Offloading Cellular Traffic through Opportunistic Communications: Analysis and Optimization

Transcription

1 1 Offloaing Cellular Traffic through Opportunistic Communications: Analysis an Optimization Vincenzo Sciancalepore, Domenico Giustiniano, Albert Banchs, Anreea Picu arxiv: v1 [cs.ni] 14 May 24 Abstract Offloaing traffic through opportunistic communications has been recently propose as a way to relieve the current overloa of cellular networks. Opportunistic communication can occur when mobile evice users are (temporarily) in each other s proximity, such that the evices can establish a local peer-to-peer connection (e.g., via Bluetooth). Since opportunistic communication is base on the spontaneous mobility of the participants, it is inherently unreliable. This poses a serious challenge to the esign of any cellular offloaing solutions, that must meet the applications requirements. In this paper, we aress this challenge from an optimization analysis perspective, in contrast to the existing heuristic solutions. We first moel the issemination of content (injecte through the cellular interface) in an opportunistic network with heterogeneous noe mobility. Then, base on this moel, we erive the optimal content injection strategy, which minimizes the loa of the cellular network while meeting the applications constraints. Finally, we propose an aaptive algorithm base on control theory that implements this optimal strategy without requiring any ata on the mobility patterns or the mobile noes contact rates. The propose approach is extensively evaluate with both a heterogeneous mobility moel as well as real-worl contact traces, showing that it substantially outperforms previous approaches propose in the literature. I. INTRODUCTION Following the huge popularization of smartphones an the ensuing explosion of mobile ata traffic [1], cellular networks are currently overloae an this is foreseen to worsen in the near future [2]. A recent promising approach to alleviate this problem is to offloa cellular traffic through opportunistic communications [3]. The key iea is to inject mobile application content to a small subset of the intereste users through the cellular network an let these users opportunistically sprea the content to others intereste upon meeting them. By exploiting opportunistic communications in this way, such an approach has the potential to substantially relieve the loa of the cellular infrastructure. Among other mobile applications, this can be use for news [4], roa traffic upates [5], social ata [6] or streaming content [7]. Inee, as shown by our performance evaluation results, the loa of the cellular network can be reuce between 50% an 95%, epening on the application. Opportunistic networking exploits the aily mobility of users, which enables intermittent contacts whenever two mobile evices are in each other s proximity. These contacts are use to transport ata through the opportunistic network, which may introuce substantial elays. However, the type of content concerne by cellular offloaing may not always be entirely elay-tolerant. In many applications, it is inee critical that the content reach all users before a given ealine, lest it lose its relevance or its usability. Therefore, the esign of opportunistic-base cellular offloaing techniques faces serious challenges from the intermittent availability of transmission opportunities an the high ynamics of the mobile contacts. In orer to fin the best trae-off between the loa of the cellular network an the elay until the content reaches the intereste users, any opportunistic-base offloaing esign must answer crucial questions such as, how many copies of the content to inject, to which users an when. While a number of techniques have been propose in the literature to offloa cellular traffic through opportunistic communications, previous approaches are either base on heuristics (an hence o not ensure that the loa of the cellular network is minimize) [5] [7] or fail to provie elay guarantees [4], [7]. In contrast to the above approaches, in this paper, we propose the HYPE (HYbri opportunistic an cellular) technique, which minimizes the loa of the cellular network while meeting the constraint in terms of elay guarantees. To our best knowlege, we are the first to provie such features. The key contributions of our work are as follows: 1) Builing on the founations of epiemic analysis [8], we propose a moel to unerstan the funamental trae-offs an evaluate the performance of a hybri opportunistic an cellular communication approach. Our moel reveals that content tens to isseminate faster through opportunistic contacts when a sufficient, but not excessive, number of noes have alreay receive the content; in contrast, issemination is slower when either few users have the content or few users are missing it. 2) Base on our moel, we erive the optimal strategy for injecting content through the cellular network. In line with our previous finings, this strategy uses the cellular network when low spee of opportunistic propagation is statistically expecte, an lets the opportunistic network sprea the content the rest of the time. 3) We esign an aaptive algorithm, base on control theory, that implements the optimal strategy for injecting content through the cellular network. The key strengths of this algorithm over previous approaches are that it aapts to the current network conitions without monitoring the noes mobility an that it incurs very low signaling overhea an complexity. Both features are essential features for a practical implementation. The rest of the paper is structure as follows. After thoroughly reviewing relate work in Section II, we outline the basic esign guielines of our approach an theoretically analyze its performance in Section III. Base on this analysis, in Section IV, we then erive the optimal strategy an present our aaptive algorithm, which implements this optimal strategy. The algorithm s performance is extensively evaluate in Section V, using mobility moels as well as experimental contact traces. Finally, Section VI closes the paper with some

2 2 final remarks. II. RELATED WORK The problem of the unsustainable increase in cellular network traffic an how to offloa some of it has become more an more popular. Two types of solutions can be istinguishe, on the basis of the outlet chosen for part of the cellular traffic: (i) offloaing through aitional (new or existing) infrastructure, an (ii) offloaing through a hoc communication. Our proposal, HYPE, falls into the secon category. In the first category, many solutions [9], [10] are aiming to exploit the relatively large number of existing WLAN access points, as well as cellular iversity. A ifferent approach, base on new infrastructure, is introuce in [11], in the context of vehicular networks. In that paper, the authors avocate the eployment of fixe roasie infrastructure units an stuy the performance of the system in offloaing traffic information from the cellular network. In the secon category, along with our stuy, an increasing boy of work is investigating the use of infrastructure-free opportunistic networking as a complement for the cellular infrastructure. In particular, the stuies in [4] [7], [12] propose solutions base on this iea. In [4], the authors propose to push upates of ynamic content from the infrastructure to subscribers, which then isseminate the content epiemically. The istribution of content upates over a mobile social network is shown to be scalable, an ifferent rate allocation schemes are investigate to maximize the ata issemination spee. A substantial ifference between this work an HYPE is that [4] oes not minimize the loa incurre in the cellular network an oes not provie any elay guarantees, which are central objectives in our approach. Moreover, the solution introuce in [4] results in higher resource consumption for the most central users (i.e., highest contact rates) an/or the most social users. Han et al. investigate, in [6], which initial subset of users (who receive the content through the cellular) will lea to the greatest infection ratio. A heuristic algorithm is propose, that uses the history of user mobility of the previous ay to ientify a target set of users for the cellular eliveries. HYPE iffers significantly from this, in the following aspects: (i) the solution in [6] is heuristic an thus oes not guarantee optimal performance, (ii) it requires to know the mobility patterns of all users, which may not be realistic in most scenarios, an (iii) it only investigates which users to choose, but not how many of them. In [7], an architecture is implemente to stream vieo content to a group of smartphones users within proximity of each other, using both the cellular infrastructure an WLAN a-hoc communication. The ecision of who will ownloa the content from the cellular network is base on the phones ownloa rates. In contrast to our work, the focus of [7] is on the implementation rather than the moel an the algorithm. Inee, the algorithm propose is a simple heuristic, which oes not guarantee optimal performance. Another stuy where opportunistic networking is use to offloa the mobile infrastructure is [12]. Here, some chosen users, name helpers, participate in the offloaing, an incentives for these users are provie by using a micropayment scheme. Alternatively, the operator can offer the participants a reuce cost for the service or better quality of service. Thus, the focus of [12] is on incentives, which is out of the scope of our work. Most similar to HYPE is the Push-an-Track solution, presente in [5]. There, a subset of users initially receive content from a content provier an subsequently propagate it epiemically. Upon reception of the content, every noe sens an acknowlegment to the provier, which may ecie to reinject extra copies to other users. Upon reaching the content ealine, the system enters into a panic zone an pushes the content to all noes that have not yet receive it. The most prominent ifference between this approach an ours is that Push-an-Track relies on a heuristic to choose when to fee more content copies into the opportunistic network, which oes not guarantee that the loa on the cellular network is minimize. In contrast, we buil on analytical results to guarantee that performance is optimal. An aitional rawback of Push-an-Track is that it incurs a very high signaling overhea, which compromises the scalability with the number of subscribe users. Results in Section V confirm that our theory-riven algorithm outperforms the heuristics propose in [5] both in terms of cellular loa an of signaling overhea. Finally, from a ifferent perspective, HYPE is also relate to content issemination solutions in purely opportunistic networks [13], [14]. However, most of these stuies focus on fining the best ways to collaborate or contribute to the issemination, uner various constraints (e.g., limite public buffer space). Evaluation is usually base on the elay incurre to obtain esire content or the equivalent metric of average content freshness over time. In contrast, our metric is the loa incurre in the cellular network. However, when eveloping our initial moel, we o use a similar moeling metho as in purely opportunistic issemination (e.g., [13], [15]). Like all the previous works on offloaing cellular networks through opportunistic communications [4] [6], with our approach all the transmissions over the cellular network are unicast. There are several key reasons that limit the usage of multicast messages in a cellular network. First, multicast cannot be easily combine with opportunistic transmissions, as this woul require that the Content Server is aware of the cell of each noe an can ynamically select the subset of noes at each cell that receives the multicast message, which is not possible with current cellular multicast approaches. Secon, in urban scenarios users will likely be associate to ifferent base stations (there are hunres/thousans of them in the city, each covering some sector, an in ense urban areas femtocells have starte to be eploye). Thus, there is a low probability that users subscribe to a specific content are associate to the same cells at the same time, an hence multicast may collapse to unicast. Finally, transmissions with multicast woul occur at the lowest rate to preserve users in the ege of the cell, which egraes the resulting performance. III. THE HYPE APPROACH In this section, we present the basic esign guielines of the HYPE (HYbri opportunistic an cellular) approach. HYPE is a hybri cellular an opportunistic communications approach

3 3 that elivers content to a set of users by (i) sening the content through the cellular network to an initial subset of the users (which we will call see noes), an (ii) letting these initial users or see noes share the content opportunistically with the other noes. We aim at esigning HYPE so as to combine the cellular an opportunistic communication paraigms in a way that retains the key strengths of each paraigm, while overcoming their rawbacks. HYPE consists of two main builing blocks: (i) the Content Server, an (ii) the Mobile Applications. The Content Server runs insie the network infrastructure, while the Mobile Applications run in mobile evices that are equippe with cellular connectivity, as well as able to irectly communicate with each other via short range connections (e.g., via WLAN or Bluetooth). The Content Server monitors the Mobile Applications an, base on the feeback receive from them, elivers the content through the cellular network to a selecte subset of Mobile Applications (the see noes). When two mobile evices are within transmission range of each other, the corresponing Mobile Applications opportunistically exchange the content by using local (short-range) communications. A. Objectives The funamental challenge of the HYPE approach is the esign of the algorithm that ecies which mobile evices an when they shoul receive the content through the cellular network. The rest of this paper is evote to the esign of such an algorithm. The key objectives in the esign are: (i) Maximum Traffic Offloa: Our funamental objective is to maximize the traffic offloae an thus reuce the loa of the cellular network as much as possible. This is beneficial both for the operators (who may otherwise nee to upgrae their network, if the cellular infrastructure is not capable of coping with current eman), as well as for the users (who must pay for cellular usage, either irectly or by seeing their ata rate reuce). (ii) Guarantee elay: Most types of content have an expiration time, arising either from the content s usefulness to the user (e.g., roa traffic information), its valiity after an upate (e.g., aily news) or its play-out time (e.g., streaming). Therefore, a key requirement for our approach is that the content reaches all the intereste users before its ealine. (iii) Fairness among users: In orer to make sure that all users benefit from HYPE, it is important to guarantee a goo level of fairness both in terms of cellular usage (for which users have to pay), as well as in terms of opportunistic communications (which may increase the energy consumption of the evice). 1 (iv) Reuce signaling overhea: The signaling overhea between the Content Server an the Mobile Applications nees to be low. This is important for two reasons: first, to ensure that HYPE scales with the number of mobile evices (otherwise the signaling traffic woul overloa the cellular network); secon, to avoi using 1 Inee, an important rawback of certain existing solutions is that they ten to over-exploit the users with high contact rates [4], [6], thus iscouraging the participation of such users. the cellular interface for small control packets (which is highly energy inefficient ue to the significant tail consumption after a cellular transmission [16]). The above objectives involve some trae-offs, making it very challenging to satisfy all of them simultaneously. For instance, to maximize the traffic offloa, we may consier a greey approach, where the Content Server sens the content to users with the highest contact rates; however this woul (i) eteriorate the fairness among users, an (ii) increase the signaling overhea to gather ata on user mobility patterns. Another approach may instea minimize the signaling overhea by injecting content as long as there is enough banwith available, avoiing thus any signaling; however, this will not maximize the traffic offloa. In the following, we set the basic esign guielines of an approach that satisfies all these objectives. B. Basic esign guielines In orer to satisfy the above objectives, a key ecision of HYPE is how to eliver a certain piece of content (hereafter referre to as ata chunk) through the cellular network. In particular, this ecision involves the selection of the noes to which the ata chunk is elivere via cellular, as well as the times when to perform these eliveries. In HYPE, a ata chunk is initially elivere to one or more users through the cellular network; aitional copies may be injecte later if neee. The ecision of when to inject another copy of the chunk is riven by the number of users that have alreay receive it. As long as the ealine has not expire, any user with a copy of the chunk will opportunistically transmit it to all the users it meets, that o not have the chunk. Finally, upon reaching the ealine of the content, the remaining users that have not yet receive the chunk, ownloa it from the cellular network; 2 this ensures that the elay guarantees are met an thus we satisfy objective (ii) from Section III-A. In orer to provie a goo level of fairness among users, which is objective (iii), HYPE selects each of the see noes uniformly at ranom. Over the long term, this ensures that, on the one han, all users have the same loa in terms of cellular usage an, on the other han, they also share fairly well the loa incurre in opportunistic communications. This is confirme by the simulation results presente in Section V, which show that HYPE provies a goo level of fairness while paying a small price in terms of performance. 3 The approach sketche above meets objectives (ii) an (iii). In the following, we first present a moel for the opportunistic issemination of content injecte by a cellular network. Base on this moel, in Section IV we erive the optimal strategy for the elivery of a single ata chunk, that minimizes the loa of the cellular network fulfilling objective (i), an then we esign an algorithm to implement this strategy, that incurs very low signaling overhea thus also satisfying objective (iv). 2 An ae avantage of this architecture is that the mobile noes only nee to keep the ata chunks for forwaring until their ealine an no longer. The buren on the mobile noes buffers is thus kept very low. 3 This is also supporte by the results of [5], which show that the ifference in terms of performance between the ranom selection an other strategies is very small.

4 4 C. Moel In orer to erive the optimal strategy, with the above approach, for the elivery of ata chunks through the cellular network, we nee to etermine: The total number of copies of the ata chunk to be elivere by the cellular network. This is not trivial: for example, an overly conservative approach, that elivers too few copies before the ealine, may have the sieeffect of overloaing the cellular network with a large number of copies when the ealine expires. The optimal instants for their elivery. The ecision of when to eliver a copy of a ata chunk through the cellular network is base on the current status of the network, which is given by the number of users that alreay have the chunk. In the following, we moel the opportunistic issemination of content injecte by a cellular network an analyze the loa of the cellular network as a function of the strategy followe. Then, base on this analysis, in Section IV we obtain the optimal strategy, that minimizes the loa of the cellular network for a given content ealine. Let N be a set of mobile noes subscribe to the same content, with N = N the size of this set (total number of noes). All noes have access to the cellular network. Any two noes also have the ability to setup pairwise bi-irectional wireless links, when they are in each other s communication range (in contact). Thus, opportunistic communication happens via the store-carry-forwar metho, through the sequences of intermittent contacts establishe by noe mobility. At time 0, a ata chunk is injecte in the (opportunistic) network, i.e., copies of the chunk are pushe via the cellular interface to a small subset of N, the see noes. Throughout the moel escription, we follow the epiemic issemination of this chunk of content. We enote by M(t) the number of mobile noes holing the chunk at time t (we refer to such noes as infecte ). The elivery ealine assigne to a ata chunk is given by T c (its value epens on the mobile application s requirements). 1) Opportunistic communication: In the opportunistic phase of HYPE, ata are exchange only upon contacts in the network N, therefore a mobility moel base on contact patterns is sufficient for our analysis. We assume every pair of noes (x, y) in the network N meets inepenently of other pairs, at exponentially istribute time intervals 4 with rate β xy 0. Then, the opportunistic network N can be represente as a weighte contact graph using the N N matrix B = {β xy }. We further assume that the inter-contact rates β xy are samples of a generic probability istribution F (β) : (0, ) [0, 1] with known expectation µ β (various istribution types for F (β) an their effects on aggregate inter-contact times are investigate in [20]). Aitionally, we assume that the uration of a contact is negligible 4 Though all pairwise inter-contact rates may not always be exactly exponential (preliminary stuies of traces [17] suggeste that this is true for subsets of noe pairs only), the most in-epth an recent stuies [18], [19] conclue that inter-contact time intervals o feature an exponential tail. This is supporte by the recent results of Passarella et al. [20], which show that the non-exponential aggregate inter-contacts iscovere in the preliminary trace stuies [17] can, in fact, be the result of exponentially istribute pairwise inter-contacts with ifferent rates. Fig. 1. Markov chain for HYPE communication, assuming homogeneous noe mobility. Transitions can be cause either by (i) a contact between two noes, or (ii) injection of the chunk to one noe through the cellular network (instantaneous transition, represente with rate in the figure). Fig. 2. Markov chain for epiemic spreaing, assuming heterogeneous noe mobility. HYPE specific transitions (i.e., chunk injection by cellular) are left out for clarity. This Markov chain is very complex an intractable for large scenarios; in Theorem 1 we can then reuce it to an equivalent Markov chain that is much simpler an for which we can erive a close-form solution. in comparison to the time between two consecutive contacts, an that the transmission of a single chunk is instantaneous in both the cellular an the opportunistic network. The assumptions of exponential inter-contact an negligible contact uration are the norm in analytical work ealing with opportunistic networks [21] [23]. Stuies base on looser assumptions (generic inter-contact moels, non-zero contact uration) have, so far, only resulte in broa, qualitative conclusions (e.g., infinite vs. finite elay), while we aim at obtaining more concrete, quantitative results. In aition, all our simulations feature non-zero contact uration an some of them also have non-exponential inter-contact times, thus testing the applicability of our results outsie the omain of these assumptions. Epiemic issemination in opportunistic networks is typically escribe with a pure-birth Markov chain, similar to the one in Fig. 1 (slightly aapte from, e.g., [21]). This type of chain only moels the number of copies of a chunk in the network N at any point in time, regarless of the specific noes carrying those copies. This is only possible when consiering noe mobility to be entirely homogeneous (i.e., all noe pairs meet at a unique rate: β xy = λ for all x, y N ), which allows all noes to be treate as equivalent. However, as state in the beginning of this subsection, we consier noe mobility to be heterogeneous, with noe pairs meeting at ifferent rates β xy with x, y N. In this case, not only the number of sprea copies must be moele, but also the specific noes carrying those copies. This results in more complex Markov chains, as illustrate in Fig. 2 for a 4-noe network N = {a, b, c, }. Transition rates in Markov chains like the one shown in Fig. 2 epen on the noes infecte in each of the eparture an the arriving states. For example, in Fig. 2, the transition between state a an state ab can happen if noe a meets noe

5 5 b. Therefore, the transition time between these two states is exponential with rate given by the meeting rate of the (a, b) noe pair, β ab. Similarly, the transition between state ab an state abc can happen if noe a meets noe c, or if noe b meets noe c (whichever meeting happens first). Thus, the transition time for this transition is the minimum of two exponential variables with rates β ac an β bc. Since inter-contact times are exponential, this minimum is also exponential with rate β ac + β bc, as shown in Fig. 2. 2) Cellular communication: The ecision to eliver a copy of the chunk through the cellular network is base on the current issemination level, i.e. the number of noes that alreay have the chunk. We say that the HYPE process or its associate Markov chain (similar to Fig. 2) is at level i, when i mobile noes are infecte, i.e., M(t) = i. Each level i correspons to a set of ( ) N i states {K i 1, K i 2,..., K i ( N i ) } in the Markov chain. For instance, in our 4-noe network from Fig. 2, the HYPE process is at level 3, when the chain is in any of the states K 3 1 = abc, K 3 2 = ab, K 3 3 = ac or K 3 4 = bc. The strategy to transmit copies of the chunk over the cellular network is given by the levels at which we inject a copy. We enote these levels by C = {c 1, c 2,..., c }: as soon as we reach one of these levels c i C before the ealine T c, a copy of the chunk is sent to a ranomly chosen noe. With this, the transitions in the HYPE Markov chain can be cause either by: (i) a contact between two noes (one infecte, the other uninfecte), which occurs at rates inicate in the previous subsection, or (ii) the injection of the chunk to one noe through the cellular network. The latter correspons to an instantaneous transition (since the chain instantly jumps to a state of the next issemination level), an is represente in Fig. 1 with rate 5. Finally, upon reaching the ealine T c, the chunk is sent through the cellular network to those noes that o not have the content by that time. D. Analysis Base on the above moel, in the following, we analyze the loa of the cellular network (which is the metric that we want to minimize) as a function of the strategy followe to inject content (which is given by C = {c 1, c 2,..., c }). The cellular network loa correspons to the number of copies elivere through the cellular network, which we enote by D. Let p i (t) = P[M(t) = i] enote the probability of being at level i at time t. Then, D is given by: N D = ( i + i )p i (T c ) (1) i=1 where i is the number of eliveries through the cellular network that take place until level i is reache ( i = {1, 2,..., i} C ) an i is the number of copies elivere upon reaching the ealine T c, if it expires at level i ( i = N i). In orer to compute p i (T c ), we first analyze the case C = {c 1 } 6, i.e., when we only inject one copy of the ata chunk 5 Note that, for clarity, the Markov chain of Fig. 2 oes not moel transitions cause by chunk injection through the cellular network. This type of transition woul be the same as in Fig. 1 (i.e., rate). 6 Note that c 1 must necessarily be equal to 0. at the beginning an o not inject any other until we reach the ealine. Let p c1 i (T c) enote the probability that, in this case, the system is at level i at time T c. In orer to compute p ci i (T c), we moel the transient solution of our Markov chain as shown in the following theorem. (The formal proofs of the theorems are provie in the Appenix.) Theorem 1: Accoring to the HYPE Markov chain for heterogeneous mobility (similar to Fig. 2), the process {M(t), t 0} is escribe by the following system of ifferential equations: t pc1 1 (t) = λ 1p c1 1 (t), i = 1 t pc1 i (t) = λ ip c1 i (t) + λ i 1p c1 i 1 (t), 1 < i < N (2) t pc1 N (t) = λ p c1 (t), i = N where λ i = i(n i)µ β. (Recall that µ β is the known expectation of the generic probability istribution F (β) : (0, ) [0, 1], from which the inter-contact rates escribing our network are rawn: {β xy } = B.) Theorem 1 has effectively reuce our complicate Markov chain for heterogeneous mobility back to a simpler Markov chain, like the one in Fig. 1 (the λ factor being replace by µ β ). In the simpler chain, each state represents a level of chunk issemination (i.e., number of noes holing a copy of the chunk). This is possible, as shown in the proof, thanks to the fact that our heterogeneous contact rates β xy are all rawn from the same istribution, F (β) : (0, ) [0, 1], which means that all the states of a certain issemination level i: {K i 1, K i 2,..., K i } are, in fact, statistically equivalent. ( N i ) Applying the Laplace transform to the above ifferential equations, an taking into account that p c1 i (0) = δ i1, leas to sp c1 1 (s) = λ 1P c1 1 (s) + 1, i = 1 sp c1 i (s) = λ i P c1 i (s) + λ i 1 P c1 i 1 (s), 1 < i < N (3) sp c1 N (s) = λ P c1 (s), from which P c1 i (s) = 1 i 1 λ j, s + λ i s + λ j P c1 N (s) = 1 s j=1 j=1 λ j s + λ j, i = N i < N i = N In case we eliver the ata chunk through the cellular network at the levels C = {c 1, c 2,..., c }, then the transitions corresponing to those levels are instantaneous, an the Laplace transforms of the probabilities P i (s) are compute as: 1 λ j, i < N, i / C s + λ i s + λ j j S i 1 P i (s) = 0, i < N, i C (5) 1 λ j, i = N s s + λ j j S where S i 1 is the set of levels up to level i 1, without incluing those that belong to set C, i.e., S i 1 = {1, 2,..., i 1} \ ({1, 2,..., i 1} C). For the levels i C, we simply have Pi C (s) = 0, since we will never be at these levels. (4)

6 6 From Eq. (5), we can obtain a close-form expression for the probabilities p i (t) as follows. The polynomial P i (s) is characterize by first an secon orer poles which have all negative real values. Let {s = λ n } be the poles of P i (s). Then, p i (t) for i < N, i / C is compute as: ( ) p i (t) = j S i 1 λ j {s= λ n} Res e st j S i (λ j + s) where Res inicates the resiue, which is given by: e λnt s= λ n e st Res = (λ j + s) j S i (λ j λ n ), λ n is a 1 st orer pole j S i j n e λnt t j S i λ r λ n (λ j λ n ) j S i λ j λ n 1 (λ r λ n ) (6), λ n is a 2 n orer pole Aitionally, for i < N, i C we have p i (t) = 0, an for i = N, p N (t) = 1 i=1 p k(t). By evaluating p i (t) at time t = T c an applying Eq. (1), we can compute the average number of eliveries over the cellular network, D. IV. OPTIMAL STRATEGY AND ADAPTIVE ALGORITHM In this section, we first leverage on the above moel to etermine the optimal strategy for the elivery of ata chunk, an then we esign an aaptive algorithm to implement this strategy. A. Optimal strategy analysis Our goal is to fin the best strategy C = {c 1, c 2,..., c } for injecting chunk copies over the cellular network, that minimizes the total loa D of the cellular network while meeting the content s ealine T c. To solve this optimization problem, we procee along the following two steps: 1) We show that the optimal strategy is to eliver the content through the cellular network only at the beginning an at the en of the ata chunk s perio, an never in-between. The ata chunk s perio is efine as the interval between t = 0 (when we first start istributing the content) an t = T c (when the content s ealine expires). 2) We obtain the optimal number of copies of the chunk to be elivere at the beginning of the perio such that the average loa of the cellular network, D, is minimize. The following theorem aresses the first step. Theorem 2: In the optimal strategy, the ata chunk is elivere through the cellular network to see noes at time t = 0, an to the noes that o not have the content by the ealine at time t = T c. Accoring to Theorem 2, the optimal strategy is to: (i) eliver a number of copies through the cellular network at the beginning of the perio, (ii) wait until the ealine without elivering any aitional copy, an (iii) eliver a copy of the chunk to the mobile noes missing the content at the en of the perio. The intuition behin this result is as follows. When few users have the content, information spreas slowly, since it is unlikely that a meeting between two noes involves one of the few that have alreay the content. Similarly, information spreas slowly when many users have the content, as a meeting involving a noe that oes not yet have the content is improbable. The strategy given by Theorem 2 avois the above situations by elivering a number of chunk copies through cellular communication at the beginning (when few users have the content) an at the en (where few users miss the content). As a result, the strategy lets the content isseminate through opportunistic communication when the expecte spee of issemination is higher, which allows to minimize the average loa of the cellular network. The secon challenge in eriving the optimal strategy is to compute the optimal number of copies of the chunk to be elivere at the beginning of the perio, which we enote by. To that en, the following proposition efines the notion of gain an computes it: Proposition 1: Let us efine G as the gain resulting from sening the ( + 1) th chunk of chunk copy at the beginning of the perio (i.e., G = D D +1, where D +1 an D are the values of D when we eliver + 1 an copies at the beginning, respectively). Then, G can be compute from the following equation: G = j= λ j λ p j (T c ) 1, (7) Builing on the above notion of G, the following theorem provies the optimal point of operation: Theorem 3: The optimal value of is the one that satisfies G = 0. The rationale behin the above theorem is as follows. When G > 0, by sening one aitional copy at the beginning, we save more than one copy at the en of the perio an hence obtain a gain. Conversely, when G < 0, we o not benefit from increasing. The proof shows that G is a strictly ecreasing function of, which implies that, to fin the optimal point of operation, we nee to increase as long as G > 0 an stop when we reach G = 0 (after this point, G < 0 an further increasing yiels a loss). B. Aaptive algorithm for optimal elivery While the previous section aresse the elivery of a single ata chunk, in this section we focus on the elivery of the entire content, e.g., a flow of roa traffic upates, news fees or a streaming sequence. We consier that the istribution of content in mobile applications is typically performe by inepenently elivering ifferent pieces of content in a sequence of ata chunks. For instance, a streaming content of 800 MB may be ivie into a sequence of chunks of 1 MB. When elivering chunks in sequence, we nee to aapt to the system ynamics. For instance, inter-contact time statistics

7 7 may vary epening on the time of the ay [24], which means that the optimal value obtaine by Theorem 3 nees to be aapte accoringly. Similarly, the number of mobile noes N subscribe to the content may change with time, e.g., base on the content popularity. To aress the above issues, we esign an aaptive algorithm base on control theory, that ajusts the number of chunk copies elivere at the beginning of each perio to the behavior observe in previous rouns (hereafter we refer to the sequence of perios as rouns). For instance, in the example above we woul have a total of 800 rouns. In the following, we first present the basic esign guielines of our aaptive algorithm. Builing on these guielines, we then esign our system base on control theory. Finally, we conuct an analysis of the system to guarantee its stability an ensure goo response times. C. Aaptive algorithm basics In orer to evise an aaptive algorithm that rives the system to optimality, we first nee to ientify which variable we shoul monitor an what value this variable shoul take in optimal operation. To o this, we buil on the results of the previous section to esign an algorithm that: (i) monitors how many aitional infecte noes we woul have at the en of a roun, if we injecte one extra copy at the beginning of that roun; an (ii) rives the system to optimality by increasing or ecreasing epening on whether this number is above or below its optimal value. To efficiently monitor the number of aitional infecte noes, we apply the following reasoning. Accoring to Theorem 3, in optimal operation, one extra elivery at the beginning of a roun leas to one aitional infecte noe at the en of that roun. If we focus on a single copy of the chunk elivere over the cellular network an consier it as the extra elivery, the noes that woul receive the content ue to this one extra elivery are those that receive this specific copy an coul not have receive the chunk from any other source. Since this hols for each of the copies elivere over the cellular network, in optimal operation there are on average a total of noes at the en of the roun, which receive the chunk from one source an coul not have receive it from any other source. Our algorithm focuses on this aggregate behavior of the eliveries rather than on a single copy, as this provies more accurate information about the epiemic issemination of the ata chunk. Base on this, each roun of the aaptive algorithm procees as follows (see Fig. 3 for an example): 1) Initially, copies of the ata chunk are transmitte to a ranom set of see noes over the cellular network. Each of the copies is marke with a ifferent ID that uniquely ientifies the source of the copy. 2) When a noe that oes not have the chunk receives it from another noe opportunistically, it recors the ID of the copy receive. 3) If two noes that have copies with ifferent IDs meet, they mark this event, to recor that they coul have a b c e c a b e f f a b c e f c a f b e a b c e a, b, c, e f f Ú Ö Ð ÓÔÝ Á Ò Ð ÓÔÝ Á ÒÓ ÓÔÝ Á Fig. 3. Example of chunk issemination in optimal operation. Noe a an b receive a copy of the chunk from the Content Server ( = 2). At the en of the roun, there are two noes with a single copy ID, that is, s = 2. potentially receive the chunk from ifferent sources. 7 We say that such noes have several copy IDs, while those that keep only one ID have a single copy ID. 4) If a noe who oes not have any copy or has a single copy ID meets with another noe who recore the several copy IDs event, the first noe also marks its copy with the several copy IDs mark. 5) At the en of the roun, the noes whose chunk comes from a single source (i.e., no several copy IDs mark) sen a signal to the Content Server. By running the above algorithm, we count the number of noes whose copy of the chunk comes from a single source (i.e., who have a single copy ID at the en of the roun), which we enote by s. As argue at the beginning of this section, in optimal operation this number is equal to the number of see noes. This implies that, at this operating point, the number of ata chunks injecte through the cellular network at the beginning of the roun is equal, in expectation, to the number of signals receive at the en of the roun, i.e., s =. A key feature of the above algorithm esign is that it oes not require gathering any complex statistics on the network, such as the behavior of the mobile noes, their mobility or social patterns, or their contact rates. Instea, we just nee to keep track of the number of chunks injecte at the beginning of each roun an the signals receive at the en, an this is sufficient to rive the system to optimal operation. As a result, the propose algorithm involves very reuce signaling overhea, which fulfills one of the objectives that we ha ientifie in Section III-A, namely objective (iv). D. System esign Base on the above esign guielines, our aaptive algorithm shoul (i) monitor the number of signals receive at the en of each roun, an (ii) rive the system to the point of operation where this value is equal to the number of copies injecte at the beginning of the roun. To o this, in this paper we rely on control theory, which provies the theoretical basis for monitoring a given variable (the output signal in control 7 Note that, for a noe with several copy IDs, we only mark the event an o not keep the IDs of the copies, since (i) we are only intereste in signaling the number of noes with a single copy ID, an (ii) this leas to more efficient operation, requiring fewer communications an less protocol overhea.

8 8 R + C(z) Controller (t) z 1 H(z) System O(t) + W (t) + Fig. 4. Our system is compose by two moules: the controlle system H(z), that moels the behavior of HYPE, an the PI controller C(z), that rives the controlle system to the optimal point of operation. theory terminology) an riving it to some esire value (the reference signal). Following a control theoretic esign, we propose the system epicte in Fig. 4. This system is compose from a controller C(z), which is the aaptive algorithm that controls the chunk elivery, an the controlle system H(z), which represents the HYPE network. Furthermore, the component z 1 provies the elay in the feeback-loop (to account for the fact that the value use in the current roun is compute from the behavior observe in the previous roun). For the controller, we have ecie to use a Proportional-Integral (PI), because of its simplicity an the fact that it guarantees zero error in the steay-state. The z transform of the PI controller is given by: C(z) = K p + K i (8) z 1 where K p an K i are the parameters of the controller. Here, the variable that we want to optimally ajust is the number of eliveries at the beginning of the roun (i.e., ). Following classical control theory [25], this variable is the control signal provie by the controller. In each roun, the controller monitors the system behavior (an in particular the output signal, which we will efine later), given the value that is currently use. Base on this behavior, it ecies whether to increase or ecrease in the next roun, in orer to rive the output signal to the reference signal. A key aspect of the system esign is the efinition of the output an reference signals. On the one han, we nee to enforce that by riving the output signal to the reference signal, we bring the system to the optimal point of operation. On the other han, we also nee to ensure that the reference signal is a constant value that oes not epen on variable parameters, such as the number of noes or the contact rates. Following the arguments expose in Section IV-C, we esign the output signal O(t) an the reference signal R of our controller as follows: { O(t) = s(t) (t) (9) R = 0 where (t) is the number of eliveries at the beginning of a given roun t, an s(t) is the number of signals receive at the en of this roun. Note that, with the above output an reference signals, by riving O(t) to R we bring the system to the point of operation given by s =, which, as iscusse previously, correspons to the optimal point of operation. Following classical control theory, we represent the ranomness of the system by aing some noise W (t) to the output signal, as shown by Fig. 4. E. Control theoretic analysis The behavior of the propose system (in terms of stability an response time) epens on the parameters of the controller C(z), namely K p an K i. In the following, we conuct a control theoretic analysis of the system an, base on this analysis, calculate the setting of these parameters. Note that this analysis guarantees that the algorithm quickly converges to the esire point of operation an remains stable at that point. In orer to analyze our system from a control theoretic stanpoint, we nee to characterize the HYPE network with a transfer function H(z) that takes as input an provies s as output. In orer to erive H(z), we procee as follows. Accoring to the efinition given in Proposition 1, G is the gain resulting from sening an extra copy of the chunk. In one roun, by sening one extra copy of the chunk at the beginning, there are on average s/ aitional noes that have the chunk at the en. Inee, s is the total number of noes that receive the chunk from only one of the initial see noes, which means that on average each see noe contributes with s/ to this number. This yiels to: from which: G = s/ 1, (10) s = G. (11) The above provies a nonlinear relationship between an s, since G (given by Eq. (7)) is a non-linear function of. To express this relationship as a transfer function H(z), we linearize it at the optimal point of operation. 8 Then, we stuy the linearize moel an ensure its stability through appropriate choice of parameters. Note that the stability of the linearize moel guarantees that our system is locally stable. 9 To obtain the linearize moel, we approximate the perturbations suffere by s at the optimal point of operation, (s ), as a linear function of the perturbations suffere by,, (s ) (s ), (12) which gives the following transfer function for the linearize system: H(z) = (s ). (13) Combining the above with Eq. (11), we obtain the following expression for H(z): H(z) = (s ) = G + G. (14) Evaluating H(z) at the optimal point of operation (G = 0) yiels: H(z) = G. (15) 8 This linearization provies a goo approximation of the behavior of the system when it suffers small perturbations aroun the stable point of operation [26]. Note that the approximation only affects the transient analysis an not the analysis of the stable point of operation at which the system is brought by the algorithm. 9 A similar approach was use in [27] to analyze the Ranom Early Detection (RED) scheme from a control theoretic stanpoint.

9 9 To calculate the above erivative, we approximate λ i (given by λ i = i(n i)µ β ) by its first orer Taylor polynomial evaluate at level i = ˆ, where ˆ is the average value of i at time T c (i.e., the average number of noes that have the chunk at the ealine). Since the Taylor polynomial provies an accurate approximation for small perturbations aroun ˆ, an the number of noes that have the chunk at time T c is istribute aroun this value, we argue that this approximation leas to accurate results. The first orer Taylor polynomial for λ i at i = ˆ is: λ i λ ˆ (i ˆ)(2 ˆ N)µ β. (16) Substituting this into Eq. (7) yiels G = 1 λ N i=1 ( p i (T c ) λ ˆ (i ˆ)(2 ˆ ) N)λ 1 = λ ˆ λ 1 = ˆ(N ˆ)µ β (N )µ β 1 (17) Since at the optimal point of operation we have G = 0, this implies that (at this operating point) = ˆ. Moreover, from Theorem 3 we have that, when operating at the optimal point, if we eliver one aitional copy at the beginning (i.e., increase by one unit), this leas to one aitional noe with the chunk at the en (i.e., ˆ also increases by one unit). Therefore, at the optimal operating point we also have ˆ/ = 1. Accounting for all of this when performing the partial erivative of G yiels: from which: G = 2(2 N) (N ), (18) H(z) = G 2) = 2(N N. (19) Having obtaine the transfer function of our HYPE network, we finally aress the configuration of the controller parameters K p an K i, that will ensure a goo trae-off between our system s stability an response time. To this en, we apply the Ziegler-Nichols rules [28], which have been esigne for this purpose. Accoring to these rules, we first obtain the K p value that leas to instability when K i = 0; this value is enote by K u. We also calculate the oscillation time T i uner these conitions. Once the K u an T i values have been erive, K p an K i are configure as follows: K p = 0.4K u, K i = K p 0.85T i. (20) Let us start by computing K u, i.e., the K p value that ensures stability when K i = 0. From control theory [25], we have that the system is stable as long as the absolute value of the closeloop gain is smaller than 1. The close-loop transfer function T (z) of the system epicte in Fig. 4 is given by: H(z)C(z) T (z) = 1 z 1 H(z)C(z). (21) To ensure that the close-loop gain of the above transfer function is smaller than 1, we nee to impose H(z)C(z) < 1. Doing this for K i = 0 yiels: H(z)C(z) = 2) 2(N N K p < 1. (22) The above inequality gives the following upper boun for K p, at which the system turns unstable: K p < N 2(N 2). (23) We want to ensure that the system is stable inepenently of N an, that is, the above inequality hols for any N an values. Since the smallest possible value that the righthan sie of Eq. (23) can take is 1/2 (when 0), the system is guarantee to be stable as long as K p < 1/2, an may turn unstable when K p excees this value. Accoringly, we set K u = 1/2. Furthermore, when the system becomes unstable, the control signal may change its sign up to every roun, yieling an oscillation perio of two rouns, which gives T i = 2. With these K u an T i values, we set K p an K i following Eq. (20), K p = 0.4 2, K 0.4 i = , (24) which terminates the configuration of the PI controller. While the Ziegler-Nichols rules aim at proviing a goo trae-off between stability an response time, they are heuristic in nature an thus o not guarantee the stability of the system. The following theorem proves that the system is stable with the propose configuration. Theorem 4: The HYPE control system is stable for K p = 0.2 an K i = 0.4/3.4. V. PERFORMANCE EVALUATION In this section, we evaluate HYPE for a wie range of scenarios, incluing several instances of a heterogeneous mobility moel, as well as real-worl mobility traces. We show that: The analytical moel provies very accurate results. The optimal strategy for ata chunk elivery effectively minimizes the loa incurre in the cellular network. The propose aaptive algorithm is stable an quickly converges to optimal operation. HYPE outperforms previously propose heuristics in terms of the cellular loa, signaling loa an fairness among users. From the four esign objectives introuce in Section III-A, our evaluation focuses on the traffic offloa, fairness an signaling overhea. Note that, since the elay guarantees are satisfie by esign, we meet the objective on the elay. a) Simulation setting: To evaluate the performance of HYPE, we use both real mobility traces an a heterogeneous mobility moel. For the evaluation with real mobility traces, we select the contact traces collecte in the Haggle project for 4 ays uring Infocom 2006 [24], an the GPS location traces of San Francisco taxicabs, 10 collecte through the Cabspotting project [29]. The number of users for the Infocom 2006 an San Francisco traces are 78 an 536, respectively. As for the heterogeneous mobility moel, we generate contacts as follows. For any given noe pair (x, y), the pairwise inter-contact times are exponentially istribute with rate β xy. The pairwise contact rates, β xy, are rawn from 10 We assume two taxicabs are in contact when they are within 100 meters of each other.