CACHING popular video contents at the network edge,

Transcription

1 1 Mobility and Popularity-Aware Coded Small-Cell Cachin Emre Ozfatura and Deniz Gündüz arxiv: v1 [cs.it] 21 Nov 2017 Abstract In heteroeneous cellular networks with cachin capability, due to mobility of users and storae constraints of small-cell base stations (SBSs), users may not be able to download all of their requested content from the SBSs within the delay deadline of the content. In that case, the users are directed to the macro-cell base station (MBS) in order to satisfy the service quality requirement. Coded cachin is exploited here to minimize the amount of data downloaded from the MBS takin into account the mobility of the users as well as the popularity of the contents. An optimal distributed cachin policy is presented when the delay deadline is below a certain threshold, and a distributed reedy cachin policy is proposed when the delay deadline is relaxed. Index Terms Heteroeneous cellular network, content cachin, user mobility. I. INTRODUCTION CACHING popular video contents at the network ede, closer to the end users, is a promisin method to cope with increasin video traffic. In heteroeneous cellular networks, popular video files are stored at SBS caches to reduce latency as well as backhaul traffic [1] [3]. The oal in these works is to maximize the hit rate, or equivalently, to minimize the amount of data downloaded from MBSs under iven SBS cache capacity constraints. A common assumption is that a user stays connected to the same set of SBSs durin the whole duration of video download. However, in ultra dense networks, where there is a lare number of operatin SBSs with limited coverae areas, it is indispensable to take user mobility into consideration to meet the prescribed quality of service (QoS) requirements [4]. Our aim in this paper is to provide a cachin policy, which, for iven video popularity profile and user mobility patterns, minimizes the averae amount of data downloaded from the MBS. We adopt the delayed offloadin scheme in heteroeneous networks; that is, each video content request has a deadline, and if the mobile user cannot download all the framents of the content from the SBSs it has connected to by the deadline, the remainder of the request is satisfied by the MBS [5]. We consider maximum distance separable (MDS) codin for storin the video contents in the SBS caches. To the best of our knowlede, [6] is the only work that considers mobility-aware cachin with delayed offloadin. The oal in [6] is to minimize the probability of a request bein served by the MBS. In this work, as in [7], we assume that the MBS serves the remainin framents as MDS coded packets at a hiher cost. Hence, unlike [6], our oal is to minimize the Emre Ozfatura and Deniz Gündüz are with the Information Processin and Communications Lab, Department of Electrical and Electronic Enineerin, Imperial Collee London, London SW7 2AZ, UK. This work was supported by EC H2020-MSCA-ITN-2015 project SCAV- ENGE under rant number , and by the European Research Council project BEACON under rant number amount of data downloaded from the MBS. We show that, if the request deadline is below a certain threshold there is an optimal distributed solution. If the request deadline does not meet this condition, we introduce a sub-optimal reedy cachin policy. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider a heteroeneous network that consists of one MBS and N SBSs, SBS 1,..., SBS N, with disjoint coverae areas. A mobile user (MU) is served by only one SBS at any particular time. We consider a video library V = {v 1,..., v K consistin of K distinct video files, each of lenth B bits, indexed accordin to their popularity profile, i.e., file v k is the kth most popular file with request probability p k. Further, we assume that video files are encoded by rateless MDS codin [3], so that a video file can be retrieved when B parity bits are collected in any order, from any SBSs. We consider equal-lenth time slots, whose duration corresponds to the minimum time that a MU remains in the coverae area of a SBS. Therefore, althouh a MU cannot connect to more than one SBS durin one time slot, it may stay connected to the same SBS over several consecutive time slots. We assume that SBS n is capable of transmittin R n bits within a time slot to a MU within its coverae area, and it has a storae capacity of C n bits. Due to the QoS requirement, a video file must be downloaded within T time slots once it is requested. Thus, if a MU is not able to collect B bits from SBSs in T slots, the remainin bits are provided by the MBS at a hiher cost. We define the mobility path of a user as the sequence of small-cells visited over T time slots after requestin a video file. For instance, for T = 5, SBS 1, SBS 2, SBS 2, SBS 3, SBS 4 is a possible mobility path. We remark that the MU may remain connected to the same SBS more than one time slot, and can connect to at most T different SBSs. Hence, there is a finite number of distinct mobility paths, denoted by M. We denote the mth mobility path by I m, and its realization probability by q m. Realization probabilities can be obtained from empirical observations, or via modelin mobility paths as random walks on a Markov chain. Our aim is to minimize the expected amount of data downloaded from the MBS for iven SBS storae capacities C {C n N, data transmission rates R {R n N, video popularity profiles P {p k K, and mobility paths with realization probabilities I T {(I m, q m ) m=1 M. Let = { N,K xn,k denote the cachin policy over T time slots, where n= x n,k indicates the number of parity bits for file v k stored in SBS n. A cachin policy is feasible if K x n,k C n, n. The averae amount of data downloaded from the MBS is denoted by d av (, I T ), or simply by d av. Let d k,m denote the amount of coded data downloaded from

2 2 the MBS for video v k followin mobility path I m. For iven C, R, P, I T, and cachin policy, we have { ( d k,m = max B min { ) x n,k, R n S m,n, 0, (1) where S m,n denotes the total number of time slots the MU connected to SBS n in mobility path I m. Then, takin the averae over all mobility paths and video files, d av can be written in terms of d k,m as d av = M K m=1 q m p k d k,m. Our oal is to find the optimal feasible cachin policy XT that minimizes d av, formulated as follows: P1: min d av subject to: x n,k C n, n. (2) x n,k 0, n, k. (3) In order to avoid the max operator in Eqn. (1), problem P1 is reformulated as follows. We treat d k,m as decision variables, and add the followin constraint: { ( d k,m max B min { ) x n,k, R n S m,n, 0. (4) We note that, for a iven feasible cachin policy the objective function is monotonically increasin with d k,m ; hence, for the optimal solution, constraint (4) must be satisfied with equality. The equivalent optimization problem is obtained as follows. P2: min,d T d av subject to: x n,k C n, n, (5) min { x n,k, R n S m,n + dk,m B, k, m, (6) d k,m 0, k, m, (7) where D T { K,M d k,m. Notice that constraints (6) and (7) k=m=1 toether imply constraint (4). Recall that a MU cannot connect to more than T different SBSs. Hence, for each (k, m) pair, constraint (6) can be replaced by at most 2 T linear constraints. To clarify, let T = 4 and N = 6, and consider the mobility path I m = {SBS 1, SBS 1, SBS 2, SBS 2. For this specific mobility path and file v k, (6) can be written as min { x 1,k, 2R 1 + min { x2,k, 2R 2 + dk,m B. (8) Equivalently, (8) can be replaced by the followin set of linear constraints: x 1,k + x 2,k + d k,m B, (9) x 1,k + 2R 2 + d k,m B, (10) 2R 1 + x 2,k + d k,m B, (11) 2R 1 + 2R 2 + d k,m B. (12) Consequently, the initial optimization problem P1 can be cast into a linear optimization problem. However, the number of constraints are exponential in time constraint T. In the next subsection we show that, under a certain assumption on T, P1 can be solved in a distributed manner. III. DISTRIBUTED SOLUTION B In this section, we consider the case T T min R ma x, where R max is the maximum data rate across all the cells, i.e., R max max {R 1,..., R N. This special case is also instrumental in hihlihtin the distinction between our problem formulation and that of [6], whose oal is to minimize the probability of downloadin any data from the MBS. We note that, with the formulation of [6], when T < T min all cachin policies are equivalent since it is not possible to collect B bits in T slots. While [6] inores the mobility paths when T < T min, each cachin policy will induce a different d av. Hence, an optimal cachin policy in [6] may lead to a suboptimal d av. Instead, we present the optimal cachin alorithm that minimizes d av when T T min. We also propose a reedy cachin policy, for T > T min. A. Optimal Distributed Solution When T T min, (1) simplifies to d k,m = B min { x n,k, R n S m,n. (13) Then, our objective d av can be rewritten as: d av = B M m=1 q m p k min { x n,k, R n S m,n. (14) {{ d av Note that minimizin d av is equivalent to maximizin d av, which denotes the averae amount of data downloaded from the SBSs. We chane the order of the summations in (14): d av = p k m=1 M q m min { x n,k, R n S m,n, (15) {{ d av, n we observe that the optimal cachin policy can be obtained via maximizin d av,n, defined above, for each SBS n separately. Let XT n denote the cachin policy for SBS n. For SBS n, we have the followin optimization problem: P3: max X n T subject to: d av,n x n,k C n. (16) x n,k 0, k. (17) If we roup the mobility paths accordin to the time spent in cell SBS n, d av,n can be written as d av,n = t=1 m:s m, n =t p k q m min { x n,k, tr n. (18) {{ d k av, n The term min { x n,k, tr n can be expanded as follows: min { t 1 x n,k, tr n = max { min { x n,k ir n, R n, 0, (19) i=0

3 3 Alorithm 1: Alorithm for optimal distributed cachin (-based policy) Input : R,C,{ n N Output: X T 1 for,...,n do 2 x n,k 0, k {1,..., K; 3 while C n > 0 do 4 ń k, t max n ; 5 x x n,ḱ n,ḱ { + min(c n, R n ); 6 n n \ ń ; k, t 7 C n C n R n ; 8 end 9 end and d k av,n can be rewritten as: d k av,n = Equivalently, d k av,n = t=1 m:s m, n t p k q m max { min { x n,k (t 1)R n, R n, 0. (20) n k,t max { min { x n,k (t 1)R n, R n, 0, (21) t=1 where n k,t p k P(S m,n t). We observe that for each k, d av,n k is a monotonically increasin piecewise linear function of x n,k, and its slope is n k,t for x n,k ((t 1)R n, tr n ). Consequently, the objective function in P3 is the sum of N monotonically increasin piecewise linear functions, and sum of its variables are bounded by constraint (16). Accordinly, it is maximized by maximizin the variable that corresponds to the linear function with the maximum slope. We propose Alorithm 1 to maximize the objective function that follows a straihtforward procedure usin n values for each SBS k,t n. The alorithm starts with increasin the variable x n,k that corresponds to the maximum slope, until the slope of d av,n k chanes, then it aain searches for the maximum slope, and repeats this process until the sum of the variables satisfies (16) with equality. From a computational point of view, proposed alorithm sorts the elements of set Γ n { nk,t : k = 1,..., K; t = 1,..., T for each n N. Since Γ n = KT, the complexity of Alorithm 1 is O(NKT lo(kt)). The optimality of the alorithm follows from the fact that n k,t n for any (k, t) pair, which implies k,t 1 that d av,n k is a concave function for each k. The cachin policy constructed accordin to Alorithm 1 is called the -based policy, and denoted by X T. We note that when T T min, X T is the optimal policy, i.e., X T = X T. B. Distributed Greedy Cache Allocation Scheme When T > T min it is not possible to predict the performance of X T, or ensure that d av(x T, I T ) d av (X T min, I Tmin ). We note that X T min is the -based cachin policy explained above for T min. However, we know that for any T > T min, d av (X T min, I T ) d av (X T min, I Tmin ). Hence, our aim is to provide a reedy distributed cachin policy X T that performs Alorithm 2: Greedy alorithm for storae reallocation Input : C, R, I T, P, XT min Output: X T 1 for,...,n do 2 + k, k NULL : k {1,..., K, Vred {; 3 V + {,V {; 4 x max = max { x n,1,..., x n,k ; 5 6 while x max > 0 do ḱ = max { k : x n,k x max ; 7 if ḱ V then 8 V V { ḱ, calculate k ; 9 end 10 if ḱ + 1 V + then 11 V + V + { ḱ + 1, calculate + k ; 12 end 13 x max x max R n ; 14 end 15 + max max +, max max ; if + max > max then ḱ max k + k, `k max k k ; 18 +`k, + NULL; ḱ 19 x x n,ḱ n,ḱ + R n, x n, `k x n, `k R n; 20 Go back to line 3 21 end 22 end better than X T min, i.e., d av (X T, I T ) d av (X T min, I T ). Our proposed method to construct X T consists of two steps. In the first step, we obtain the optimal cachin policy XT min by executin Alorithm 1. In the second step, we follow a reedy method for cache reallocation for each cell separately, which is performed by Alorithm 2. Assume that we are reallocatin the cache for SBS n, Alorithm 2 first identifies candidate video files for cache capacity increment and reduction. In this identification process, the main criteria is the popularity of the video files, e.., if there are several video files that have been allocated the same cache capacity, then the most popular file amon those is a candidate for cache capacity increment, whereas the least popular one is a candidate for cache capacity reduction. Accordinly, let V + and V denote the sets of indices of the video files that are candidates for cache capacity increment and reduction, respectively. After identification of a candidate file v k, we calculate k if k V, or + k if k V +, whose initial values are is NULL. + k and k denote the amount of chane in d av when the cache capacity of video file v k is increased by R n, or decreased by R 1 n, respectively. In the last step, alorithm compares the values of + max max k + k and max max k k. The condition + max > max implies that d av can be reduced via storae capacity reallocation. Then, the alorithm performs the followin task, storae capacity of video file v k, where + k = + max, is increased by R n, and the storae capacity of 1 Storae reallocation can be done with smaller sizes to improve the performance of the policy with a cost of complexity; however, due to limited space, we do not study this tradeoff in this letter.

4 min Cache size(% of file library), C (a) T = 5 and T min = min Delay deadline (slots), T (b) C = 300 (files) and T min = min Data transmission rate (file per slot), R (c) C = 300 (files) and T = 5. Fi. 1: The averae amount of data served by the MBS (normalized by the file size) for different values of: (a) normalized cache size C, (b) delay deadline T, and (c) data transmission rate R = 1/T min. video file v k where k = max is decreased by R n. IV. NUMERICAL RESULTS For numerical simulations we consider K = 1000 files in the library, and assume that their popularities follow a Zipf distribution with parameter 0.56 [8]. There are 16 SBSs located in a 2D square rid. We fix the transmission rate of each SBS to R, accordin to parameter T min, i.e., R = 1/T min file per slot. We consider the followin Markov mobility model: a MU connected to SBS n remains connected to the same SBS with probability f n, or connects to one of the neihborin SBSs with equal probability. In the experiment, we consider f 4 = f 13 =, f 7 = f 9 = 0.5, and for the all other SBSs f n =. As a performance benchmark, we also consider the policy, which simply caches the most popular N files at each SBS. We remark that, when T > T min the value of T has no impact on the performance of the policy since it caches the files as a whole. On the other hand, when T T min, the value of d av decreases linearly with increasin T. In the first experiment, we set T min = 2, T = 5 and consider the normalized SBS cache sizes (as portion of the entire file library) 10%, 20%, 30%, 40%, 50%. The reedy alorithm X T provides up to 40% further reduction in the amount of data downloaded from the MBS compared to -based policy X T as depicted in Fi. 1(a). We also observe that the ap between the performances of X T and X T widens with increasin cache sizes. Finally, note that the policy performs quite poorly in eneral as it inores the mobility patterns. In the second experiment, we set C = 300, T min = 2, and the delay deadline T takes values from 2 to 6 time slots. Performance of the cachin policies for different T values are plotted in Fi. 1(b). The key observation from the fiure is that, althouh the averae portion of the video file downloaded from the MBS monotonically decreases with increasin T under policy X T and X T min, this is not the always the case for X T. Note that X T mainly depends on the sojourn statistics, P(S m,n t), over all possible paths, and when T > T min those statistics miht be misleadin because in certain paths the MU miht collect all the parity bits before connectin to SBS n. In that case, storae capacity of the popular files miht be increased due to sojourn statistics even thouh it is not required. In the third experiment we set T = 5, C = 300, and the transmission rate takes values 1/2, 1/3, 1/4, 1/5, 1/6 file per slot, which correspond to T min values of 2, 3, 4, 5, 6 slots respectively. Althouh it is expected that the MBS usae decreases with the increasin transmission rate, Fi 1(c) illustrates that after a certain point the amount of data downloaded from the MBS increases with the transmission rate under all policies. This is because, T min decreases when the rate increases and the difference between T min and T widens, as a result of which the performance becomes worse. V. CONCLUSIONS In this letter, we studied mobility and popularity aware content cachin for a heteroeneous network with MDScoded cachin at the SBSs. Assumin a maximum download time requirement T, for each request, we first defined the threshold T min on T, below which some bits of the request must be downloaded from the MBS. Then, we obtained the optimal distributed cachin policy when T T min, called the -based policy, which minimizes the amount of data that need to be downloaded from the MBS. Then, we utilized the parameter T min and the -based policy for T = T min to obtain a reedy cachin policy for T > T min. Consequently, we showed how to desin a coded cachin policy accordin to T min and performed various simulations to demonstrate that the utilization of T min improves the performance sinificantly. REFERENCES [1] W. Jian, G. Fen, and S. Qin, Optimal cooperative content cachin and delivery policy for heteroeneous cellular networks, IEEE Trans. Mobile Comput., vol. 16, May [2] K. Poularakis, G. Iosifidis, and L. Tassiulas, Approximation alorithms for mobile data cachin in small cell networks, IEEE Trans. Commun., vol. 62, Oct [3] K. Shanmuam, N. Golrezaei, A. G. Dimakis, A. F. Molisch, and G. Caire, Femtocachin: Wireless content delivery throuh distributed cachin helpers, IEEE Trans. Inf. Theory, vol. 59, Dec [4] R. Wan, X. Pen, J. Zhan, and K. B. Letaief, Mobility-aware cachin for content-centric wireless networks: Modelin and methodoloy, IEEE Communications Maazine, vol. 54, Au

5 [5] K. Lee, J. Lee, Y. Yi, I. Rhee, and S. Chon, Mobile data offloadin: How much can WiFi deliver? IEEE/ACM Trans. Netw., vol. 21, Apr [6] K. Poularakis and L. Tassiulas, Code, cache and deliver on the move: A novel cachin paradim in hyper-dense small-cell networks, IEEE Trans. Mobile Comput., vol. 16, no. 3, Mar [7] J. Liao, K. K. Won, M. R. A. Khandaker, and Z. Zhen, Optimizin cache placement for heteroeneous small cell networks, IEEE Commun. Lett., vol. 21, Jan [8] M. Zink, K. Suh, Y. Gu, and J. Kurose, Characteristics of YouTube network traffic at a campus network - measurements, models, and implications, Comput. Netw., vol. 53, no. 4, Mar