throughput is at least Sn = nTn . ♦ III. M AIN R ESULTS We have: Theorem 1: For the caching wireless D2D network defined before, the optimal throughput satisfies the scaling laws: α−β if α − β ∈ (0, 1], (2) Tn = Ω n− 2 − where > 0 is arbitrarily small. Furthermore, we have: Theorem 2: Consider the caching wireless D2D network defined before and assume that demands follow a Zipf popularity distribution6 with exponent γ ∈ (0, 1). Then the throughput of any scheme must satisfy  0 if α − β > 1, α−β Tn = (3) − + O n 2 if α − β ∈ (0, 1], where > 0 is arbitrarily small. The rest of the paper is dedicated to proving Theorems 1 and 2 . For the case of α −β > 1, we only need to derive an upper bound, which is given in Section VI-A. In this case, the total caching memory size in the network is less than the number of files in the library, i.e., M n < m. Therefore, there may be a node to request a file not stored in the nodes’ memories whp, thus resulting in a non-zero outage probability. Since Tn is defined as a rate with no outage, it gives a zero pernode throughput. For α − β ∈ (0, 1], we prove Theorem 1 4 If β ≥ α, then each node is able to store the entire files in F , which make the delivery phase trivial. 5 In the following, an event A n is said to occur “whp” if limn→∞ P(An ) = 1. 6 The Zipf distribution with exponent γ [13] is defined by p (i) = r −γ Pmi for 1 ≤ i ≤ m. −γ j

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j=1


√

.. .

ah

√

TDMA cell size Transmission pair √ .. ah

ac

•

.

• Source Destination

...

...

√

...

5ah

...

√ (1 + ∆) 5ah

.. . (a)

√ ≤ 2⌈(1 + ∆) 5ah ⌉ + 1

•

.. . (b)

Fig. 1. a) The proposed multihop routing protocol after the source node selection. b) TDMA cell size from the protocol model.

by presenting and analyzing a specific achievable scheme in Sections IV and V, respectively, and Theorem 2 by finding a converse upper bound in Section VI-B. IV. ACHIEVABLE S CHEMES In this section, we present a file placement policy and a transmission protocol for α − β ∈ (0, 1].

•

Divide each traffic cell into square hopping cells of area n ah = 2 log n . Define the horizontal data path (HDP) and the vertical data path (VDP) of a SD pair as the horizontal line and the vertical line connecting a source node to its destination node, respectively. Each source node transmits the requested file to its destination by first hopping to the adjacent hopping cells on its HDP and then on its VDP.7 Time Division Multiple Access (TDMA) scheme is used with reuse factor T for which each hopping cell is activated only once out of T time slots. A transmitter node in each active hopping cell sends a file (or fragment of a file) to a receiver node in an adjacent hopping cell. Round-robin is used for all transmitter nodes in the same hopping cell.

In this scheme, each hopping cell should contain at least one node for relaying as in [11], [14], which is satisfied whp since n (see Lemma 1 (a)). ah = 2 log n Lemma 1: The following properties hold whp: (a)

A. α − β ∈ (0, 1)

(b)

In this regime, a distributed file placement and a local multihop protocol are proposed as follows. 1) Distributed file placement: Each node u stores M distinct files in its local memory Mu , chosen uniformly at random from F, independently of other nodes. 2) Local multihop protocol: We first explain how each node finds its source node having the requested file (source node selection): • Divide the entire network into square traffic cells of area ac = n−η for some η ≥ 0, where η will be determined later on. • Each node chooses one of the nodes having the requested file in the same traffic cell as its source node. If there are multiple candidates, choose one of them uniformly at random. From Definition 1 and the above source node selection, all nodes should find their source nodes within their own traffic cells whp, in order to achieve a non-zero Tn . Lemma 2 below characterizes such a condition of the area of traffic cell ac (i.e., η) such as η ∈ [0, 1 − (α − β)). For the ease of exposition, we call the pair of a node and its source node source–destination (SD) pair. Notice that in our model, each SD pair is located in the same traffic cell while in the conventional ad hoc network, SD pairs are randomly located over the entire network. Thanks to caching, we can reduce the distance of each SD pair (see Lemma 2). Also, differently from the conventional model, each node can be a source node of multiple destinations, which make the throughput analysis more complicated (see Lemma 4). Next, we explain the proposed multihop transmission scheme for the file delivery between n SD pairs, see also Fig. 1 (a) (multihop transmission):

Partition the network area [0, 1]2 into cells of area 2 log n n . Then the number of nodes in each cell is between 1 and 4 log n. Partition the network area [0, 1]2 into cells of area n−a , where a ∈ [0, 1). For any δ > 0, the number of nodes in each cell is between (1 − δ)n1−a and (1 + δ)n1−a .

Proof: The proofs of first and second properties are given in [14, Lemma 1] and [15, Lemma 4.1], respectively. Lemma 2: Suppose that α − β ∈ (0, 1). If η ∈ [0, 1 − (α − β)), then all nodes are able to find their source nodes within their traffic cells whp. Proof: Let Ai denote the event that node i establishes its source node within its traffic cell, where i ∈ [1 : n]. Then, we have: Pr ∩i∈[1:n] Ai = 1 − Pr ∪i∈[1:n] Aci (1−δ)nac X whp m−M c ≥1− Pr (Ai ) ≥ 1 − n m i∈[1:n]

=1−n

1 − 1/n

α−β α−β n

(1−δ)n1−η−α+β ,

(4)

where the first inequality follows from the union bound and the second inequality is due to the fact that the number of nodes in each traffic cell is lower bounded by (1 − δ)nac whp whp (1−δ)nac (see Lemma 1 (b)) and hence, Pr(Aci ) ≤ m−M . m Therefore, from the fact that lim 1 − 1/nα−β

n→∞

nα−β

= 1/e,

(5)

Pr ∩i∈[1:n] Ai → 1 as n → ∞, since η < 1 − α + β is assumed in this lemma. This completes the proof. 7 If a source node and its destination node are in the same hopping cell, then the source node directly transmits to its destination.

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B. α − β = 1

Then, we have:

In this case, the total number of files that can be stored by n nodes (i.e., the total number of files stored in the network) is equal to the number of files in the library (i.e., nM = m), which might not be the interesting regime in practice. We study this regime for completeness. In this regime, we propose a centralized file placement and a globally multihop protocol schemes. 1) Centralized file placement: It can be seen that a distributed file placement might result in an outage, as seen from the analysis in Lemma 2. Instead, we employ a simple centralized file placement for which all distinct m files (in the library) are randomly stored in the total memories of n nodes. Hence, the network can contain all m files, thus being able to avoid an outage. 2) Globally multihop protocol: As explained before, the traffic cell should be equal to the entire network (i.e., η = 0 in Section IV-A), in order to avoid an outage. Namely, n SD pairs are located over the entire network . Hence, we can expect the same scaling result with the conventional wireless ad hoc network in [11], namely, no caching gain is expected. V. ACHIEVABLE T HROUGHPUT We derive an achievable throughput of the proposed schemes in Section IV. A. α − β ∈ (0, 1)

In this subsection, we prove that Tn = n−

α−β 2 −

(6)

Pr ∩i∈[1:n] Bi (k) = 1 − Pr ∪i∈[1:n] Bic (k) j n −j n1 X whp n1 M M 1 ≥ 1−n 1− j m m j=k

≥ 1 − n exp (−n1 D (k/n1 kM/m)) , M m

k n1

(7)

a log( ap ) + (1 − a) log( 1−a 1−b )

< < 1, where D(akb) = if denotes the relative entropy for a, b ∈ (0, 1). Here the first inequality follows from the union bound and holds whp since the number of nodes in each traffic cell is upper bounded by n1 whp from Lemma 1 (b), and the second inequality is due to the fact that for X ∼ B(n, p), Pr(X ≥ k) ≤ exp (−nD (k/nkp)) if p < k/n < 1.

(8)

Suppose that k = nτ for τ > 0. Then the condition M m < k 1−η−(α−β) τ 1−η < 1 is given by (1 + δ)n < n < (1 + δ)n , n1 which is satisfied as n increases if 1 − η − (α − β) < τ ≤ 1 − η.

(9)

Hence, from (7), we have: τ whp n τ

M Pr ∩i∈[1:n] Bi (n ) ≥ 1 − n exp −n1 D n1 m τ −1+η+(α−β) n τ = 1 − n exp −n log 1+δ {z } | :=A α n ((1 + δ)n1−η − nτ ) · exp −((1 + δ)n1−η − nτ ) log (1 + δ)n1−η (nα − nβ ) | {z } :=B

is achievable if α − β ∈ (0, 1). From Lemma 2, we assume η ∈ [0, 1 − (α − β)) to achieve a non-zero Tn by the proposed scheme in Section IV-A. Then, we provide the following useful lemmas for the proof. Lemma 3: Suppose that α − β ∈ (0, 1) and η ∈ [0, 1 − (α − β)). Let Rn denote the aggregate rate achievable for any √ 2 hopping cell. If T ≥ 2d(1 + ∆) 5e + 1 , then Rn = W T is achievable. Proof: Consider an arbitrary transmission pair consisting of a transmitter node and its receiver node illustrated in Fig. √ 1 (b). Clearly, the hopping distance is upper bounded √ by 5ah and hence, we choose the transmission range r = 5ah in the protocol model. Thus, the transmission is successful if there is no node √ simultaneously transmitting within the distance of (1 + ∆) 5ah from the receiver node. This is √ 2 satisfied if T ≥ 2d(1 + ∆) 5e + 1 . That is, the aggregate √ 2 rate of W T is achievable if T ≥ 2d(1 + ∆) 5e + 1 . Since this holds for all hopping cells, Rn = W T is achievable if √ 2 T ≥ 2d(1 + ∆) 5e + 1 . This completes the proof. Lemma 4: Suppose that α − β ∈ (0, 1) and η ∈ [0, 1 − (α − β)). For > 0 arbitrarily small, each node can be a source node of at most n1−η−(α−β)+ nodes in its traffic cell whp. Proof: Let Bi (k) denote the event that node i becomes a source node for less than k nodes. Denote n1 = (1 + δ)n1−η .

provided that (9) is satisfied. Since τ > 0, τ −1+η+(α−β) − ln(2) n A = n exp (−nτ ) 1+δ converges to zero as n increases. Furthermore B = exp −((1 + δ)n1−η − nτ ) α − ln(2) n ((1 + δ)n1−η − nτ ) · (1 + δ)n1−η (nα − nβ )

(10)

(11)

converges to zero as n increases if τ ≤ 1 − η. In summary, Pr ∩i∈[1:n] Bi (nη ) converges to zero as n increases if (9) holds. Therefore, Pr ∩i∈[1:n] Bi (n1−η−(α−β)+ ) → 0 as n → ∞ by setting τ = 1 − η − (α − β) + for > 0 arbitrarily small, implying that each node becomes a source node of at most n1−η−(α−β)+ nodes whp. This completes the proof. Based on Lemma 4, we will derive an upper bound on the number of data paths that should be carried by each hopping cell in the following lemma. Lemma 5: Suppose that α − β ∈ (0, 1) and η ∈ [0, 1 − (α − β)). For > 0 arbitrarily small, each hopping cell is required 3(1−η) to carry at most n 2 −(α−β)+ data paths whp. Proof: First consider the number of HDPs that must be carried by an arbitrary hopping cell, denoted by Nhdp . By

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assuming that all HDPs of the nodes in the hopping cells located at the same horizontal line pass through the considered hopping cell, we have an upper bound on Nhdp . Since the total area of these cells is given by r 1−η √ 1 2 log n 2 log n ac ah = n−η =n 2 √ , (12) n 2 log n n the number of nodes in that area is upper bounded by 1−η 1−η p 1 n 2 √ 2 log n 4 log n = n 2 2 log n

(13)

0

for > 0 arbitrarily small. The same analysis holds for VDPs. In conclusion, each hopping cell carries at most 3(1−η) n 2 −(α−β)+ data paths whp for > 0 arbitrarily small, which completes the proof. We are now ready to prove our main result in (6). Let 0 > 0 be an arbitrarily small constant satisfying 1 − (α − β) − 0 > 0, which is valid since α − β ∈ (0, 1). Set η = 1 − (α − β) − 0 . From Lemma 2, every node can find its source node within its traffic cell whp. From Lemma 3, setting T = √ 2 2d(1 + ∆) 5e + 1 , each hopping cell is able to achieve the aggregate rate of 2 √ (15) Rn = W/ 2d(1 + ∆) 5e + 1 . Furthermore, from Lemma 5, the number of data paths that each hopping cell needs to perform is upper bounded by 3(1−η) −(α−β)+0 2

=n

α−β 5 0 2 +2

W − α−β − 5 0 − α−β − (17) √ 2 n 2 2 ≥ n 2 2d(1 + ∆) 5e + 1

for > 0 arbitrarily small. In conclusion, (6) holds. B. α − β = 1

In this regime, the following rate is achievable: Tn = n

− 12 −

.

µ

n→∞

(18)

We briefly explain how to achieve the above rate, since the procedures of proof are almost similar to the case of α − β ∈ (0, 1). Similarly to Lemma 4, we can show that each node is able to be a source node of at most n nodes whp for > 0 arbitrarily small. Then, following the analysis in Section V-A, we can easily prove that (18) is achievable.

n X

pr (i) = 0.

(19)

i=1

Pq Pnµ Proof: Letting f (q) = i=1 i−γ , we have i=1 pr (i) = f (nu )/f (nα ). Then, we obtain the lower and upper bounds: Z n Z n −γ x dx ≤ f (n) ≤ 1 + x−γ dx . (20) 1

1

Using them, we have: nu(1−γ) − 1 nu(1−γ) − γ f (nu ) ≤ lim ≤ lim , n→∞ nα(1−γ) − γ n→∞ nα(1−γ) − 1 n→∞ f (nα ) (21) where both bounds converge to zero since α > u. lim

A. α − β > 1

Lemma 8: Suppose that α − β > 1. Let Nout,1 denote the number of nodes that they cannot find their requested files in whp

the entire network. Then, for any > 0, we have Nout,1 ≥ (1 − )n. Proof: The total number of files that are able to be stored by the entire network is given by nM = n1+β . Hence the probability that each node cannot find its requested file in the entire network is lower bounded by

(16)

whp, where we used η = 1 − (α − β) − 0 . Since each hopping cell serves multiple data paths using round-robin fashion, each data path is served with a rate of at least (15) divided by (16) whp. Therefore, an achievable per-node throughput is given by Tn =

Pm In this section, we assume pr (i) = i−γ /( j=1 j −γ ) (Zipf distribution), with γ ∈ (0, 1). We first introduce the following two technical lemmas. Lemma 6: Let X follow a binomial distribution with parameters n and p, i.e., X ∼ B(n, p). Then, for k ∈ [0 : np], 2 1 (np−k) . Pr(X ≤ k) ≤ exp − 2p n Lemma 7: For any µ ∈ [0, α), lim

whp from Lemma 1 (a). Moreover, each of these nodes may become a source node of multiple nodes within the same traffic cell. Therefore, from Lemma 4 and (13) whp 1−η p 0 Nhdp ≤ n1−η−(α−β)+ n 2 2 log n 3(1−η) 0p (14) = n 2 −(α−β)+ 2 log n

n

VI. C ONVERSE

pout,1 := 1 −

1+β nX

pr (i).

(22)

i=1

Then, for µ ∈ [0, pout,1 ], we have: n (a) X n i Pr(Nout,1 ≥ µn) ≥ pout,1 (1 − pout,1 )n−i i i=µn µn X n i ≥1− pout,1 (1 − pout,1 )n−i i i=1 (b) (pout,1 − µ)2 ≥ 1 − exp − n , (23) 2pout,1 where (a) follows from (22) and the fact that each node requires a file independent of other nodes and (b) follows from Lemma 6. Notice that the condition µ ∈ [0, pout,1 ] is required to apply Lemma 6. From Lemma 7, pout,1 → 1 as n → ∞. Hence, setting µ = 1 − for > 0 arbitrarily small in (23), which satisfies µ ∈ [0, pout,1 ] as n → ∞, yields that Nout,1 ≥ (1 − )n whp. This completes the proof. As said before, a non-vanishing outage probability implied by Lemma 8 yields that Tn = 0 in this regime.

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for µ ∈ [0, pout,2 ]. From Lemma 7, pout,2 → 1 as n → ∞. Hence setting µ = 1 − in (26), which satisfies µ ∈ [0, pout,2 ] as n → ∞, yields that Nout,2 ≥ (1 − )n whp. This completes the proof.

···

πr2

Source

Fig. 2.

n−

1−(α−β)+ǫ 2

Destination

VII. C ONCLUDING R EMARKS

A lower bound on the exclusive area occupied by the multihop

transmission of a SD pair with distance n−

1−(α−β)+ 2

.

B. α − β ∈ (0, 1]

From Lemma 9 below, for > 0 arbitrarily small, there are at least c1 n SD pairs whose distances are larger than 1−(α−β)+ 2 n− whp, where c1 > 0 is some constant and independent of n. Then, we restrict only on the delivery of the requests of such SD pairs, obtaining clearly an upper bound on the per-node throughput. First, we consider the exclusive area (i.e., the area to prohibit the transmission for other SD pairs) occupied by the multihop transmission of a SD pair 1−(α−β)+ 2 . In order to obtain a lower bound with distance n− of such area, we assume that ∆ = 0 and each receiver node is located at the distance of r from its transmitter node along with the SD line (see Fig. 2). Then, the exclusive area is lower bounded (i.e., only taking the shaded areas in Fig. 2) such as 2 −

2πr n

1−(α−β)+ 2

−

/2r = πrn

1−(α−β)+ 2

.

(24)

Hence, the maximum number of SD pairs guaranteeing a rate of W over the entire network of a unit area is upper bounded 1−(α−β)+ 1 W 1−(α−β)+ 2 2 by πr whp. As a result, Sn ≤ πr and n n −1−(α−β)+ Sn W 2 accordingly, we have Tn ≤ n ≤ πr n . Notice that the above bound on Tn increases as r decreases. On the other hand, it was shown in [11, Section V] that the absence of isolated nodes is a p necessary condition for a nonzero Tn requiring that r ≥ c2 log n/n for some constant c2 > 0 independent of n. Therefore, we have an upper bound on the −(α−β) per-node throughput as Tn ≤ n 2 + for > 0 arbitrarily small, when α − β ∈ (0, 1]. Lemma 9: Suppose that α−β ∈ (0, 1]. For > 0 arbitrarily small, let Nout,2 denote the number of nodes that they cannot 1−(α−β)+ 2 find their requested files within the distance of n− whp

from their positions. Then, we have Nout,2 ≥ (1 − )n. Proof: For simplicity, denote ζ = 1−(α−β)+ . Let Nfile 2 denote the total number of files that are able to be stored by the area of radius n−ζ . From Lemma 1 (b), the number of nodes in that area is upper bounded by (1 + δ)n1−2ζ whp. Hence Nfile ≤ (1 + δ)n1−2ζ M = (1 + δ)n1−2ζ+β whp. Then the probability that each node cannot find its requested file within the radius of n−ζ is lower bounded by (1+δ)n1−2ζ+β

pout,2 :=1 −

X i=1

(1+δ)nα−

pr (i) = 1 −

X

pr (i) (25)

i=1

whp. Then similarly to (23), we have: (pout,2 − µ)2 n Pr(Nout,2 ≥ µn) ≥ 1 − exp − 2pout,2

We considered a wireless ad hoc network in which nodes have cached information from a library of possible files. For such network, we proposed an order-optimal caching policy (i.e., file placement policy) and multihop transmission protocol. Interestingly, we showed that a distributed uniform random caching is order-optimal for the parameter regimes of interest as long as the total number of files in the library is less than the overall caching memory size in the network. i.e., α−β ∈ (0, 1]. Also, it was shown that a multihop transmission provides a significant throughput gain over one-hop direct transmission as in the conventional wireless ad hoc networks. ACKNOWLEDGEMENT This work has been supported in part by the Basic Science Research Program through NRF funded by MEST [NRF2013R1A1A1064955]. R EFERENCES [1] “Cisco visual networking index: Global mobile data traffic forecast update, 2014–2019,” Cisco Public Information, 2015. [2] N. Golrezaei, K. Shanmugam, A. G. Dimakis, A. F. Molisch, and G. Caire, “Femtocaching and device-to-device collaboration: A new architecture for wireless video distribution,” IEEE Communication Magazine, vol. 51, pp. 142–149, 2013. [3] M. Ji, G. Caire, and A. F. Molisch, “Optimal throughput-outage tradeoff in wireless one-hop caching networks,” in Proc. IEEE Int. Symp. on Information Theory (ISIT), Istanbul, Turkey, Jul. 2013. [4] ——, “Fundamental limits of distributed caching in D2D wireless networks,” in Proc. IEEE Information Theory Workshop (ITW), Seville, Sept. 2013. [5] M. A. Maddah-Ali and U. Niesen, “Fundamental limits of caching,” in Proc. IEEE Int. Symp. on Information Theory (ISIT), Istanbul, Turkey, Jul. 2013. [6] ——, “Decentralized coded caching attains order-optimal memory-rate tradeoff,” arXiv preprint arXiv:1301.5848, 2013. [7] M. Ji, A. Tulino, J. Llorca, and G. Caire, “On the average performance of caching and coded multicasting with random demands,” in Proc. IEEE Int. Symp. on Wireless Communications Systems (ISWCS), Barcelona, Spain, Aug. 2014. [8] ——, “Caching and coded multicasting: Multiple groupcast index coding,” arXiv preprint arXiv:1402.4572, 2014. [9] N. Karamchandanil, U. Niesen, M. A. Maddah-Ali, and S. Diggav, “Hierarchical coded caching,” arXiv preprint arXiv:1403.7007, 2014. [10] J. Hachem, N. Karamchandanil, and S. Diggavi, “Multi-level coded caching,” arXiv preprint arXiv:1404.6563, 2014. [11] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Inf. Theory, vol. 46, pp. 388–404, Mar. 2000. [12] S. Gitzenis, G. S. Paschos, and L. Tassiulas, “Asymptotic laws for joint content replication and delivery in wireless networks,” IEEE Trans. Inf. Theory, vol. 59, pp. 2760–2776, May 2013. [13] L. Breslau, P. Cao, L. Fan, G. Phillips, and S. Shenker, “Web caching and zipf-like distributions: Evidence and implications,” in Proc. IEEE INFOCOM, New York, NY, Mar. 1999. [14] A. El Gamal, J. Mammen, B. Prabhakar, and D. Shah, “Optimal throughput–delay scaling in wireless networks—Part I: The fluid model,” IEEE Trans. Inf. Theory, vol. 52, pp. 2568–2592, Jun. 2006. ¨ ur, O. L´evˆeque, and D. N. C. Tse, “Hierarchical cooperation [15] A. Ozg¨ achieves optimal capacity scaling in ad hoc networks,” IEEE Trans. Inf. Theory, vol. 53, pp. 3549–3572, Oct. 2007.

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