Understanding Practical Limitations of Network Coding ... - IEEE Xplore

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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 33, NO. 2, FEBRUARY 2015

Understanding Practical Limitations of Network Coding for Assisted Proximate Communication Alexander Pyattaev, Student Member, IEEE, Olga Galinina, Student Member, IEEE, Sergey Andreev, Member, IEEE, Marcos Katz, and Yevgeni Koucheryavy, Senior Member, IEEE

Abstract—In next-generation wireless networks, device-todevice (D2D) communication represents a feasible way for mobile users to offload their cellular traffic demand without extra costs for deploying additional infrastructure from the network operators. Cellular (e.g., 3GPP LTE) network assistance can automate user/service discovery and connection establishment procedures, as well as enable secure D2D connectivity between proximate users. Currently, assisted direct connectivity is only available in the form of unlicensed-band protocols (e.g., WiFi Direct), which motivates research on understanding its practical limitations with realistic distributions of users and content. Whereas there are concerns that D2D communication alone may not be efficient due to limited content availability, in this paper, we advocate the use of network coding to upgrade assisted proximate communication and make it realize its full potential. In particular, we demonstrate that even simpler network coding techniques are capable to significantly improve the degrees of content availability for communicating users and thus enhance offloading performance under realistic constraints. Inspired by the recent popularity of wireless content distribution systems over D2D caches, we contribute a practical methodology for assisted data caching and distribution, mindful of the state-of-the-art D2D technology. Index Terms—Device-to-device, network coding, cellular network, LTE, assisted offloading, unlicensed bands, WiFi-Direct, content distribution.

I. I NTRODUCTION AND R ELATED W ORK

A

S research on future fifth generation (5G) cellular technology is decisively getting momentum, the related wireless industry and academia are shifting their attention to what comes beyond the state-of-the-art broadband communication systems. Currently, most pressing technology requirements are clear: next-generation networks will need to provide orders of magnitude more capacity, ubiquitous user connectivity experience, and efficiently satisfy diverse characteristics imposed by various applications, from machine-to-machine to multimedia-

Manuscript received April 19, 2014; revised September 18, 2014; accepted November 7, 2014. Date of publication December 18, 2014; date of current version March 9, 2015. This work was supported by GETA, by TISE, by the Academy of Finland, and by Tekes through the Internet of Things Program of DIGILE. The work of S. Andreev was supported in part by the Academy of Finland under a Postdoctoral Researcher Grant and in part by the Nokia Foundation under a Jorma Ollila Grant. A. Pyattaev, O. Galinina, S. Andreev, and Y. Koucheryavy are with the Department of Electronics and Communications Engineering, Tampere University of Technology, Tampere 33720, Finland (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). M. Katz is with the Centre for Wireless Communications, University of Oulu, Oulu 90014, Finland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JSAC.2014.2384232

over-wireless. The latter actually threatens the networks of today with impeding capacity crunch, given that mobile data traffic already grew over 80 percent in 2013 and is expected to further increase nearly 11-fold between 2013 and 2018 [1]. A. Potential of Assisted Proximate Communication Reacting to the acceleration in data traffic demand, the mobile network operators begin to deploy increasingly dense networks of smaller cells across their service areas. Whereas network densification indeed relieves network congestion up to a certain extent, it at the same time requires additional capital/operational expenditures (CAPEX/OPEX) to install and manage these new base stations. A viable alternative for wireless users to offload traffic without the cost of extra infrastructure is by exploiting direct device-to-device (D2D) connections, whenever associated users are in proximity [2]. Proximate D2D communication generally promises its users higher data rates, lower transfer delays, and higher energy efficiency with applications ranging from multimedia content sharing, group multicast, and local voice services to gaming, context-aware services, and public safety [3]. The attractive benefits of coupling short-range communication with a certain degree of cellular control has attracted significant research attention in [4] as well as past literature [5]. Generally, cellular network may assist direct D2D connectivity in many ways, from user/service discovery and connection establishment, to radio resource management and service continuity. With moderate degrees of device cognition and user cooperation, promising benefits have already been indicated in energy consumption [6] and, in some cases, in data rate [7] of a cooperative cluster (i.e., group of devices in close proximity). Presently, direct connectivity can only be available in two distinct flavors [8]: licensed-band D2D technology (sometimes named LTE-Direct) and unlicensed-band protocols (e.g., WiFi and Bluetooth). See our recent work in [9] for a comprehensive up-to-date survey on the topic. In a nutshell, much research effort has been invested into the characterization of D2D connections as part of LTE cellular technology by 3GPP in licensed bands, where a license permits the network operator to use spectrum exclusively. Driven by a wealth of potential practical applications, the concept of D2D communication as an underlay to a cellular network has been developed by the seminal work in [10] and numerous subsequent papers. As in cognitive radio, D2D underlay is operating on the same resources as the cellular network and D2D users control their transmit power to suppress the resultant

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PYATTAEV et al.: UNDERSTANDING LIMITATIONS OF NC FOR ASSISTED PROXIMATE COMMUNICATION

interference to the cellular users [11]. Given its growing importance, the licensed-band D2D is becoming an attractive research area, where many fundamental questions still remain open including the information-theoretic capacity of the D2D underlay. However, the corresponding standardization efforts are developing slowly [12], such that the respective products employing the D2D underlay may not be the first to meet the market. Alternatively, unlicensed bands can be used freely, which gives opportunity to leverage D2D benefits almost immediately. Whereas there already exists a plethora of unlicensed spectrum protocols to technically enable direct connectivity, there is neither centralized control of radio resources to manage quality-of-service on D2D links nor is there any scalable device discovery solution [13]. Augmenting the current technology, we envision that devices be continually associated with the cellular network and use this connectivity to help manage their D2D connections in unlicensed bands. Therefore, in the near-term we expect that the majority of gains will come from advanced architectures and protocols that would leverage the unlicensed spectrum [2]. Despite the fact that connectivity in unlicensed spectrum is already available in modern multi-radio devices, it suffers from stringent session continuity limitations, cumbersome manual setup/security procedures, and excessive user contention (in case of WiFi). Furthermore, existing mechanisms for coupling cellular (e.g., 3GPP LTE) and short-range (e.g., WLAN) technologies have in the past been limited to integration at the application layer or within the core network (such as ANDSF: Access Network Discovery and Selection Function), which provides only loose coordination between the radio access technologies and does not allow to leverage the full potential of cellular assistance. However, with the latest 3GPP efforts on “3GPP/WLAN RAN Interworking,” efficient integration solutions are sought at the radio access network (RAN) layer enabling full control of involved radio technologies [14]. Augmented with the recent introduction of a more flexible WiFiDirect technology, RAN-level coupling between 3GPP LTE and WLAN promises non-incremental performance gains and thus becomes a feasible opportunity for the operators to relieve congestion on future 5G networks [15], [16]. We have recently completed an in-depth simulation-based study of LTE-assisted WiFi-Direct offloading, identifying scenarios where proximate communication would be most beneficial for both network and users [17], [18]. With cellular assistance, the proposed D2D technology has the potential to automate user/service discovery and connection establishment procedures, as well as enable secure D2D connectivity between proximate users that are currently outside each other’s social spheres [2]. This is expected to further broaden the use of assisted proximate communication, as well as enable novel ways of interaction between users, particularly those not known to each other previously (e.g., communication between untrusted users). However, as this technology is not mature enough (as ratified by 3GPP Release 12 in late 2014), additional research is required on understanding its practical limitations with realistic spatial distributions of both users and content, which is the key focus of this paper.

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B. Benefits of Coupling D2D With Network Coding Presently, due to user mobility translating into unpredictable opportunistic contacts, a concern remains that D2D communication may not be efficient enough to relieve congestion on practical wireless deployments. However, there is an exciting innovation that we believe has the potential to mitigate limited D2D content availability and thus enable beyond-additive gains of proximate connectivity. Ever since its first introduction in [19], network coding has revolutionized the entire communication paradigm from store-and-forward to compute-and-forward. Correspondingly, coding is not restricted anymore to communication endpoints, as has been the case for source or channel codes, but any network node may participate in the encoding, recoding, or decoding processes. The use of network coding may thus facilitate the distribution of information between the D2D users and make proximate communication more efficient and reliable. Over the last years, a vast literature on network coding has been accumulated, spanning across the areas of mathematics, physics, and biology [20]. In information science alone, the range of available publications is extremely broad and differs mainly in the type of code considered, from e.g., simple XOR techniques [21] to more elaborate Fountain codes [22] and Random Linear Network Coding (RLNC) [23]. More broadly, information modeling and virtualization aspects of NC have also been addressed in [24]. More specifically, the applicability of network codes for D2D-based networks has been recently discussed in [25], where the practical aspects of the respective deployments have been outlined. Multiple applications of network coding have been thoroughly evaluated, including those in multi-hop wireless networks [26], [27], wireless cooperative networks [28], multi-way relay networks [29], [30], and delay-tolerant networks [31], [32], at the physical layer [33] (also known as analog network coding) and upper layers [34], in real-time broadcasting systems [35] and storage area networks [36], as well as many other systems. Insightful conclusions on coding scalability have also been made [37] and effects of multi-packet reception have been studied [38], as well as challenges of joint networkchannel coding [39]. For more information, the interested reader is referred to comprehensive recent surveys and tutorials on both digital and analog network coding in [33], [40], and [41]. Some initial efforts have also been made in suggesting network coding (NC) for short-range direct communication [42]–[44]. In particular, these important papers touched upon WLAN-based cooperative file-sharing services, D2D-aware resource allocation, and coupling between D2D and physicallayer NC. Whereas offering crucial insights, these singular papers only address legacy D2D technologies without accounting for the full capabilities of network assistance. In this work, we bridge this gap by thoroughly investigating an LTE-assisted WiFi-Direct system augmented with NC (see Fig. 1). The use of network assistance allows to convert the D2D concept from a theoretical experiment widely studied in academia to a practical solution by addressing its key issues related to security and management of the proximate connections.

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Fig. 1. Envisioned network-coded D2D offloading system.

More specifically, Fig. 1 gives an example of envisioned system operation illustrating the base station, together with its associated mobile users, which demand, download, and store in their caches multiple coded data segments of various content (black/gray squares). Such segments may be downloaded either from the network or from the other devices in proximity by using the direct communication capability. We intend to demonstrate that even simpler NC techniques (e.g., RLNC type of NC [45]) have the potential to significantly improve on the degrees of content availability for communicating users and thus increase offloading performance under realistic constraints. To the best of our knowledge, this is the first study addressing practical operation of NC over assisted D2D communication having as its goal to reveal the available performance gains and resolve any impediments to them. As advocated in [46] and [47], the consideration of NC-aided D2D systems is increasingly important in practice given the current structure of mobile data traffic. Indeed, mobile video traffic exceeded 50 percent for the first time in 2012 [1], but increasing popularity of video-on-demand services often leads to asynchronous content consumption by users [48], [49]. This, in turn, precludes operators from relying on synchronous mechanisms, such as LTE broadcast, to relieve congestion. All of the above creates fertile soil for employing user mobile devices as intermediate caches of wireless content, especially given the fact that the lion’s share of data traffic comes from a few popular files and disk storage capacity grows faster than wireless network capacity [47]. However, crucial aspects of content distribution in D2D caching systems have only been addressed very recently [50], including the considerations of NC for file storage [51], but from a more theoretical, technology-agnostic perspective. In this paper, we complement these promising efforts with a practical methodology for assisted data caching and distribution, mindful of the state-of-the-art D2D technology. C. Focus and Contributions of This Paper In summary, there has been a considerable divide between theoretical efforts on various flavors of NC and its actual practical applications. Important concerns remained as to whether the decoding complexity of known NC techniques would hinder their practical on-line implementation in real hardware. Fortu-

nately, real-life measurements and system prototype implementations in [52]–[54] confirm that computational complexity is not an issue and NC-based solutions can be efficiently implemented in existing mobile devices. The work in [54] has also resulted in development of a software library named KODO [55], openly offering efficient NC solutions for integration into current user equipment. Inspired by these latest results, we seek to deepen existing research on NC for assisted D2D communication and develop understanding behind practical limitations of such systems. In particular, the contributions of this paper are clearly indicating the significant benefits of network-coded content distribution in the considered scenarios and include the following. 1) A novel wireless data dissemination model based on proximate content distribution that captures the dynamic arrival and departure processes characteristic of popular content in the area of interest. 2) A new wireless content delivery mechanism employing network-coded operation and capable of handling bulk arrivals of highly demanded traffic through assisted D2D communication. 3) An advanced analytical methodology suitable for assisted D2D-based data dissemination systems with and without network coding, mindful of the spatial distribution of users as well as the content arrival/departure processes. 4) A detailed characterization of practical content distribution performance over assisted proximate communication with advanced system-level simulator capturing realistic features of 3GPP LTE and WiFi-Direct radio technologies. The rest of this text is structured as follows. The proposed model for D2D-based wireless content distribution is summarized in Section II. It is followed by our analytical solution described in Section III based on a fluid approximation of the content distribution systems with and without network coding. Further, in Section IV we provide extensive system-level evaluation results for several representative content distribution scenarios and discuss them in detail. II. M ODELING P ROXIMATE C ONTENT D ISTRIBUTION In this section, we describe our proposed system model to characterize dynamic content distribution over assisted proximate communication. Below we begin with more general remarks on the system in question and then continue with a list of principal assumptions making our analysis feasible, as well as with our corresponding comments on their validity. A. General Statements We consider an integrated two-tier wireless network, where cellular infrastructure in licensed frequency bands (e.g., 3GPP LTE) is coupled with network-assisted D2D system in unlicensed bands (e.g., WiFi-Direct). In particular, wireless users may either acquire their desired content by downloading it from the cellular base station (BS), or establish connections with other proximate users directly in order to retrieve available content via the D2D links. In practice, the required coordination


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Fig. 3. Example system operation: during the lifetime of e.g., content C4 the users randomly generate requests Ri for the given content.

B. Assumptions of the System Model

Fig. 2.

Illustration of an integrated cellular + D2D network: area of interest.

to enable efficient operation of the assisted D2D scheme could be implemented by the provider of the respective content (e.g., by using the Proximity Services function in the cellular network [2]). Further, we concentrate on a particular area of interest, which, for the sake of an example and without the loss of generality, is represented as an isolated cell of the cellular network with the radius R centered at the BS. All the effects outside the selected area are disregarded (see Fig. 2 illustrating the communication process between the users acquiring their needed data). This scenario is characteristic of mass events (i.e., concerts, sports performances, festivals, etc.), where D2D communication is expected to realize its full potential due to high user density and relatively tight correlation of popular content. The impact of outside network in such a scenario may be abstracted away, as cellular systems are typically planned to minimize inter-cell interference coming from the neighboring cells, whereas for the D2D links the effects of farther users vanish rapidly with increasing distance. For the purposes of this paper, we focus primarily on the downlink cellular channel, while the D2D links naturally constitute direct two-way transmissions. We further emphasize that the frequency channels used by the cellular and D2D systems are non-overlapping, such that the respective user transmissions do not interfere with each other. However, the transmissions of proximate users on the D2D tier may interfere (as WiFi-Direct employs random-access based communication) and we discuss it in more detail further on. Finally, as far as the content of the exchanged data is concerned, we concentrate only on the scenarios, where copyprotection techniques are not required, as DRM mechanisms naturally contradict peer-to-peer. Such content is very likely to be abundant during sports events, and may be protected from “escaping” the event area by something as simple as the preshared key represented as a quick response (QR) code.

Focusing on the considered two-tier network comprising cellular and D2D connections, we further investigate the process of dynamic content generation, as well as its acquisition and distribution. To this end, in order to capture the dynamic behavior of the realistic integrated wireless network, we aim at building an analytical model for the process evolution of the content within the area of interest, as well as for the dynamics of the data acquisition requests by the users. In order to mathematically characterize the wireless content evolution, we introduce the following principal assumptions. Assumption 1. Content Life Cycle: New instances of demanded content arrive into the system according to the input process with the rate λc and spend there random finite time intervals (termed content lifetime), with the average duration of µ−1 c , independently of the arrival process. During its lifetime, the content is considered popular and leaves the system (becomes unpopular) immediately upon lifetime’s expiration. For the sake of exposition, all instances of the content are assumed to have the same size of Z. We emphasize here that lifetimes and inter-arrival times are assumed to be i.i.d. However, hereinafter we refer to the Poisson arrival process and exponentially-distributed inter-arrival times as an example of the considered processes, which can easily be generalized for more practical cases by means of using the corresponding expressions and/or approximations from the queuing theory. Fig. 3 illustrates the lifetimes of several instances of content, which arrive and leave the system according to Assumption 1, as well as shows new requests Ri for the content C4 . Further, in order to address the structure of the content, we consistently employ the term fragment representing a part of the object of content (e.g., a data file). To this end, we make the following assumption. Assumption 2. Content Structure: Each instance of content C is divided into S fragments of equal size, such that the original object can be reconstructed upon obtaining all of the fragments: C = (C1 , . . . ,CS ). Equivalently, a user may consider the content as acquired i.f.f. it has collected all of its fragments. Over the random lifetime τ of an arbitrary content C, any user may generate a request to acquire the corresponding object and thus become a consumer. A particular user becomes a consumer only once for the same content. Assumption 3. Content Request: The time intervals between the consecutive acquisition requests by the users are i.i.d. and

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follow the exponential distribution with the parameter λ. Whenever the time interval τ elapses, new requests for the object C are not generated anymore. A user may share a previously acquired fragment of content with its D2D neighbors within a certain proximity radius R0 . Upon obtaining all the required fragments, the user assembles the original object and considers the acquisition successful. Importantly, a user can download multiple fragments of the same content by connecting to several proximate neighbors (only one of those connections would be used for data transfer at any given time). Assumption 4. User Departures: Additionally, we assume that any user (still in need of or having acquired the content) may leave the system with the rate θ. In either case, the departing user removes all the fragments of a particular content, and thus no longer participates in the distribution. A natural course of action for a user interested in the content C is to collect all of its fragments from the available sources (either the BS or the D2D neighbors) during some random time with the average of θ−1 . Here, direct D2D retrieval is assumed to be preferred (e.g., due to higher data rate and/or energy efficiency [17], [18]), but might in turn require communication with several neighbors in proximity to complete the acquisition. Importantly, we neither model particular users in the area of interest, nor consider their locations explicitly in our analysis. Instead, we assume that each coordinate of a user with the newly arrived request is a random variable, which may in practice be parametrized by a corresponding user mobility pattern. Hence, to characterize the topological randomness of a random user deployment and the opportunistic nature of the D2D links, we formulate the following important assumption. Assumption 5. Spatial Distribution of Users: The users with the newly generated acquisition requests are distributed uniformly over the considered area of interest, i.e., within a circle of radius R. However, we assume that users may also retrieve data from their D2D neighbors outside the area of interest to avoid the consideration of the border effects. By contrast to past works on content distribution in e.g., wired networks [56], spatial randomness introduces an interesting consideration in wireless systems resulting in unequal data rates for communicating users depending on the distance between them. Assumption 6. Data Transfer Rate: The data rate ri, j of a wireless link i → j and the corresponding transmit power pi are coupled by the “generalized ” Shannon’s formula: γi, j ri, j = w log 1 + pi , (1) N0 + I where w is the effective channel bandwidth (of a D2D or a cellular link, alternatively), N0 is the noise level, I is the interference level at the receiver, and γi, j is the average path gain between the communication endpoints, which depends only on the distance di, j separating them: γi, j =

G di, j κ

,

(2)

where di, j is the distance between the communicating endpoints, κ is the propagation exponent, and G is the propagation

constant, which may be established by following, e.g., the recommendations in [57]. We emphasize the fact that the effective bandwidth w can be adjusted to take into account the corresponding MAC and PHY overheads of any particular radio technology used (i.e., 3GPP LTE and WiFi-Direct in our system). The parameters κ and G are defined by a particular radio access technology and antenna types. We note that the effects of fast fading are not taken into account directly, but may be implicitly incorporated into our model as part of the (raised) noise level. The effects of slow fading have minimal statistical impact on the model, as user positions are already random, and thus are included into the parameter G. In order to impose the limits on the achievable data rate due to the predefined practical set of modulation and coding schemes, we fix the maximum rate at rlim for d < d0 . This effectively means that further increase in the SNR/SINR levels does not yield the unbounded growth of the data rate function after a certain distance of d0 = d(rlim ), which may be easily obtained via (1) and (2). C. Considerations on Data Dissemination Strategy Within the proposed dynamic model of user content distribution, we intend to contrast several characteristic examples of content dissemination strategies. Inspired by the ideas in [45] and many other works, we envision the importance of the following two strategies, termed respectively baseline and coded. 1) Baseline (“BitTorrent,” or BT) strategy: the user simply stores all the acquired fragments of the content C1 , . . . ,CS in its memory, until all of them have been collected, so that the initial object can be assembled. The user Ui communicates with its proximate neighbors (within the (i) radius R0 ) retrieving the available fragments C j , j = 1, S. Additionally, each user also has an option to contact the BS and download the fragments missing at the proximate neighbors. 2) Coded (Network Coding, NC) strategy: instead of storing and distributing independent fragments {Ci }, the users disseminate linear combinations of those, i.e., ∑Si=1 αiCi , where αi , i = 1, . . . , S are independent and uniformly random coefficients chosen from a finite field. With this type of network coding, any S linearly independent coded symbols are sufficient to reconstruct the entire original object. The motivation behind comparing these particular dissemination policies results from their expected different sensitivity to the fragment availability—from more sensitive (with BT) to less sensitive (with NC). In particular, the use of RLNC (random linear network coding) is an attractive strategy to disseminate the fragments of content across spatially distributed users. In the proposed scheme, a random linear combination of the content fragments is stored in the memory (cache) of every user device. The RLNC has been proposed as an effective mechanism to improve performance in peer-to-peer and cooperative applications [53], [58]. The choice of RLNC in this paper is motivated


by the fact that this rateless scheme is flexible and exhibits low latency compared to other block oriented coding schemes. Moreover, RLNC is better suited to mesh (irregular) topologies, particularly when users and their associated communication channels are dynamic in nature, being nearly optimum without the need of a predetermined plan as in the case of deterministic coding approaches. Hence, random linear network codes lend themselves as a very desirable candidate for the hybrid cellular and proximate communication architecture considered here, where user cooperation via D2D links is assumed [59]. Compared to deterministic network coding, no knowledge on the individual received packets is needed for efficient system operation, though the implementation of RLNC schemes could be an issue. Since coding, recoding, and decoding is carried out at the source, destination, and intermediate nodes, computation complexity of the RLNC approach may be reasonably high. However, in light of the recent real-life measurements and system prototype implementations in [52]–[54] we are hopeful that NC-based solutions can be efficiently implemented in existing mobile devices. Basing on the assumptions above, we further develop an approximation of the system evolution, similar in the spirit to the fluid model in [56]. One important feature of our proposed model is that it enables capturing the dynamics of the content distribution process as well as the core trade-offs that arise as the content arrives and leaves. Additionally, we characterize important spatial interactions between multiple communicating D2D users competing for the shared channel resource. As final parameters of interest, we evaluate how many users have had success retrieving their desired content before the deadline, as well as the percentage of traffic that was served over D2D (termed offloading gain). III. M ATHEMATICAL C HARACTERIZATION OF E NVISIONED C ONTENT D ISTRIBUTION S YSTEM In this section, we develop a fluid approximation for the process of content distribution described in Section II. First, for tractability, we decouple the following processes: content life cycle and content dissemination. Further, we consider the evolution of requesting users in our system within the content lifetime. Importantly, we cannot refer to the steady-state system operation since the content lifetime cannot be assumed long enough to reach the stationary state. Therefore, we aim at investigating the process in question in its transient mode. In order to capture the key properties of the system in its transient phase, we replace the actual discrete process of the number of users acquiring the content with the equivalent continuous process. We analyze the phases (i) while the content is still being popular (t ∈ [0, T ]) and (ii) after it becomes unpopular and leaves the system (t ∈ [T, ∞)), separately. We refer to the first phase as the growth process, while the second phase is termed the reduction process. Hence, the stationary state (t → ∞) is reached when the system empties completely. We describe both phases by employing continuous processes. To this end, we construct several systems of differential equations under the corresponding boundary conditions. As the result, by solving the respective systems of differential equations,

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we obtain the closed-form approximation, i.e., compact expressions for the number of users retrieving the content, as well as for the number of users with the complete content for both baseline and network-coded schemes. Importantly, the overall system is relatively complex to be solved straightforwardly, even though it might be possible to do so. We thus intend to propose a simple and practical analysis by decomposing the entire compound process into two distinct parts, the content lifetime process and the content exchange process. We continue by detailing the former. A. Content Lifetime Process Given the Poisson process of content arrivals and its exponential departures, our system corresponds to the M/M/∞ type. Hence, the distribution of the number of content entities in the system is a well-known expression, which can be taken from the corresponding literature on the elementary queuing theory. Here, we may assume that this simple system has been operating long enough to be observed in its steady state. Hence, the average number of simultaneously active content entities is delivered by E[Nc ] = λc /µc . We note here that the considered queuing model may be extended to yield insights for a broader class of practical systems. Having defined the average number of simultaneously active content entities, we continue by looking into the detailed operation of a specific content. Specifically, we thoroughly investigate a significantly more complex dissemination process of a particular (tagged) object of content. Along these lines, we are conditioning on the fact that there are in total E[Nc ] such processes running simultaneously in our system, or, in other words, the integrated network is sharing its resources between E[Nc ] user sessions on average. B. Content Exchange Process We remind that during the tagged content lifetime the interested users are activated in arbitrary locations, so that the data rate between the user and the BS constitutes a random variable with the distribution: ⎧ 2 r − 2 r κm ⎪ pBS Gm κm 2e wm wm −1 ⎪ e , 0 ≤ r ≤ rlim , ⎪ ⎨ κm wm R2 N0 +I 2

fr (r)= κm ⎪ pBS Gm ⎪ 1 ⎪ , r = rlim , r ⎩ R2 (N +I) e wm −1 0

(3) which may be easily obtained from the distribution of the distances mindful of the upper limit on the achievable data rate. Here, pBS is the radiated power at the BS, wm is the available bandwidth, and Gm , κm are the constants related to the signal propagation. We note that a similar approach applies to the data rate distribution within the D2D proximity radius, but in practice R0 < d0 and we may for the sake of exposition assume a fixed data rate r0 (see below) for D2D connections (at e.g., its maximum value rlim ). This corresponds to the practical intuition on that direct links either work very well, or not at all. Otherwise, in the system-wide context, we could follow the approach similar to how it has been done in (3).

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Further, let us denote the number of users currently retrieving the content as x(t) and the number of users, which already have this content, as y(t). Then, for any tagged acquiring user out of x(t), the potential D2D partners are located in R0 -proximity of it. For the sake of tractability, we take into account the averaged number of neighbors and assume that the tagged user observes in its proximity the same (typical) density of proximate users as what we have in the entire network. Therefore, on average it R20 [x(t) + y(t)] R2

possible D2D partners, as well as might observe maintain its link to the cellular BS. Following [56], we introduce the parameter η indicating the effectiveness of content sharing, such that in case of identical users the data rate would be defined as a multiplier of the indiR2

We note that y(t) evolution in both cases starts at the point t0 = Z/rBS , when the first user from the set x completes its download. Until that moment, the evolution of the process x may be described by the equation dx(t) = λdt − θx(t)dt as: x(t) =

λ (1 − e−θt ), t ∈ [0,t0 ], θ

(6)

which would tend to the linear growth λt if θ → 0. Therefore, for each dissemination strategy we consider the problem at hand either within the interval [t0 , T ] or [T, ∞). The initial conditions detailed below are given as follows: x(0) = 0, y(t0 ) = 1, and x(t0 ) = x(t) (6).

vidual rate and the share of the complete content as η · R02 x(t) +

C. Baseline (“BitTorrent,” BT) Dissemination Strategy

in user proximity. However, in our model, the data rate in the proximity is determined by the distances between the transmitter and the receiver as µi, j = ri, j /(Z · E[Nc ]), where Z is the content object size, E[Nc ] is the number of simultaneously active content sharing processes, and ri, j is the instantaneous data rate as defined by (1). Therefore, the total data rate on D2D and cellular links, respectively, may be calculated as: R20 R20 r0 rD2D = η · 2 x(t) + 2 y(t) , Z R R ri,BS , (4) rcell = ZE[Nc ]

We remind that for the baseline (uncoded) strategy, the content dissemination is characterized by means of distributing S fragments. This results in relatively low efficiency η < 1, which translates to probability that connected neighbors have exactly the same fragments and cannot retrieve new information. A more detailed discussion on η for the popular BitTorrent networks may be found in [56], where its approximation is delivered as a function of the number of the fragments. Proposition 1: Consider the evolution of the number of users x(t) and y(t), which are interested in a given content C with lifetime T and possessing incomplete and complete data, respectively. 1) Evolution in the growth phase t ∈ [t0 , T ] is defined by a solution to the Cauchy problem (5) under initial conditions x(t0 ) = x0 , y(t0 ) = y0 : x = c˜ 1 e−θ(t−t0 ) + c˜ 2 e(µ(1−η)−θ)(t−t0 ) + c3 , (7) η(µ0 θ+λµ) y = − θ(µ(1−η)−θ) − η˜c1 e−θ(t−t0 ) − c˜ 2 e(µ(1−η)−θ)(t−t0 ) ,

R20 y(t) R2

where r0 is the individual data rate of D2D links within the proximity radius R0 , and ri,BS is the data rate on the cellular r link to the BS averaged over the area of interest as 0 lim r fr (r)dr according to (3). Assuming that the channel resources are being equally shared between receivers on both cellular and D2D tiers (due to fair 3GPP LTE scheduling and random-access nature of WiFi-Direct), the total D2D rate per system would R2

be proportionally decreased as r0 R02 [ηx(t) + y(t)]. We also remind that the content lifetime is relatively short, so that the process of content distribution does not reach its stationary state. We therefore study the joint evolution of both x(t) and y(t) in a transient mode, assuming θ < µ(1 − η). Below we consider the two phases of the process in question: a phase of growth during the content lifetime t ∈ [0, T ], when new requests from users are being generated, and a phase of reduction t > T , when users lose their interest in the content and both processes decrease to the zero level. We aim to derive the closed-form expressions for the x(t) and y(t) evolution by solving two variations of the following system of differential equations for the baseline and coded content dissemination strategies, respectively: ⎧ ⎨ dx(t) = λdt − θx(t)dt − µηx(t) − µy(t)dt − µ0 dt, (5) dy(t) = µηx(t)dt + µy(t)dt + µ0 dt − θy(t)dt, ⎩ under initial conditions : x(t0 ) = x0 , y(t0 ) = y0 ,

where t0 , x0 , and y0 are defined by the initial conditions µ0 θ+λ(µ−θ) and (6), whereas c3 is given as θµ(1−η)−θ 2 and: ⎧ ⎨ c˜ 1 = −λ/θ+x0 +y0 , (1−η) ⎩ c˜ 2 = λ/θ−ηx0 −y0 − c3 . (1−η) 2) Evolution in the reduction phase t ∈ [T, ∞) is defined by a solution to the Cauchy problem (5) under initial conditions x(T ) = xT , y(T ) = yT , where xT , yT are defined by the process (7): T +˜yT ) −θ(t−T ) T +η˜xT θ1 (t−T ) x = (˜x(1−η) e − y˜(1−η) e + θµ01 , (8) T +˜yT ) −θ(t−T ) T +η˜xT θ1 (t−T ) e + y˜(1−η) e − θµ01 , y = −η (˜x(1−η) where θ1 = (µ(1−η)−θ), x˜ T = xT − θµ01 , and y˜ T = yT + θµ01 . Proof: The proofs of the first and the second parts of this proposition are given in Appendix, sections A1 and B1, respectively.

R2

where µ = rZ0 R02 is a constant introduced for the sake of brevity r and µ0 = 0 lim r fr (r)dr/(Z · E[Nc ]) is the data rate spatially averaged by (3). The initial conditions x0 and y0 are specified by the phase of the process.

D. Coded (Network Coding, NC) Dissemination Strategy As for the network-coded operation, the BS distributes the linear combinations of the fragments, so that the connected


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neighbors have useful information with the probabilities close to one, and thus we assume η = 1. Therefore, the solution is given by the following proposition. Proposition 2: Consider the evolution of the number of users x(t) and y(t), which are interested in a given content C with lifetime T and possessing incomplete and complete data, respectively. 1) Evolution in the growth phase t ∈ [t0 , T ] is defined by a solution to the Cauchy problem (5) under initial conditions x(t0 ) = x0 , y(t0 ) = y0 : ⎧ ⎨ x = (˜c1 + c˜ 2t)e−θ(t−t0 ) − µ0 θ+λ(µ−θ) , θ2 (9) λµ+µ θ c ˜ −θ(t−t 0 2 0), ⎩y = − η˜c1 + µ + c˜ 2t e θ2 where c˜ 1 = x0 − c3 + µy0 + µx0 − λµ ˜2 = θ t0 and c

−µy0 − µx0 + λµ θ . The parameters t0 , x0 , and y0 are given by the corresponding initial conditions. 2) Evolution in the reduction phase t ∈ [T, ∞) is defined by a solution to the Cauchy problem (5) under initial conditions x(T ) = xT , y(T ) = yT , where xT , yT are defined by the process (9): x = (˜xT − µ(η˜xT + y˜ T )(t − T )) e−θ(t−T ) − µθ0 , (10) y = (˜yT + µ(η˜xT + y˜ T )(t − T )) e−θ(t−T ) + µθ0 , where x˜ T = xT + µθ0 and y˜ T = yT − µθ0 . Proof: The proofs of the first and the second parts of this proposition are given in Appendix, sections A2 and B2, respectively. In summary, Propositions 1 and 2 provide us with the closedform expressions for the baseline system evaluation (7) and (8), as well as for the coded system evaluation (9) and (10) in the phases before and after content leaves the system, respectively. The obtained expressions allow for understanding the dynamic system behavior under various practical parameters and even for determining the regions of system stability. However, we note that our solution describes the system dynamics under some constraints, such as θ < µ(1 − η) and x(t), y(t) > 0. In the case when the latter condition is violated, we may translate the number of users into max(x, 0) and rewrite the corresponding equation for dy(t) = −θy under respective initial conditions. In Fig. 4(a), we compare the proposed analytical model with the system-level simulation results (methodology detailed in the following section) for the evolution of an isolated item of content in the system with the baseline content distribution policy. The presented plot compares the analytical estimation of such evolution (detailed in this section) with a realization obtained through simulation under similar conditions. It is important to note that the presented simulation results come from a single realization, and are not averaged, as to illustrate what degree of agreement the predictions typically have with respect to actual observations. Based on the presented results, one can clearly see that the proposed model captures all of the major factors that affect the content distribution process, namely, the initial swarm formation (solid line), the acquisition of the last fragment (first part

Fig. 4. Comparison of analytical (A) and simulation (S) results in similar conditions: BT (a) vs. NC (b).

of the solid curve, for BT only), the dissemination process when other members of the initial swarm acquire the last remaining fragments, as well as the decay stage when users begin to leave the system. Similarly, Fig. 4(b) showcases the dynamics of the network-coded content distribution system captured with the proposed analytical framework. Given the obtained results, one can apply our proposed analysis further towards evaluation of more complex systems with interactions of multiple content dissemination processes. Correspondingly, the NC system shows a much more gradual transition towards completed downloads than the BT system. The reason is that with BT there could be a large number of users (around 30 in the figure) that have almost completed the download and are waiting for the very last fragment, which none of them have. The NC system never has such situation, as the “last fragment” download does not make sense there, and thus the system switches to the dissemination stage much sooner. This earlier transition is what makes the NC system more “agile” under heavy load. IV. S YSTEM -W IDE P ERFORMANCE E VALUATION OF P ROXIMATE C ONTENT D ISTRIBUTION System-level evaluation of the envisioned content distribution system represents an interesting challenge, requiring the respective simulation to support dynamic channel and content formation, as well as necessary signaling to set up and dismantle D2D links in a realistic manner. Augmenting our WINTERsim simulation framework [60], which has been thoroughly calibrated and verified in our previous publications on D2D technology [2], [17], [18], we have constructed an appropriate scenario which we discuss in what follows. A. Scenario-Related Considerations Our simulation scenario mostly follows the system model assumptions in Section II, except for the following features: • In the analysis, we assume that the users are assigned content acquisition sessions irrespective of the total number

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TABLE I S YSTEM E VALUATION PARAMETERS

of users in the system and their past history. In practice, we only need to assign a content acquisition session to a user device that does not have that particular content yet in its memory (cache), until all users have the content they require. This is exactly what is done in the simulation. • The realistic interference situation between the ongoing D2D sessions is typically more complex than simple division of the channel resources among the users in one collision domain. Our simulation allows capturing the exact interference dynamics using the logic commonly employed for unlicensed-band connections over WiFiDirect, that is, CSMA/CA with the binary exponential backoff. • The D2D links take some time to initialize, and therefore we model the corresponding delay every time a user decides to retrieve data from a D2D neighbor. There is, obviously, similar delay when using cellular connection, but it is typically negligibly small. In more detail, the parameters used for the simulation are summarized in Table I. The simulation scenario specified in this section is modeled after the situation we expect to be typical in a massive social/ public event such as concert hall, or during a sports event, where masses of people move around and exchange small fragments of multimedia from that very event. Our parameters are based on a small hockey match, where instant replays are to be broadcast. One can expect the stadium to be roughly 200 meters across, and it would likely have its own cellular base station for conventional users. Naturally, cellular network capacity is fairly limited for the purposes of on-demand video streaming, and thus the owners of the stadium would look for affordable alternatives. One such alternative is assisted D2D, which could be efficiently deployed by the stadium’s owners and/or the operator that owns the cell covering the stadium. Our primary interest is to evaluate how efficiently the integrated cellular + D2D network handles the offered load and how well the users are served. However, before comparing the two data dissemination strategies by the users (baseline and coded, see Section II), we comment on the expected individual performance of the cellular network without the D2D tier.

One can easily calculate that a single cellular base station would have difficulty serving the expected traffic demand in the area of interest. In more detail, the capacity needed by the system is on the order of 80 Mbps to serve the average load, with about 80/0.7 ≈ 120 Mbps needed to serve peak loads reliably. It effectively means that with our estimated bandwidth of 10 Mbps/cell, it would require around 12 cells to serve all of the users employing nothing but cellular connections, assuming optimistically that our network capacity scales linearly with the number of base stations. While that is of course possible, in dense deployments such solutions are rather challenging due to interference issues. Indeed, partitioning a 100 m radius cell into 12 smaller cells, one would arrive at LTE cells with the radius of just 25 meters, which is significantly below what LTE is designed to work with today. With such tiny cells, achieving the frequency reuse factor of 1/3 would probably be the best feasible option, and this is what we used in our comparison. In what follows, however, we employ an alternative way to boost the system capacity, which relies on the shorter and lower-to-the-ground D2D links, that operate at very short ranges by design. One can argue here that an appropriate application of Multimedia Broadcast Multicast Service (MBMS) in 3GPP LTE network could easily resolve the capacity problem by serving all users at once. For instance, if there are some video clips being produced by sports event authorities, the network could be multicasting them to all supported devices quite efficiently. The key issue with MBMS, however, is that the user’s device does not know which content to download in advance, before the user starts watching it. Once the user decides, it may already be too late to download the needed content, as its beginning might have been broadcast already. Hence, MBMS would need to repeat all of the videos reasonably often for as long as they remain in demand. Consequently, such an approach is not particularly applicable for the considered use case. On the other hand, certain video fragments could actually be disseminated in a multicast fashion for further distribution over MBMS-like service, but we leave the respective considerations out of scope of this paper. B. Baseline (“BitTorrent,” BT) Dissemination Strategy In order to employ D2D links in the content dissemination process, we first establish a protocol for content distribution. Following the description given in Section II, each object of content (here, a file) is split into a number of fragments (similarly to chunks in the popular “BitTorrent” systems). If all fragments are automatically shared by the devices that have acquired them, we should have a significant potential for D2D link creation in the area of interest. In this study, we base the D2D links on a high-efficiency short-range wireless technology, WiFi-Direct, which enables high degrees of spatial reuse. With an effective length of R0 = 30 m, the WiFi-Direct D2D links should easily deliver the necessary capacity that we lack in the cellular system. The results of the actual performance evaluation are shown in Figs. 5 and 6 on the left side (a), (b), (c). In particular, Fig. 5 demonstrates how different systems respond to user arrivals, and most importantly how they handle


Fig. 5.

Overall system performance comparison: BT (a), (b), (c) vs. NC (d), (e), (f).

Fig. 6.

Comparison of per-content system responses: BT (a), (b), (c) vs. NC (d), (e), (f).

the high loads. In summary, using the BT system quite often means having to “disappoint” the user, as the number of users that have completed the download does not grow quickly enough, as compared to the NC system where this number steadily increases. In Fig. 6, on the other hand, we witness the contributions of individual pieces of content as they enter and leave the system. One can see that each new file follows the same evolution pattern—first it is downloaded mostly over cellular, and then over D2D. The key difference is that for the NC system the transition to D2D downloads happens sooner, and thus more capacity becomes available. One can clearly see that even though the D2D links are short enough to enable the required degrees of spatial reuse (aside from using more bandwidth and having four times the bitrate of cellular links), the overall system performance is still quite poor. Whereas the vast majority of the fragments are indeed served by the D2D connections, the total system capacity is still insufficient, and the proportion of losses is thus very significant,

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with as low as 34% of the sessions actually getting served on time. There are two reasons for such inefficient behavior. First, the actual D2D links do not share the resources as efficiently as one would expect them to. In addition, the practical D2D distribution is always limited by the rate at which the cellular system is injecting new fragments into the system. That is, if there are 9/10 fragments for a given file available in a D2D cluster, none of the nodes can actually complete the acquisition. As the result, all of them will request the cellular system to deliver that last missing fragment, and still it may be that none of them will obtain it on time, hence wasting expensive cellular resources. On top of that, the D2D dissemination rate during the first seconds of content’s existence is proportional to the cellular rate (as the initial fragments are delivered through the cellular connections anyway), and once the cellular system becomes overloaded the entire integrated network collapses. In other words, the subsequent content arrivals will observe the cellular system already occupied, which would further delay the fragment

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TABLE II S YSTEM P ERFORMANCE C OMPARISON S UMMARY

acquisition times, and congest the system even further. Overall, the baseline D2D offloading system does deliver some additional capacity, but it generally fails to realize the full potential performance that unlicensed bands are expected to deliver. C. Coded (Network Coding, NC) Dissemination Strategy In order to overcome the key limitations of the baseline D2D offloading solution, we apply RLNC (random linear network coding) on all fragments, such that their exact identity does not matter during file assembly. This allows the users to exchange relevant data fragments in much higher percentage of chances, improving the efficiency of D2D communication. We choose a simple network coding scheme to explore the available performance improvements with low degrees of complexity and overhead. The results of the envisioned system operation with network coding are presented in Figs. 5 and 6 on the right side (d), (e), (f). One can observe that the NC-enhanced system copes with the load spikes associated with new content arrivals much better, and its D2D tier has almost double the throughput. As the result, it suffers much less from the congestion of the cellular system, since most of the time the users are utilizing their D2D links for content distribution. Overall, this translates into lower latencies and better user experience, with the NC system loss ratio of just about 20%, which is quite acceptable for an opportunistic service such as D2D offloading. Of course, by introducing additional cellular capacity (e.g., in the form of pico or femto cells) even those losses could be easily mitigated. In this example, our results clearly indicate the advantages of the network coding for wireless content dissemination: when the initial fragments are very valuable, it is imperative to share them as quickly as possible with as many neighbors as possible, which is very difficult with the BT system due to potential duplicate fragments. Reducing the fragment size would have solved this problem to a point, but then the D2D link setup takes time, which means that using D2D for small pieces of content is not very efficient. Therefore, using larger fragments seems to be a natural option to compensate for the connection setup overhead. For comparison, the system performance parameters under different operation modes are presented in Table II. It is very clear that the network-coded D2D distribution system delivers roughly the same performance as superdense (non-practical) LTE deployment of 12 cells,1 but unlike LTE it requires no additional capital expenses from the mobile network operator. In addition, unlike the small cell based solution, the NC system has practically zero deployment time, as one does not need to 1 It corresponds to the equivalent effective bandwidth as for other systems under comparison.

Fig. 7.

Network coding applicability as function of system load.

install and manage the pico BS nodes, as well as configure the cumbersome interference management and frequency reuse mechanisms. D. Some Practical Considerations Interestingly, with better BS service the difference between the NC and the BT systems becomes less noticeable, going to nearly zero when the system capacity becomes sufficient to handle all of the arriving traffic. Therefore, in practice, one might want to use the BT system for as long as the loss rates are acceptable, and then switch to the coded operation whenever capacity becomes an issue. Similar policy switch can be made when system becomes idle again to minimize the overheads needed to perform decoding on the user devices. The particular details of the system delay and success rate performance as functions of the system load are shown in Fig. 7. One can learn that the compared systems have similar ranges of usable loads, yet react to them very differently. In particular, the BT system overall has worse success rates, and higher delays, while the NC system scales almost as well as the cellular system in everything, except for the transfer delay. Based on this, we conclude that the network-coded D2D offloading system is a reasonable substitute to ultra-dense cellular deployments over a wide range of conditions, assuming that the content is sufficiently correlated among users in the network. Finally, it should be noted that the real-world D2D solutions, such as WiFi-Direct, actually operate on several 20 MHz channels. In particular, it is reasonable to expect a frequency reuse factor of 1/3 from WiFi in 2.4 GHz band, as well as 23 extra channels in the 5 GHz band, potentially adding up to already impressive capacities (on the order of 5 GHz of effective radio bandwidth). Challenging these capacities with licensed-spectrum technologies is next to impossible today, making solutions similar to the one proposed in this paper the most natural choice for high-density multimedia-heavy usecases, such as sports events, concerts, and festivals. V. S UMMARY AND C ONCLUSION Notably, the unified framework for network-assisted D2D produced by this study combines a rigorous mathematical model with extensive system-level simulations. From the obtained results we learn that D2D-based content distribution, with or without network coding enhancements, provides a viable alternative to the conventional multicast-based media


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A. Growth Process Let us consider the content acquisition process as the joint evolution of the processes x(t) and y(t). In order to obtain the sought solution to the considered problem, we formulate the following Cauchy problem within the interval [t0 , T ]: ⎧ ⎨ dx(t) = λdt − θx(t)dt − µηx(t) − µy(t)dt − µ0 dt, (11) dy(t) = µηx(t)dt + µy(t)dt + µ0 dt − θy(t)dt, ⎩ under initial conditions : x(t0 ) = x0 , y(t0 ) = y0 , Fig. 8. Scheme of the proposed solution illustrating two-phase approach: within the content lifetime (growth process) and after the content leaves the system (reduction process).

distribution services (e.g., MBMS), with the added feature of on-demand content delivery. To this end, network coding has excellent potential to improve system-wide performance when the network is about to reach its peak capacity by enhancing content availability and thus allowing more D2D links to be established. Whereas there are still decoding costs associated with the use of NC, applying random linear network coding is not excessively complicated for practical scenarios. Based on our rigorous evaluation, it is confirmed that feasible NC enhancements for D2D almost always visibly improve network performance, and thus it would be natural to employ these for multimedia distribution during e.g., mass events. Overall, there is a lot of activity around proximate communication, D2D connectivity, and other similar efforts to augment cellular network capacity on the way to 5G systems. Currently, the considered network-assisted D2D technology is already a part of the 3GPP Release-12 specifications for LTE, and therefore will eventually be supported by most platforms in the very near future. We believe that the present study, explicitly taking into account the dynamics of an actual content distribution application, can shed light on how much actual gain we should expect by employing this new and promising technology. In summary, there are two practical outcomes of this research. First, we offer an efficient content distribution policy that allows to organize the dissemination and caching of popular content, so as to employ the maximum available number of potential D2D connections. Second, we quantify a certain limit that defines how much capacity can actually be leveraged by a content distribution service within a certain network deployment, which provides an important insight for service planners determining whether D2D would be a feasible option to use. A PPENDIX In this appendix, we provide detailed calculations which have been referred to in the proofs of Propositions 1 and 2. For the sake of clarity, we sequentially solve the system first for the growth process (within the content lifetime), t ∈ [t0 , T ], and then also for the reduction process (after the content leaves the system), t ∈ [T, ∞). We thus provide closed-form solutions for η < 1 and η = 1 for both cases. The general structure of our solution is illustrated in Fig. 8, where the sketch of the x or y defines the growth and the reduction processes, arriving at four technical components of the solution as below.

where the value x0 = x(t0 ) in the initial conditions may be obtained from (6). We further concentrate on solving the problem at hand subject to the general initial conditions (11). Omitting the index t for brevity, we express the variable y from the first equation: y = 1µ [λ − θx − µηx − x − µ0 ], (12) y = µηx + µy + µ0 − θy. Then, we proceed with the second equation, substituting the expression for y obtained from (11): x + x (θ + µη + θ − µ) + x [θ(θ−µ)+θµη]= − [µ0 θ+λ(µ−θ)] . (13) The solution of corresponding homogeneous differential equation is represented by: x = c1 eα1 t + c2 eα2 t , where α1 = −θ, α2 = (µ(1 − η) − θ) are the roots of characteristic equation. Further, we consider two separate cases: when roots are distinct α1 = α2 , and when they are equal α1 = α2 . 1) Distinct Roots. Case α1 = α2 : With the particular solution of the equation, the solution to the differential equation (13) may be obtained as: x = c1 e−θt + c2 e(µ(1−η)−θ)t +

µ0 θ + λ(µ − θ) . θµ(1 − η) − θ2

In order to find a solution to Cauchy problem with the initial values x(0) = 0, y(0) = 0, we firstly derive x : x = −θc1 e−θt + (µ(1 − η) − θ) c2 e(µ(1−η)−θ)t . Therefore, the corresponding solution to the system (11) is given as: µ0 θ+λ(µ−θ) , x = c1 e−θt + c2 e(µ(1−η)−θ)t + θ(µ(1−η)−θ) (14) η(µ0 θ+λµ) y = − θ(µ(1−η)−θ) − ηc1 e−θt − c2 e(µ(1−η)−θ)t . Taking into account initial conditions x(t0 ) = x0 , y(t0 ) = y0 , we obtain the system of equations for the coefficients c1 , c2 : c1 + c2 eµ(1−η)t0 = (x0 − c3 )eθt0 , η(µ0 θ−λµ) ηc1 + c2 eµ(1−η)t0 = θ(µ(1−η)−θ) − y0 eθt0 , where c3 is the particular solution obtained above: µ0 θ + λ(µ − θ) . θµ(1 − η) − θ2

(15)

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The system may be easily solved, and the sought coefficients are obtained as: ⎧ ⎨ c1 = −λ/θ+x0 +y0 eθt0 = c˜ 1 eθt0 , (1−η) (16) ⎩ c2 = λ/θ−ηx0 −y0 −c3 e(θ−µ(1−η))t0 = c˜ 2 e(θ−µ(1−η))t0 . (1−η) Therefore, substituting the coefficients (16) into the expressions (14), we may collect together the final solution of (11): x = c˜ 1 e−θ(t−t0 ) + c˜ 2 e(µ(1−η)−θ)(t−t0 ) + c3 , (17) η(µ0 θ+λµ) y = − θ(µ(1−η)−θ) − η˜c1 e−θ(t−t0 ) − c˜ 2 e(µ(1−η)−θ)(t−t0 ) , where constants c˜ 1 , c˜ 2 , and c3 are given above, whereas t0 , x0 , and y0 are defined by the initial conditions. The formula (17) is the sought expression for the growth phase evolution if α1 = α2 . Further, we continue with the case of equal roots α1 = α2 . 2) Indistinct Roots. Case α1 =α2 : We note that α1 =α2 i.f.f. µ(1−η) = 0. Since the request arrival rate µ > 0 by definition, this case fully corresponds to the case of linear network coding . η = 1. The coefficient c3 then would be defined as − µ0 θ+λ(µ−θ) θ2 Hence, the solution to the (13) may be given in the form: x = (c1 + c2t)e−θt −

µ0 θ + λ(µ − θ) , θ2

where the derivative is: x = −θ(c1 + c2t)e−θt + c2 e−θt . Hence, the solution to the system (11) may be written as: −θt x = (c1 + c2t)e + c3 , (18) c2 0θ y = λµ+µ − c + + c t e−θt . 1 2 2 µ θ Applying the initial conditions x(t0 ) = x0 , y(t0 ) = y0 , we arrive at the system of equations for the coefficients c1 , c2 : θt0 c1 + c2t0 = (x 0 − c3)e , (19) λµ+µ0 θ 1 c1 + µ + t0 c2 = − y0 eθt0 , θ2 which results in the following values for the coefficients: ⎧ ⎨ c1 = x0 − c3 + µy0 + µx0 − λµ t0 eθt0 = c˜ 1 eθt0 , θ (20) ⎩ c2 = −µy0 − µx0 + λµ eθt0 = c˜ 2 eθt0 . θ Substituting the coefficients into the expression (18), we obtain the final solution for the case η = 1: ⎧ ⎨ x = (˜c1 + c˜ 2t)e−θ(t−t0 ) − µ0 θ+λ(µ−θ) , θ2 (21) λµ+µ θ c ˜ −θ(t−t 0 2 0), ⎩y = − η˜c1 + µ + c˜ 2t e θ2 where c˜ 1 =x0 −c3 + µy0 +µx0 − λµ ˜ 2 = −µy0 −µx0 + λµ θ t0 and c θ . B. Reduction Process Now let us consider the process when the content becomes unpopular after time moment T and users stop generating new requests. The only change in the evolution is the absence of

new arrivals λ = 0. Therefore, we may reuse the solution (14) and (18) on the interval [T, ∞) under new initial conditions. We again differentiate between the cases of distinct and indistinct roots. 1) Distinct Roots. Case α1 = α2 : Adding new initial conditions to the problem, we rewrite the system (11) as: ⎧ µ0 −θt (µ(1−η)−θ)t + ⎪ ⎨ x = c1 e + c2 e µ(1−η)−θ , µ0 (22) − ηc1 e−θt − c2 e(µ(1−η)−θ)t , y = − µ(1−η)−θ ⎪ ⎩ under initial conditions : x(T ) = xT , y(T ) = yT , where xT and yT are described by the evolution of the growth process. By the analogy, we define the coefficient c1 , c2 for (22): ⎧ µ0 θT = ηc + c eµ(1−η)T , ⎨ xT − 1 2 µ(1−η)−θ e µ0 θT = −ηc − c eµ(1−η)T . ⎩ yT + 1 2 µ(1−η)−θ e The coefficients may be easily calculated as follows: T +˜yT ) θT c1 = (˜x(1−η) e , T +η˜xT θT −µ(1−η)T e e , c1 = − y˜(1−η)

µ0 µ0 and y˜ T = yT + µ(1−η)−θ . Finally, the where x˜ T = xT − µ(1−η)−θ solution for the reduction process is defined by: T +˜yT ) −θ(t−T ) T +η˜xT θ1 (t−T ) e − y˜(1−η) e + θµ01 , x = (˜x(1−η) (23) T +˜yT ) −θ(t−T ) T +η˜xT θ1 (t−T ) e + y˜(1−η) e − θµ01 , y = −η (˜x(1−η)

where θ1 = (µ(1 − η) − θ). We note, that the latter expression corresponds to both x(t) and y(t) process evolutions after the content leaves the system and until the processes reach zero level. Starting from the point any of the processes touches the zero level, the equation for another one changes respectively, similar to (6). 2) Indistinct Roots. Case α1 = α2 : Finally, we consider the case of indistinct roots (or η = 1) for the reduction phase. Here, the problem (11) transforms into the following system: ⎧ µ0 −θt ⎪ ⎨ x = (c1 +c2t)e − θ , (24) y = µθ0 − ηc1 + cµ2 + ηc2t e−θt , ⎪ ⎩ under initial conditions : x(T ) = xT , y(T ) = yT . Applying the initial conditions, we again arrive at the system of linear equations for the coefficients: c1 = xT eθT + µ(xT + yT )eθT T, (25) c2 = −µ(xT + yT )eθT , where x˜ T = xT + µθ0 and y˜ T = yT − µθ0 . As the result, the final expression may be delivered as: x = (˜xT − µ(˜xT + y˜ T )(t − T )) e−θ(t−T ) − µθ0 , (26) y = (˜yT + µ(˜xT + y˜ T )(t − T )) e−θ(t−T ) + µθ0 . Therefore, we have sequentially obtained the expressions for the growth phase (including cases η < 1 as well as η = 1) and for the reduction phase (also η < 1 and η = 1). Noteworthy, the case of η = 1 corresponds to linear network coding.


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Alexander Pyattaev (S’13) received the B.Sc. degree from the Saint Petersburg State University of Telecommunications, Saint Petersburg, Russia, and the M.Sc. degree from the Tampere University of Technology, Tampere, Finland, where he is currently working toward the Ph.D. degree with the Department of Electronics and Communications Engineering. He has publications on a variety of networking-related topics in internationally recognized venues, as well as several technology patents in collaboration with Intel Labs. His primary research interest lies in the area of future wireless networks: shared spectrum access, smart RAT selection, and flexible and adaptive topologies.

Olga Galinina (S’13) received the B.Sc. and M.Sc. degrees in applied mathematics from the Department of Applied Mathematics, Faculty of Mechanics and Physics, Saint Petersburg State Polytechnic University, Saint Petersburg, Russia. She is currently working toward the Ph.D. degree with the Department of Electronics and Communications Engineering, Tampere University of Technology, Tampere, Finland. Her research interests include applied mathematics and statistics; queuing theory and its applications; wireless networking and energy efficient systems; machine-to-machine and device-to-device communication.

Sergey Andreev (S’10–M’12) received the Specialist degree in 2006 and the Cand.Sc. degree in 2009 from the Saint Petersburg State University of Aerospace Instrumentation, Saint Petersburg, Russia, and the Ph.D. degree in 2012 from the Tampere University of Technology, Tampere, Finland. He is a Senior Research Scientist with the Department of Electronics and Communications Engineering, Tampere University of Technology. He (co)authored more than 90 published research works on wireless communications, energy efficiency, heterogeneous networking, cooperative communications, and machine-to-machine applications.

Marcos Katz received the M.S.E.E. and Dr.Tech. degrees from the University of Oulu, Oulu, Finland. He was with Nokia during 1987–2001, Samsung Electronics during 2003–2005, and the Technical Research Centre of Finland during 2006–2009. Since 2010, he has been a Professor at the Centre for Wireless Communications, University of Oulu. He is currently working on cooperative and cognitive communications, particularly in the area of mobile clouds. He is also working on visible light communications.

Yevgeni Koucheryavy (M’02–SM’09) received the Ph.D. degree in 2004 from the Tampere University of Technology (TUT), Tampere, Finland. He is a Full Professor and Lab Director at the Department of Electronics and Communications Engineering, TUT. He is the author of numerous publications in the field of advanced wired and wireless networking and communications. His current research interests include various aspects in heterogeneous wireless communication networks and systems, the Internet of Things and its standardization, as well as nanocommunications. He is an Associate Technical Editor of IEEE Communications Magazine and an Editor of IEEE Communications Surveys and Tutorials.