Coded Caching for Hierarchical Networks with a Different Number of Layers

Transcription

1 Coded Caching for Hierarchical Networks with a Different Number of Layers Makoto Takita, Masanori Hirotomo, Masakatu Morii Kobe University, Saga University November 20, 2017 ASON 17@Aomori

2 Outline 1 1. Introduction 2. Caching scheme 3. Coded caching scheme 4. Coded Caching for Hierarchical Networks with a Different Number of Layers 1. Program setting 2. Three basic schemes 3. Combination scheme 4. Lower bound 5. Numerical result 5. Conclusion

3 Background: Consumer internet traffic 2 PB per Month 250, , , ,000 Internet video Web, , and data Online gaming File sharing 50, Source: Cisco VNI, 2017 IP traffic will grow at a Compound Annual Growth Rate of 24% from 2016 to IP video traffic will be 82% of all consumer Internet traffic by 2021, up from 73% in Cisco Visual Networking Index: Forecast and Methodology, (September 15, 2017)

4 Purpose: Reduce the peak network load 3 Normalized network load Time The network traffic has high time variability Caching is recognized as one solution to reduce the peak network load Users store popular contents in the cache memories during the off-peak time Users can recover their requests by the stored contents during the peak time Caching can effectively use the network resources during off-peak time

5 Caching scheme: Placement phase 4 server shared link caches user Users store popular contents in the cache memories during the off-peak time. The problem is what contents are stored in the memories. The limitation is the memory capacity.

6 Caching scheme: Delivery phase 5 server shared link 1 request 2 delivery caches user 3 recover 1 Each user requests one content in the database 2 Server delivers messages based on the users requests and the stored contents 3 Each user recovers its request by the messages and the stored contents The problem is what messages to send users and the size of message The limitation is the network load

7 Coded caching: Difference to uncoded scheme 6 N = 2 files, K = 2 users, cache size M = 1 Uncoded caching Coded caching A 1, A 2 A 1, A 2 B 1, B 2 B 1, B 2 A 2 B 2 R = 1 A 2 B 1 R = 1 2 A B A B A 1 B 1 A 1 B 1 A 1 B 1 A 2 B 2 Placement Delivery Store same content Uncoded multicast Use cache locally Store different contents Coded multicast Use cache global

8 Coded caching: Related works 7 Single layer network with nonuniform demand [5]-[7] Centralized [2] Decentralized [3] with heterogeneous cache sizes [8] with distinct file sizes[9] and more Hierarchical network Two-layer network [4] [This work] Two-layer and single layer network

9 This work 8 The network with the two layer and single layer of cache at same time. We propose three basic coded caching schemes Scheme A: divide the caching problem into three blocks Scheme B: ignore the capabilities of the mirror servers Scheme C: divide the caching problem into two blocks We also propose the combination scheme by combining the scheme B and C discuss the combining parameter α and β show comparison with lower bound α 1 α Rate R 1 N files Scheme B Scheme C β 1 β Size M 1 Rate R 2 Size M 2 combination scheme

10 Problem setting: Two-layer and single-layer network model 9 Origin server N files Error-free broadcast link? rate R 1 K 1 mirror servers? size M 1 Error-free broadcast link? rate R 2 K 1 K 2 + K 3 memories???? size M 2 K 1 K 2 + K 3 users What contents are stored in the memories? What messages is sent users in response to users requests? The relation between memory size and message size?

11 Proposed scheme: Basic scheme A 10 Divide the caching problem into three blocks Apply the single-layer caching algorithm to each block Decentralized scheme for single layer R N, K, M = K 1 M N 1 1 MΤN K KM/N (a) (c) N files K 2 R N, K 1, M 1 + R N, K 3, M 2 Rate R 1A (b) R N, K 2, M 2 Rate R 2A Rate R 2A

12 Proposed scheme: Basic scheme B 11 The mirror servers are only used to forward messages Apply the single-layer algorithm for all of user Decentralized scheme for single layer R N, K, M = K 1 M N 1 1 MΤN K KM/N N files Rate R 1B R N, K 1 K 2 + K 3, M 2 ignore the capabilities of the mirror servers R N, K 2, M 2 Rate R 2B Rate R 2B

13 Proposed scheme: Basic scheme C 12 Divide the caching problem into two blocks Block (a) need to consider different cache size Apply the zero-padding solution [8] to the block (a) (a) Decentralized scheme for single layer R N, K, M = K 1 M N 1 1 MΤN K KM/N The rate for different cache size R N, K 1, K 3, M 1, M 2 = R N, K 3, M M K 3 2 R N, K N 1, M 1 (b) Rate R 1C N files R N, K 1, K 3, M 1, M 2 + K 2 1 R N, K 1, M 1 Rate R 2C Rate R 2C R N, K 2, M 2

14 Comparison: The rate R 1 of each basic scheme 13 Number of Files N 35 Number of Mirrors K 1 5 Number of Mirror users K 2 5 Number of origin users K 3 3, 5, 10 memory size of user M 2 1 Scheme C is better than Scheme A regardless of K 3. Scheme C is better than Scheme B with large M 1. Scheme B is constant even if M 1 increases

15 Proposed scheme: Combination scheme 14 Want to have good performance with any memory size of mirror Combining the scheme C and B to use resources of the mirror. Origin server α 1 α N files Scheme B Scheme C Rate R 1 K 1 mirror servers Size M 1 β 1 β Rate R 2 K 1 K 2 + K 3 memories K 1 K 2 + K 3 users Size M 2 What is the optimal fix parameters α and β?

16 Combination scheme: How to find optimal parameter 15 Optimal parameters α, β They are minimize rate R 1 depend on other parameters N, Ks,Ms. Method 1 (Brute-force) For given parameters, the exact optimal values of α and β are obtained by the brute force method. Method 2 (formulating) For given parameters, the almost optimal values of α and β are formulated by other parameter. We determined the formulated parameters α and β empirically from some numerical simulations. α, β = M 1 N, 0 M 1, 0 M 1 + K 3 M 2 if M 1 + M 2 K 3 N otherwise

17 Comparison: Brute-force and formulating 16 Number of Files N 30 Number of Mirrors K 1 5 Number of Mirror users K 2 5 Number of origin users K 3 5 memory size of user M 2 3 The combination scheme with formulated α, β is almost same as with optimal α, β. The formulating method is getting worse with the large M 1

18 Combination scheme: Achievable rate 17 The rates of the combination scheme are given in term of the function R N, K, M. R N, K, M is the rate of the decentralized coded caching algorithm. The achievable rates of decentralized algorithm are given in [3]. R N, K, M = ቐ min K, N M 1 if M N 1 0 otherwise Achievable rate of the combination scheme with the formulated parameters α β R 1 α, β min K 1 K 2 + K 3, M 1 N min K 3, N + 1 M 1 N K 1K 2 + K 3, N M 1 M 2 M 1 N M 1 K 2 + K M 1 + K 3 M 2 M 1 + K 3 M 3 + K 3M 2 K 3 N M 1 2 M 1 + K 3 M 2 2 if M 1 + M 2 K 3 N otherwise R 2 α, β M 1 N min K 2, N + 1 M 1 N K 2, N M 1 M 2 min K 2, N M 2 if M 1 + M 2 K 3 N otherwise

19 Our problem setting: Lower bound 18 We provide lower bounds based on cut-set bound [11]. Theorem 1(Lower bound) For N N files and K 1 N mirrors each with memory of size 0 M 1 N, K 2 N mirror users and K 3 N origin users each with the memory of size 0 M 2 N, where N is the set of natural numbers, the lower bounds of the rates R 1 and R 2 are given by R 1 max s 1 0,1,,K 1 s 2 0,1,,K 2 s 3 0,1,,K 3 max s 1 s 2 + s 3 s 1M 1 + s 1 s 2 + s 3 M 2, N s 1M 1 s 1 s 2 + s 3 M 2 NΤ s 1 s 2 + s 3 N/(s 1 s 2 + s 3 ) R 2 max max t tm 2 t 1,2,,K 2 NΤt, N tm 2 NΤt [11]T. M. Cover and J. A. Thomas, Elements of Information Theory, New York, NY, USA: Wiley, 1991.

20 Comparison: Combination scheme and lower bound 19 Number of Files N 30 Number of Mirrors K 1 5 Number of Mirror users K 2 5 Number of origin users K 3 5 memory size of user M 2 3 The combination scheme shows better performance than basic scheme. But, our lower bound is loose. We focus on how to realize a coded caching scheme for our network model. So, our lower bounds are not tight.

21 Conclusion 20 The network with the two layer and single layer of cache at same time. We propose three basic coded caching schemes. Scheme B is better with small memory of mirrors. Scheme C is better with large memory of mirrors. We also propose the combination scheme by combining scheme B and C. We discuss the combining parameter α and β. We formulated α and β that have almost same performance as optimal α and β. We show comparison with lower bound. Our lower bound is loose, so tighter bound is need. Future work Derivation of the tight bound More layers of caches Realistic simulations Size M 2 β 1 β α Rate R 2 Size M 1 1 α Rate R 1 N files Scheme B Scheme C

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