The Throughput-Outage Tradeoff of Wireless One-Hop Caching Networks

Transcription

1 The Throughput-Outage Tradeoff of Wireless One-Hop Caching Networks Mingyue Ji, Student Meber, IEEE, Giuseppe Caire, Fellow, IEEE, and Andreas F. Molisch, Fellow, IEEE arxiv:3.637v [cs.it] 8 Jul 05 The authors are with the Departent of Electrical Engineering, University of Southern California, Los Angeles, CA 90089, USA. e-ail: {ingyuej, caire, olisch}@usc.edu. A short version of this work was presented at the 03 IEEE International Syposiu on Inforation Theory, ISIT 03, Istanbul, July 7-, 03. This work was supported by the Intel VAWN Video Aware Wireless Networks progra and the National Science Foundation under grants CCF-4340 and CIF-680.

2 Abstract We consider a wireless device-to-device DD network where the nodes have pre-cached inforation fro a library of available files. Nodes request files at rando. If the requested file is not in the onboard cache, then it is downloaded fro soe neighboring node via one-hop local counication. An outage event occurs when a requested file is not found in the neighborhood of the requesting node, or if the network adission control policy decides not to serve the request. We characterize the optial throughput-outage tradeoff in ters of tight scaling laws for various regies of the syste paraeters, when both the nuber of nodes and the nuber of files in the library grow to infinity. Our analysis is based on Gupta and Kuar protocol odel for the underlying DD wireless network, widely used in the literature on capacity scaling laws of wireless networks without caching. Our results show that the cobination of DD spectru reuse and caching at the user nodes yields a per-user throughput independent of the nuber of users, for any fixed outage probability in 0,. This iplies that the DD caching network is scalable : even though the nuber of users increases, each user achieves constant throughput. This behavior is very different fro the classical Gupta and Kuar result on ad-hoc wireless networks, for which the per-user throughput vanishes as the nuber of users increases. Furtherore, we show that the user throughput is directly proportional to the fraction of cached inforation over the whole file library size. Therefore, we can conclude that DD caching networks can turn eory into bandwidth i.e., doubling the on-board cache eory on the user devices yields a 00% increase of the user throughout. Throughput-outage tradeoff, scaling laws, caching wireless networks, device-to-device counications. Index Ters

3 I. INTRODUCTION Data traffic generated by wireless and obile devices is predicted to increase by soething between one and two orders of agnitude [] in the next five years, ainly due to wireless video streaing. Traditional ethods for increasing the area spectral efficiency, such as using ore spectru and/or deploying ore base stations, are either insufficient to provide the necessary wireless throughput increase, or are too expensive. Thus, exploring alternative strategies that leverage different and cheaper network resources is of great practical and theoretical interest. The bulk of wireless video traffic is due to asynchronous video on deand, where users request video files fro soe library e.g., itunes, Netflix, Hulu or Aazon Prie at arbitrary ties. This type of traffic differs significantly fro live streaing. The latter is essentially a lossy ulticasting proble, for which the broadcast nature of the wireless channel can be naturally exploited see for exaple [] [6]. The theoretical foundation of schees for live streaing relies on well-known inforation theoretic settings for one-to-any transission of a coon essage with possible refineent inforation, such as successive refineent [7] [9] or ultiple description coding [0] []. In contrast, the asynchronous nature of video on deand prevents fro taking advantage of ulticasting, despite the significant overlap of the requests people wish to watch a few very popular files. Hence, even though users keep requesting the sae few popular files, the asynchronis of their requests is large with respect to the duration of the video itself, such that the probability that a single transission fro the base station is useful for ore than one user is essentially zero. Due to this reason, current practical ipleentation of video on deand over wireless networks is handled at the application layer, requiring a dedicated data connection typically TCP/IP between each client user and the server base station, for each streaing user, as if users were requesting independent inforation. One of the ost proising approaches to take advantage of the inherent asynchronous content reuse is caching, widely used in content distribution networks over the wired Internet [3]. In [4], [5], the idea of deploying dedicated helper nodes with large caches, that can be refreshed via wireless at the cellular network off-peak tie, was proposed as a cost-effective alternative to providing large capacity wired backhaul to a network of densely deployed sall cells. An even ore radical view considers caching directly at the wireless users, exploiting the fact that odern devices have tens and even hundreds of GBytes of largely under-utilized storage space, which represents an enorous, cheap and yet untapped network resource. Recently, a coded ulticasting schee exploiting caching at the user nodes was proposed in [6]. In this

4 3 schee, the files in the library are divided in blocks packets and users cache carefully designed subsets of such packets. Then, for given set of user deands, the base station sends to all users ulticasting a coon codeword fored by a sequence of packets obtained as linear cobinations of the original file packets. As noticed in [6], coded ulticasting can handle any for of asynchronis by suitable subpacketization. Hence, the schee is able to create ulticasting opportunities through coding, exploiting the overlap of deands while eliinating the asynchronis proble. For the case of arbitrary adversarial deands, the coded ulticasting schee of [6] is shown to perfor within a sall gap, independent of the nuber of users, of the cache size and of the library size, fro the cut-set bound of the underlying copound channel. However, the schee has soe significant drawbacks that akes it not easy to be ipleented in practice: the construction of the caches is cobinatorial and the sub-packetization explodes exponentially with the library size and nuber of users; changing even a single file in the library requires a significant reconfiguration of the user caches, aking the cache update difficult. In [7], siilar near-optial perforance of coded caching is shown to be achieved also through a rando caching schee, where each user caches a rando selection of bits fro each file in the library. In this case, though, the cobinatorial coplexity of the coded caching schee is transferred fro the caching phase to the coded delivery phase, where the construction of the ulticast codeword requires solving ultiple clique cover probles with fixed clique size known to be NP-coplete, for which a greedy algorith is shown to be efficient. Our contributions: In this paper, we focus on an alternative approach that involves rando independent caching at the user nodes and device-to-device DD counication. Instead of creating ulticasting opportunities by coding, we exploit the spatial reuse provided by concurrent ultiple short-range DD transissions. Inspired by the current standardization of a DD ode for LTE the 4-th generation of cellular systes [8], we restrict to one-hop counication. Under such assuption, requiring that all users ust be served for any request configuration is too constraining. Therefore, we introduce the possibility of outages, i.e., that soe requests are not served, because of soe network adission control policy to be discussed in details later on. For the syste described in Section II, we define the throughput-outage region and obtain achievability and converses that are sufficiently tight to characterize the throughput-outage scaling laws within a sall gap of the constants of the leading ter. Furtherore, our analysis shows very good agreeent with with finite-diensional siulation results. In the relevant regie of sall outage probability, the throughput of the DD one-hop caching network The copound nature of this odel is due to the fact that the schee handles adversarial deands.

5 4 behaves in the sae near-optial way as the throughput of coded ulticasting [6], [7], while the syste architecture is significantly ore straightforward for a practical ipleentation. In particular, for fixed cache size M, as the nuber of users n and the nuber of files becoe large with nm, the throughput of the DD one-hop caching network grows linearly with M, and it is inversely proportional to, but it is independent of n. Hence, DD one-hop caching networks are very attractive to handle situations where a relatively sall library of popular files e.g., the 500 ost popular ovies and TV shows of the week is requested by a large nuber of users e.g., 0,000 users per k in a typical urban environent. In this regie, the proposed syste is able to efficiently turn eory into bandwidth, in the sense that the per-user throughput increases proportionally to the cache capacity of the user devices. We believe that this conclusion is iportant for the design of future wireless systes, since bandwidth is a uch ore scarce and expensive resource than storage capacity. Related literature: The analysis of the capacity scaling laws for large DD or ad-hoc wireless networks has been the subject of a vast body of literature. Gupta and Kuar [9] proposed a network odel where n are randoly placed on soe planar region and counicate through ultiple hops. They characterized the transport capacity scaling as n, under the sae protocol odel considered in our paper see Section II. For rando assignent of source-destination pairs, [9] showed that the per-link capacity ust vanish as O n. In addition, a ulti-hop relaying schee was shown to achieve throughput scaling Ω n log n. The sae results were confired, using a soehow sipler and ore general analysis technique, in [0]. The ulticast capacity of large wireless networks has been investigated in [], []. Finally, Franceschetti, Dousse, Tse and Thiran [3] closed the logn gap between upper bound and achievability in [9], [0] by creating an alost deterinistically placed grid sub-network with vertical and horizontal highways that relay essages with very short hops. The existence of such grid subnetwork is guaranteed with high probability by percolation theory. Given the fact that randoly placed nodes yield the sae scaling laws of nodes placed on a deterinistic squared grid, in this work we assue a grid network fro the start. This allows to focus on the essential aspect of caching at the nodes, and avoid the analytical coplication of randoly placed nodes. The sae approach is taken in [4], where ulti-hop DD counication is considered under the protocol odel for a network of nodes placed deterinistically on a squared grid. If the aggregate distributed Scaling law order notation: given two functions f and g, we say that: fn = O gn if there exists a constant c and integer N such that fn cgn for n > N. fn = o gn if li n fn gn = 0. 3 fn = Ω gn if gn = O fn. 4 fn = ω gn if gn = o fn. 5 fn = Θ gn if fn = O gn and gn = O fn.

6 5 storage space in the network is larger than the total size of the library, ulti-hop guarantees that all user requests can be served by the network. Under the sae assuption ade here of the user requests distribution, [4] finds a deterinistic replication caching schee and a ulti-hop routing schee that achieves order-optial average throughput. Besides the consideration of ultihop and single hop, there are several other key differences between our work and [4]. First, [4] considers a deterinistic caching placeent approach, which depends on the files popularity distribution. This approach is not robust when users ove between cells. In contrast, obility is easily handled by our schee which is based on independent rando caching. Next, in [4], the transission range is fixed, where each node can only reach its four neighbors. Besides the deterinistic caching placeent, the key aspect of the proble is the design of the routing protocol and analyze the traffic through the bottleneck link of the network. Our work focuses on deterining the transission range within which nodes can access their neighbors caches in one hop. This, in turn, deterines the point of the throughput-outage tradeoff at which the syste operates. Finally, [4] only gives the order of the throughput as the nuber of users n goes to infinity, but does not characterizes the ultiplicative constant of the throughput leading ter in the scaling law. Therefore, it is difficult to understand in which regie of large but finite n the scaling laws becoe relevant. In passing, we notice that this is a coon proble in several studies focused on scaling laws of large wireless networks. In our case, we provide outer bounds and inner achievable bounds to the throughput-outage tradeoff, which are tight enough to characterize also the constants of the leading ters, within a bounded gap. In particular, the analysis of our achievability schee atches well with finite-diensional siulations. Preliinary work of the present paper is given in [5], where only the su throughput was considered irrespectively of user outage probability. The analysis in [5] considers a heuristic rando caching policy, while here we find the optial rando caching distribution. More iportantly, the total su throughput is not a sufficient characterization of the perforance of DD one-hop caching networks: in certain regies of the nuber of users and file library size, it can be shown that in order to achieve a large su throughput only a sall portion of the users should be served, while leaving the ajority of the users in outage. In contrast, the throughput-outage tradeoff region considered here is able to capture the notion of fairness, since it focuses on the iniu per-user average throughput and on the fraction of users which are denied service. The paper is organized as follows. Section II introduces the network odel and the precise proble forulation of the throughput-outage tradeoff in DD one-hop caching networks. The ain results on the outer bound and achievability of the throughput-outage tradeoff are presented in Sections III and

7 6 IV, respectively. In Section V we presents soe concluding rearks. All proofs are relegated in the Appendices, in order to aintain the flow of exposition. II. NETWORK MODEL AND PROBLEM FORMULATION We consider a network deployed over a unit-area squared region and fored by n nodes U = {,..., n} placed on a regular grid with iniu node distance / n see Fig.. Each user u U akes a request to a file f u F = {,..., } in an i.i.d. anner, according to a given request probability ass function {P r f : f F}. In order to odel the asynchronous content reuse and forbid any for of for-free ulticasting by overhearing, we consider the following theoretical odel. We assue that each file is fored by a sequence of L packets. Each user deand corresponds to a file index f F and a segent of L < L consecutive packets, starting at soe initial index l, uniforly and independently distributed over {,..., L L +}. The packets of the requested segent are downloaded sequentially. We easure the cache size in files, and in order to copute the syste perforance we consider first the liit for large file size L with L finite, and then study the syste scaling laws for n,. Hence, the probability that users request overlapping segents vanishes for L for any finite n,, L, thus preventing the trivial use of naive ulticasting i.e., overhearing coon essages. In contrast, the probability that two users request segents of the sae file depends on the library size and on the request distribution P r. We hasten to say that this odel is just a way to express in precise atheatical ters the notion of asynchronous content reuse, such that the overlap of the deands and the overlap of concurrent transissions are decoupled. 3 Fig. shows qualitatively our odel assuptions. In our syste, DD counication obeys the following protocol odel [9]: a node u can receive successfully a packet fro node v if du, v R and if no other node v at distance du, v < + R is transitting. The transission range R is a design paraeter that can be set as a function of and n. We consider the following definitions: Definition : Network A network if fored by a set of user nodes U and a set of files F = {,..., }. Nodes in U are placed in the two-diensional unit-square region, and their transissions obeys the protocol odel. In general, all nn directed links between user nodes, subject to the protocol odel, define an interference conflict graph. Only the links in an independent set in the interference graph can be active siultaneously. 3 As a side note, we observe also that the segentation of large files into saller packets or chunks to be sequentially downloaded is consistent with current video streaing protocols such as DASH [6] [9].

8 7 Fig.. Qualitative representation of our syste assuptions: each user caches an entire file, fored by a very large nuber of packets L. Then, users place rando requests of segents of L packets fro files of the library, starting at rando initial points. In the figure, we have L = 4. + R s Fig.. Grid network with n = 49 nodes black circles with iniu separation s = n. The red area is the disk where the protocol odel iposes no other concurrent transission. R is the case transission range and is the interference paraeter, such that the forbidden disk around the receiver has radius + R.

9 8 Definition : Cache placeent The cache placeent Π c is a rule to assign files fro the library F to the user nodes U with replaceent i.e., with possible replication. Let G = {U, F, E} be a bipartite graph with left nodes U, right nodes F and edges E such that u, f E indicates that file f is assigned to the cache of user u. A bi-partite cache placeent graph G is feasible if the degree of each user node is not larger than the axiu user cache size equal to M files. Let G denote the set of all feasible bi-partite graphs G. Then, Π c is a probability ass function over G, i.e., a particular cache placeent G G is assigned with probability Π c G. Notice that deterinistic cache placeents are special cases, corresponding to a single probability ass equal to on the desired assignent G. In contrast, we will be interested in decentralized rando caching placeents constructed as follows: each user node u U selects its cache content in an i.i.d. anner, by independently generating at rando M indices f u,,..., f u,m according to the sae caching probability ass function {P c f : f F}. Definition 3: Rando requests At each request tie integer ultiples of L, each user u U akes a request to a segent of length L of chunks fro file f u F, selected independently with probability P r. The vector of current requests f is a rando vector taking on values in F n, with product joint probability ass function Pf = f,..., f n = n i= P rf i. In this paper, we assue that P r is a Zipf distribution with paraeter 0 < < [30], i.e., P r f = f H,, for f =,...,, and H, x, y = y i=x i. Definition 4: Transission policy The transission policy Π t is a rule to activate the DD links in the network. Let L denote the set of all directed links. Let A L denote the set of all feasible subsets of links this is a subset of the power set of L, fored by all independent sets in the network interference graph. Let A A denote a feasible set of siultaneously active links according to the protocol odel. Then, Π t is a conditional probability ass function over A given f requests and G cache placeent, assigning probability Π t A f, G to A A. We ay think of Π t as a way of scheduling siultaneously copatible sets of links subject to the protocol odel. Modeling the scheduling policy in a probabilistic anner allows the analytical convenience of defining the average per-user throughput see below as an enseble average. As a atter of fact, deterinistic link activation rules can be included by defining the average throughput as a tieaverage. For exaple, a bounded deterinistic delay per user can be guaranteed by activating groups of copatible links foring a axial independent set in the network interference graph in a deterinistic round-robin sequence, such that each user is served with a deterinistic delay. Definition 5: Useful received bits per slot For given P r, Π c and Π t, and user u U, we define the

10 9 rando variable T u as the nuber of useful received inforation bits per slot unit tie by user u at a given scheduling tie irrelevant because of stationarity. This is given by T u = v:u,v A c u,v {f u Gv} where f u denotes the file requested by user node u, c u,v denotes the rate of the link u, v, and Gv denotes the content of the cache of node v, i.e., the neighborhood of node v in the cache placeent graph G. Consistently with the protocol odel, c u,v depends only on the active link u, v A and not on the whole set of active links A. Furtherore, we shall obtain our results under the siplifying assuption usually ade under the protocol odel [9] that c u,v = C for all u, v A. The indicator function {f u Gv} expresses the fact that only the bits relative to the file f u requested by user u are useful and count towards the throughput. Obviously, scheduling links u, v for which f u / Gv is useless for the sake of the throughput defined above. Hence, we can restrict our transission policies to those activating only links u, v for which f u Gv. These links are referred to as potential links, i.e., links potentially carrying useful data. Potential links included in A are active links, at the given scheduling slot. The average throughput for user u U is given by T u = E[T u ], where expectation is with respect to the rando triple f, G, A n i= P rf i Π c GΠ t A f, G. We say that user u is in outage if E[T u f, G] = 0. This condition captures the event that no link u, v with f u Gv is scheduled with positive probability, for given requests vector f and cache placeent G. In other words, a user u for which E[T u f, G] = 0 experiences a long lack of service zero rate, as far as the cache placeent is G and the request vector is f. Definition 6: Nuber of nodes in outage The nuber of nodes in outage is given by Notice that N o is a rando variable, function of f and G. by N o = u U {E[T u f, G] = 0}. Definition 7: Average outage probability The average across the users outage probability is given p o = n E[N o] = P E[T u f, G] = 0. 3 n u U

11 0 In this work we focus on ax-in fairness, i.e., we express the throughput-outage tradeoff in ters of the iniu average user throughput, defined as T in = in u U { T u }. 4 Notice that that the ax-in fairness criterion in our setting is essential to ake the outage probability p o defined in 3 eaningful. In fact, for 0 p o < p o, consider a syste that achieves outage probability p o by serving only a fraction λ of users with outage probability p o = po λ λ, while leaving the reaining fraction λ of users peranently idle. In this case, we have T in = 0 since there are λn > 0 users with identically zero throughput. Hence, a syste that peranently excludes soe users in favor of others is certainly not optial in ters of the throughput-outage tradeoff as defined below: Definition 8: Throughput-Outage Tradeoff For a given network and request probability distribution P r, an throughput-outage pair T, p is achievable if there exists a cache placeent Π c and a transission policy Π t with outage probability p o p and iniu per-user average throughput T in T. The throughput-outage achievable region T is the closure of all achievable throughput-outage pairs T, p. In particular, we let T p = sup{t : T, p T }. Notice that T p is the result of the following optiization proble: axiize T in subject to p o p, 5 where the axiization is with respect to the cache placeent and transission policies Π c, Π t. Hence, it is iediate to see that T p is non-decreasing in p. The range of feasible outage probability, in general, is an interval [p o,in, ] for soe p o,in 0. We say that an achievable point T, p doinates an achievable point T, p if p p and T T. The Pareto boundary of T consists of all achievable points that are not doinated by other achievable points, i.e., it is given by {T p, p : p [p o,in, ]}. It is also iediate to see that the throughput-outage tradeoff region is convex. In fact, consider two achievable points T in, p o and T in, p o corresponding to caching placeents G and G, with probability assignents Π c and Π c. For λ [0, ], the caching placeent G with ixture probability assignent Π c = λπ c + λπ c achieves p o = λp o + λp o. For this value of outage probability, the best possible strategy achieves { } T p o in λt u u + λt u λ in T u u + λ in T u u, where, by definition, T in = in u T u and T in = in u T u. Hence, the segent joining any

12 two achievable throughput-outage points T in, p o and T throughput-outage region. in, p o is contained into the achievable We conclude this section by providing the intuition behind the tension between outage and throughput, and explaining through an intuitive arguent why T p is non-decreasing for p [p o,in, ]. The key tradeoff quantity here is the cooperation cluster size g, that is, the size of the set of nearest neighbor nodes aong which any node u can look for its desired file f u. On one hand, we would like to have g large, in order to take advantage of the content reuse, i.e., the larger g, the larger the probability that any user can find and retrieve its desired file. On the other hand, we would like to have g sall, in order to take advantage of the spatial reuse, i.e., the saller g, the larger the nuber of siultaneously active links that the network can support. Therefore, g describes the tradeoff between content reuse and spatial reuse. As g increases the probability that user u does not find its desired file decreases. Hence, p o is a decreasing function of g see Fig.3a. Since nodes can retrieve their desired files within a cluster of size g, then the counication range of the DD links ust be enough to counicate across such clusters. The average nuber of active links that can be activated without violating the protocol odel is of the order of the nuber of disjoint clusters in the network, i.e., n g. With a probability p o that any user cannot find its requested file, the average per-user throughput is roughly given by T po n n g = po g. Since for sall g we have p o, and for large g we have p o p o,in, where the latter is the probability that a node u does not find its requested file in the whole network, we clearly see that T ust be increasing for sall g and decreasing for large g see Fig.3b. Now, consider the constraint on the outage probability p o p. If p is sall, then the constraint ust be satisfied with equality, yielding the corresponding value of g Fig.3a which in turns yields a corresponding value of T Fig.3b. As p increases, we obtain the concave increasing part of the throughput vs outage Pareto boundary curve qualitatively depicted in Fig. 3c. However, when p becoes larger than soe threshold, the optial throughput is obtained by letting g = g, which is the size that achieves the axiu unconstrained throughput see Fig. 3b. This eans that for values of p beyond this threshold value, the throughput curve reaches a bound horizontal line, equal to the unconstrained axiu throughput, as shown in Fig. 3c. It follows fro the upper bounds developed in Section III that this intuitive arguent, even though it is developed for a cluster-based achievability strategy, holds true also for the upper bounds, despite the fact that the latter do not assue any a priori transission strategy other than the one-hop constraint and

13 outage prob. throughput throughput T p o,in p g cluster size g g cluster size p o,in outage prob. a b c Fig. 3. Qualitative behavior of the tradeoff between throughput and outage probability, by ways of the tradeoff paraeter g, which represents the size of the cluster of nodes over which any node can look for its desired requested file and download it by DD on-hop counication. the protocol odel. III. OUTER BOUNDS Under the one-hop restriction, network topology and protocol odel given in Section II, we can provide an outer bound T ub p on the throughput-outage tradeoff, such that the enseble of points {T ub p, p : p [0, ]} doinates the tradeoff region T. In what follows, the quantity α = plays an iportant role. Notice that 0 α < / under the assuption ade here that 0 < <. The following results are proved in Appendices A and B: Theore : When li n α n T ub p, p given by : T ub p = 6CM p { in + o p = 0, the throughput-outage region is doinated by the set of points, p = MgR } 6CM, f p ub ρ α + o α, Mρ α p < Mρ α f ub ρ α + o α, Mρ α p, where ρ ρ and ρ is the solution with respect to ρ of the equation ζρ = log + ζρ 8 6 7

14 3 with ζρ = + 3 ρ M, 9 g R is any function such that g R = ω α and g R in{ M, n}, and where f ub ρ = 6C ρ e ζρ. 0 Theore : When there exists a positive constant ξ such that ξ li n α n 6 ρ, the throughputoutage region is doinated by the set of points T ub p, p given by: T ub p = { } 6CM in, f p ub ρ α + o α, Mρ α p < Mρ α f ub ρ α + o α, Mρ α p, where ρ is the solution of 8, and ρ [ρ, 6 n ]. α Theore 3: When li n α n > 6 ρ doinated by the set of points T ub p, p given by : Mn T ub p = C + o ρ being the solution of 8, the throughput-outage region is n, Mn p. Notice that the range of p in Theores and 3 is liited to [p o,in, ] with p o,in = Mρ α for Theore and p o,in = Mn for Theore 3, showing that in these regies the outage probability cannot be sall. As a atter of fact, of all the regies identified by Theores, and 3, the only practically interesting one is the first regie of Theore. In particular, Theore and 3 show that, when li n α n is bounded away fro zero, any schee for the one-hop DD caching network yields outage probability that goes to, which is clearly not an acceptable. In contrast, li n n = κ <, there ight exist schees achieving soe fixed target outage probability value p [0,, as n,. Intuitively, the function g R plays the role of the size of the cooperation cluster of neighboring nodes within which each user can find its requested file. For exaple, choosing g R = β for soe constant β in { M, } κ, both conditions gr = ω α and g R in { M, n} are satisfied, for all sufficiently large n. Notice that for κ M, the choice β = /M yields that the outer bound contains points of the type OM/, 0 zero outage probability. We shall see in the next section that throughput-outage points with throughput ΩM/ and fixed p bounded away fro are achievable.

15 4 For a conventional unicast syste where users are served by a single oniscient node e.g., a base station that can store the whole library, the throughput scaling is O/n. 4 Hence, in the case of nm, the cobination of caching and DD spatial reuse yields a very large throughput relative gain with respect to a conventional syste. It is also interesting to notice that despite the first regie of Theore requires that grows ore slowly than n /α, the only practically interesting sub-regie is = on, otherwise conventional unicast fro the base station yields better throughput scaling and zero outage probability all users are served. All other regies in Theores 3 are included for copleteness, in order to prove atheatically a soehow intuitively expected negative result: unless the library size is sall with respect to the aggregate caching eory nm, caching cannot achieve significant throughput gains with respect to conventional unicast fro a single base station. This result is expected since, in this case, the asynchronous content reuse that the DD caching network tries to exploit is essentially non-existent. It is also iportant to notice that here we are considering the realistic case of a heavy tail Zipf request distribution with 0,. If >, then a finite nuber of files collects essentially all the request probability ass, and this case is siilar to the case of = O, which is a special case of = on. As a atter of fact, Zipf-distributed requests with 0, have been observed experientally [3] [33]. In the next section, we show that the upper bounds obtained here are tight in the scaling laws, and that the constants of the leading ters can be deterined within constant gaps. This is obtained by exhibiting and analyzing a specific achievability strategy. IV. ACHIEVABLE THROUGHPUT-OUTAGE TRADEOFF Consistently with the outer bounds in Theores 3 and the concluding rearks in Section III, we consider achievability only in the sall library regie li n α n = 0, for which there is hope to achieve soe target outage probability strictly less than. We obtain an achievable inner bound on the achievable throughput-outage tradeoff region by considering a transission policy based on clustering and a caching policy based on independent rando caching. Clustering: the network is divided into clusters of equal size, denoted by g c, and independent of the users deands and cache placeent realizations. A user can only look for the requested file inside 4 This is obviously achieved by TDMA, serving users on different tie slots in a round-robin fashion. Notice that even if a ore refined physical layer including a Gaussian broadcast channel is considered, the throughput scaling reains the sae.

16 5 Fig. 4. Exaple of TDMA reuse schee: each square represents a cooperation cluster. Gray squares represent the concurrently active clusters in a given tie slot. In other ties slots, other patterns of concurrently active clusters obtained by shifting the pattern in the figure are activated. In this particular exaple, the TDMA reuse paraeter is K = 9. its own cluster. For each user whose deand is found inside its cluster, we say that a potential link exists in the cluster. If a cluster contains at least one potential link, we say that this cluster is good. We use an interference avoidance transission policy Π t for which at ost one concurrent transission is allowed in each cluster, over any tie-frequency slot transission resource. Furtherore, potential links inside the sae cluster are scheduled with equal probability or, equivalently, in round robin, such that all users have the sae throughput T u = T in. To avoid interference between clusters, we use a conventional TDMA with spatial reuse schee [34, Ch. 7], very siilar to the spatial reuse schee of a cellular network, where each cluster acts as a cell. In short, a coloring schee with K colors is applied to the clusters such that clusters with the sae color can be concurrently active on the sae tie slot, without violating the protocol odel. The resulting groups of clusters are assigned to K orthogonal tie slots, and are activated in a round-robin fashion. In particular, we use K = + + in order to guarantee that concurrently active clusters do not interference with each other. Fig. 4 shows an exaple for K = 9. Rando Caching: we consider a caching policy Π c where each node independently caches M files according to a coon probability distribution Pc, given by the following result proved in Appendix C:

17 6 Theore 4: Under the syste odel assuptions and the clustering schee described above, the caching distribution P c its corresponding cluster is given by that axiizes the probability that any user u U finds its requested file inside P c f = [ ν z f ] +, f =,...,, 3 where ν =, z j = P r j Mgc, and = Θ in{ M g c, }. j= z j The following theore proved in Appendices D yields an inner bound on the achievable outagethroughput tradeoff region: Theore 5: Assue li n α n caching and clustering behaves as: C M K ρ + o/, T p = CA K M p = 0. Then, the throughput-outage tradeoff achievable by rando + o CB K α + o α, CD where we define a = M, b = a p p = e ρ, p = a gc aρ α p ab α K α + o α, ab α p,, A =, B = aρ +aρ 4, D = ab +ab and where ρ and ρ are positive paraeters satisfying ρ and ρ b. The cluster size g c is any function of satisfying g c = ω α and g c /M. In all cases, the achievable throughput scaling law both for p bounded away fro and p coincide with the outer bounds of Theore. Therefore, these throughput scaling laws are tight up to soe gap in the constants of the leading ters. In the rest of this section we copare the achievable throughput scaling law of Theore 5 with the outer bounds of Section III and with the perforance achievable by other schees. In particular, we focus on the interesting regie of sall library Theores and 5. Since α < /, even in this regie the library size can grow faster than n. However, we restrict to the practically relevant regie of = On linear or sub-linear in n. Choosing g c = β for soe β > 0, it is apparent fro the first and second line of 4 that p strictly bounded away fro. By fixing a sall but positive target outage probability, the per-user average throughput of the DD one-hop caching network with rando decentralized caching scales as T p = Θ ax { n, M }, where the scaling Θ n can be trivially achieved by letting the whole network to be a single cluster e.g., transission radius R = and serving one deand per unit tie. This scaling is equivalent to conventional unicast fro a single

18 7 oniscient node which can be regarded as the state of the art of today s single cell systes, with a base station or access point serving individual requests without exploiting the asynchronous content reuse. We notice that, when nm, the throughput of the DD caching network achieves per-user throughput that increases linearly with M. In this regie, caching in the user nodes and exploiting the dense spatial reuse of the DD network is a very attractive approach, since storage space is uch cheaper than scarce resources such as bandwidth or dense base station deployent the reader will forgive this vague stateent in this context. It is interesting to notice that our analysis is able to characterize also the constant of the leading ter within a bounded gap. This is a fortunate fact that does not happen often for the scaling analysis of wireless network capacity e.g., see [9] [3]. For exaple, upper and lower bounds Theore and 5, respectively and finite-diensional siulation results are copared in Fig. 5, which shows both theoretical solid lines and siulated dashed lines curves of the throughput y-axis vs. outage x-axis tradeoff for different values of. In this siulation, the throughput is noralized by C, so that it is independent of the link rate. In particular, the theoretical curves show the doinant ter in 4 divided by C. In these exaples we used = 000, n = 0000, M = and the spatial reuse factor K = 4. The Zipf paraeter varies fro 0. to 0.6, corresponding to the curves fro the left blue to the right cyan..5 x 0 3 Noralized Throughput per User p Fig. 5. Coparison between the noralized theoretical result solid lines and noralized siulated result dashed lines in ters of the iniu throughput per user vs. outage probability, where = 000, n = 0000, M = and spatial reuse factor K = 4. The Zipf paraeter varies fro 0. to 0.6 fro the left blue to the right cyan.

19 8 As anticipated in Section I, in order to understand the potential of the cobination of DD one-hop counication and caching in the user devices, it is instructive to copare the scaling laws achieved by the DD caching network with those achievable by other possible approaches. We have already discussed conventional unicast, achieving Θ n throughput. When the nuber of files is less than the nuber of users n, an alternative consists of broadcasting all files on orthogonal channels. In order to guarantee that any requesting user can start its playback within a delay uch shorter than the whole file duration, also in the presence of asynchronous requests, a wellknown approach consists of Haronic Broadcasting [35] [38]. This schee broadcasts continuously to all users a coon essage fored by the ost probable files, each of which is encoded in the way proposed in [35], refined in [36], [37] and whose optiality in an inforation theoretic sense was established in [38]. Without entering the details, each file of length L packets is divided into blocks of length L/N packets and encoded with bandwidth expansion factor H,, N = N i= i, such that the storage space at each user node is 0 < M < less than one entire file is stored at each given tie, and the tie that any user ust wait between the instant at which the streaing session is requested and the instant at which the playback starts is not larger than L/N packets. In order to allow the users to start playback within a finite delay while L, the ratio L/N ust be finite, i.e., it ust be N = ΘL. Furtherore, for and 0 < <, in order to have outage probability bounded away fro, we need = Θ. Therefore, the bandwidth expansion factor of haronic broadcasting in this regie is log N = Θ log L. It follows that the throughput of Haronic Broadcasting scales as Θ log L. Fro a strictly technical viewpoint, since in our syste assuptions we study the syste perforance by first letting L, and then considering n, that siultaneously grow in soe relation, the throughput of haronic broadcasting under our assuption is identically zero. In practice, for large but finite L, the gain of DD caching over Haronic Broadcasting can be appreciated by by coparing the ter M with the ter log L in the per-user throughput. It is clear that Haronic Broadcasting does not take advantage of the user nodes storage eory, and in addition suffers an arbitrarily large ultiplicative penalty as the length of the files L increases. Finally, we exaine the coded ulticasting schee of [6], already briefly described in Section I, which represent another exaple of one-hop network with caching in the user nodes, able to ake efficient and in fact, near-optial in an inforation theoretic sense use of caching. The rate analysis provided in [6] shows that the nuber of equivalent file ulticast transissions fro the base station

20 9 in order to satisfy any set of users requests is given by Nn,, M = n M + Mn such that the iniu average throughput per user is given by T = C 0 + Mn n M, where C 0 is the coon downlink rate at which the base station can send the ulticast coded essage to all users. For large, n and finite M, the scaling of the per-user throughput given again by Θ ax { n, M }, where the two ters inside the ax are realized depending on whether nm, or nm. Interestingly and soehow surprisingly, this is the sae scaling behavior of the DD caching network studied in this paper. 5, V. CONCLUSIONS In this paper we have considered a wireless device-to-device DD network where the nodes have pre-cached inforation fro a fixed library of possible files, users request files at rando and, if the requested file is not in the on-board cache, then it is downloaded fro soe neighboring node via one-hop local counication. To odel the wireless network, we have followed the siple protocol odel, widely considered in the analysis of the transport capacity scaling laws of wireless ad-hoc networks. We have proposed a odel that captures atheatically the asynchronous content reuse typical of on-deand video streaing, where the users requests have strong overlap and concentrate on a sall set of popular ovies, but the deands are copletely asynchronous, such that naive ulticasting is not effective. In our odel, a user is in outage when its requested file is not found within the allowed transission range. We have defined the optial tradeoff between iniu per-user average throughput and the average fraction of users in outage, that we refer to as outage probability. Then, we have characterized such optial tradeoff in ters of tight scaling laws in all the scaling regies of the syste paraeters, when both the nuber of nodes and the nuber of files in the library grow to infinity. The ain result of this work is that, in the relevant regie sall library, i.e., when = On and the aggregate cache capacity of the network, nm is uch larger that the library size, the throughput of 5 For a perforance coparison between the DD caching network of the present work, coded ulticasting in [6], Haronic Broadcasting in [35] and conventional unicasting under realistic assuptions on the underlying DD and cellular physical channels, please see [39].

21 0 the DD one-hop caching network is proportional to M/ and independent of n. Hence, DD one-hop caching networks are very attractive to handle situations where a relatively sall library of popular files e.g., the 500 ost popular ovies and TV shows of the week is requested by a large nuber of users e.g., 0,000 users per k in a typical urban environent. In this regie, the proposed syste is able to efficiently turn eory into bandwidth, in the sense that the per-user throughput increases proportionally to the cache capacity M of the user devices. Since the latter follows the doubling rate of Moore s law, caching in the user devices can achieve orders of agnitude throughput gains without requiring ore bandwidth. Interestingly, the sae throughput scaling law is achieved by coded ulticasting [6], [7] for a different one-hop network topology with caching at the user nodes, where a single central node e.g., a base station ulticast network-coded codewords fored by EXORing data packets. It is worthwhile to point out that, although these schees yield the sae throughput scaling law, they achieve their orderoptial caching gain according to two copletely different principles. The DD caching network exploits the dense spatial reuse provided by caching, i.e., by replicating the sae file any ties in the network, any user with high probability can find its requested file at short distance, such that any siultaneously active links can be supported on the sae tie slot. In contrast, coded ulticasting achieves its gain by using network coding in order to create a single essage which is siultaneously useful for any users. While in the DD network transissions should be as local as possible in order to exploit spatial reuse, in the coded ulticast network transissions should be as global as possible, in order to benefit the largest nuber of users. In a recent follow-up paper [40], we have investigated a decentralized version of network-coded schee of [6] for the sae DD network of the present work, without any oniscient node that has the whole file library. It turns out that coded ulticasting gain and spatial reuse gain do not cuulate. Thus, it sees that the throughput scaling law obtained here is soehow an inherent liitation of one-hop networks with caching in the user nodes. APPENDIX A PROOF OF THEOREMS AND We first provide an outline of the proof and then dig into the details. We define T su = n u= T u and let T sup, p be the solution of axiize T su subject to p o p, 5

22 where the axiization is with respect to the cache placeent and transission policies Π c, Π t. As for T p, also T sup is non-decreasing in p. Furtherore, the inequality T p n T sup follows iediately fro the definition of T sup and T p. We paraeterize proble 5 with respect to the nuber of nodes in a disk of radius R, referred to for brevity as disk size and indicated by g R, where R denotes the one-hop transission range of the protocol odel. For any value g R = g, let T sug denote the largest achievable su throughput with disk size g, and let p og denote the corresponding outage probability. While obtaining exact expressions for T sug and for p og is difficult, we shall obtain an upper bound T ub sug T sug and a lower bound p lb g p og. By the onotonicity property said above, it follows that T ub sug, p lb g doinates T sug, p og and, as a consequence, n T ub sug, p lb g doinates T p, p for p = p og. Also, we have that the set of outage probability values p og obtained by varying g includes the feasibility doain [p o,in, ] of the original proble 5. This iplies that the set of points n T ub sug, p lb g, obtained by varying g, doinates the Pareto boundary of the throughput-outage region T. 3 Finally, we shall consider separately the different regies of the outer bound, by eliinating the paraeter g R. Conceptually, this can be obtained by letting p = p lb g, solving for g as a function of p and replacing the result into T ub sug. The resulting outer bound shall be denoted siply by T ub p, p, given by Theores and. We focus first on Theore, where li n α n = 0, and consider in details step of the above outline. In the following, we shall iplicitly ignore the non-integer effects when they are irrelevant for the scaling laws. For exaple, recalling that the network has node density n we have n nodes in the unit square, the disk size is given up to integer rounding by g R = πr n = g. 6 For given disk size g, a lower bound on p o can be obtained by observing that p o is upper bounded by the axiu over the users u =,, n, of the probability that user u can be served by the DD network. A necessary condition for this to happen is that the essage f u is found in the cache of soe node inside a disk of size g centered at node u. We denote such event by F u g. 6 If g /M, then the outage probability lower bound is zero, since we can arrange the files in the caches such that at least 6 Notice: events are defined in the probability space of the triple f, G, A n i= PrfiΠcGΠtA f, G, of requests, cache placeents and transission scheduling decisions.

23 one node u finds all files in the library within a radius R. Hence, assuing g < /M, we have p o ax P F u u g a = Mg P r f = f= Mg f= f H,, H,, Mg H,,, 7 where a follows by caching all ost popular Mg files within a disk of radius R for a given user. In order to estiate the value of H,,, we have the following lea: Lea : For, then For =, then y + x H, x, y y x + x. 8 Proof: See Appendix E. logy + logx H, x, y logy logx + x. 9 Fro 7 and Lea, we have the lower bound 0 for g p og p lb g = M Mg + for g < M Next, we seek an upper bound on T sug as a function of the disk size g. According to the protocol odel see Section II, the throughput T su is given by 0 T su = C E [L], where L is the nuber of active links over any strategy with transission radius R. Letting i, j and k, l denote two distinct transitter-receiver pairs, using the triangle inequality and the protocol odel constraints, we have dj, l dk, j dk, l + R dk, l + R R = R. Hence, any two receivers ust be separated by distance not saller than R. Equivalently, disks of radius R around any receiver ust be disjoint. Since there is at least a fraction /4 of the area of such

24 3 Fig. 6. Illustration of the fact that the nuber of sall disks intersecting the union of the big disks centered at the active receivers is necessarily an upper bound to the nuber of active receivers. disks inside the unit square containing our network, the nuber of such disjoint disks in the unit square is upper bounded by 6 π R. We wish to upper bound the nuber of siultaneously active receivers L. In order to do so, consider the situation in Fig. 6, where the potentially active receivers those that can receive according to the protocol odel are at the centers of utually exclusive disks of radius R. Now, any of these receivers u can effectively receive only if F u g occurs. Fro 0, we have PF u g p lb g. Now, consider a disk of radius + R around each active receiver shown as filled dots in Fig. 6, and let UR,, L denote the union of all such disks. It is clear that the nuber of active receivers L is less than or equal to the nuber of sall disks of radius R with non-epty intersection with UR,, L. Since, as argued before, there are at ost 6 π R such disks, we can write L 6 π R i= {disk i UR,, L}. 3