Resource Placement and Assignment in Distributed Network Topologies Accepted to: INFOCOM 2013 Yuval Rochman, Hanoch Levy, Eli Brosh

Motivation: Video-on-Demand service  Video-on-Demand (VoD) internet service  Large collection of movies  Highly-variable  Geo-distributed demand  Use Content Distribution Network 2 Rochman, Levy, Brosh April 2013

Motivation: Content Distribution Network  Multi-region server structure (e.g., terminal based service, cloud)  Service costs: intra-region < inter-region < central 3 Rochman, Levy, Brosh April 2013 Intra-region Low cost Inter-region Medium cost Central High cost Region 2 Central video server Region 1 User terminals - demand Request type Disk

System and Objective  Players: users + content servers (local, central)  Objective: Reduce service costs  Replicating content at regions 4 Central video server Region 1 Region 2 User terminals - demand Low cost Medium cost High cost  Problem: Which movies to place where? Rochman, Levy, Brosh April 2013

 Tewari & Kleinrock [2006]  Proposed the Proportional Mean Replication.  Zhou, Fu & Chiu [ 2011]  Proposed the RLB (Random with Load Balancing) Replication. Related Work 5 Rochman, Levy, Brosh April 2013

The Multi-Region Placement Problem 6 Available resource Local storage Input:  Region j storage size: S j  Stochastic demand distribution N i j, random variable.  Service costs Rochman, Levy, Brosh April 2013 S 1 =4 S 2 =2 ? ? ? ? ? ? ? ? Pr(N 1 1 <=x) Pr(N 2 1 <=x) Pr(N 1 2 <=x)Pr(N 2 2 <=x) ? ? ? ? Stochastic demand E.g., high-variability, correlated Local < Remote < Server

Pr(N 1 1 <=x) Pr(N 2 1 <=x) Pr(N 1 2 <=x)Pr(N 2 2 <=x) S 1 =4 The Multi-Region Placement Problem 7 Local < Remote < Server Input: Storage S j, demand N i j, service costs Allocation: Place resources at regions Cost of allocation: expected cost of optimal assignment (over all demand realizations) Goal: find allocation with minimal cost Rochman, Levy, Brosh April 2013 Actual demand S 2 =2 ? ? ? ? Stochastic demand Available resource ? ? ? ? ? ? ? ? Local storage

Challenge and principles Challenge: Combinatorial problem based on multi- dimensional stochastic variables Keys of solution: Semi-Separability, Concavity, Reduction to Min-cost Flow problem. 8 Rochman, Levy, Brosh April 2013 Local storage S 1 =4 S 2 =2 ? ? ? ? ? ? ? ? Pr(N 1 1 <=x) Pr(N 2 1 <=x) Pr(N 1 2 <=x)Pr(N 2 2 <=x) ? ? ? ? Stochastic demand Available resource Exponential number of allocations Large database!

Single Region: Matching Demand realization to resources Observed Demand Resources A profit formula! 9 Rochman, Levy, Brosh April 2013

Single region: Revenue Formulation  Lemma: optimal matching maximizes revenue of a realization 10 Random Demand Type-i replicas  Hence: we have to maximize For any placement and demand Rochman, Levy, Brosh April 2013

Multi-Region: Matching Match local first, then remote, then server. 11 Rochman, Levy, Brosh April 2013 Available resource

Multi-Region: Revenue formulation  Thm: maximize revenue to find opt placement {L i j } : Local revenue Global revenue Type-i resources at region j 12 Rochman, Levy, Brosh April 2013 Local=3 Global=4

Separability and semi-Separability  Definition: function is separable iff  Sum of separated marginal components  Definition: function is semi-separable iff  “Almost” separated components 13 Rochman, Levy, Brosh April 2013 Where

Key 1: Revenue is Semi-separable  Revenue function  Revenue function is semi-separable.  Sum of local replicas = # global replicas. 14 Rochman, Levy, Brosh April 2013 Local replicas Global replicas

Key 2: Concavity  Partial expectation  Partial Expectations are concave!  Cumulative(cdf) is monotonic  Thus, Partial expectation is concave 15 Rochman, Levy, Brosh April 2013 Tail formula:

Placement Optimization Problem 16  Find {L i j } allocation of type-i movie at region j ({L i j } ) maximizing:  Under: capacity bound in each region Concave in placement vars {L i j } Rochman, Levy, Brosh April 2013

The Multi-Region Problems 17  Symmetric bounded– QEST 2012, low complexity  Greedy algorithm, max-percentile based  Asymmetric bounded– INFOCOM 2013, higher complexity  Reduction to min-cost flow problem Rochman, Levy, Brosh April 2013

Key 3: Min cost flow 18 Rochman, Levy, Brosh st 11/13 12/12 15/20 1/41/4 4/94/9 7/77/7 4/44/4 8/13 11/ Flow/Capacity 0 Weight April 2013 Flow value Flow weight (cost)

The Min-Cost Flow Problem 19Rochman, Levy, Brosh  Input:  A positive capacity function C on the edges, C: E  R +  A positive weight function W on the edges, W: E  R +  Required Flow value r  Output: : an s-t flow f, with flow value= r, which minimizes weight Σf(e) W(e). st 11/13 15/20 1/41/4 4/94/9 7/77/7 4/44/4 8/13 11/ /12 April 2013

Main theorem 20Rochman, Levy, Brosh Theorem :  Assume: - concave & semi-separable  Then, there is effective solution for  Solution uses min cost flow algorithm  On 7-layer graph!  Correctness at the paper. April 2013

7-layer graph: Local part S Region 21 Region, Movie type Region, Movie, # replicas Capacity, Weight Rochman, Levy, Brosh April 2013 Capacity of region Local weight

7-layer graph: Global part t Region, Movie type, # items Movie type, # items Movie type 22 April 2013 Rochman, Levy, Brosh Global weight

Min-Cost Flows  Standard solution to min-cost flow using Successive Shortest Path (SSP).  Complexity of SSP (standard solution) is  s= total storage in the system  k= # regions  m= # movie types  High complexity! 23Rochman, Levy, Brosh April 2013

Other proposed algorithms  Bipartite algorithm (INFOCOM 2013) in complexity of (instead of )  Idea: use only Region and movie type nodes  Clique algorithm -complexity of  Online algorithm. 24Rochman, Levy, Brosh s= total storage in the system k= # regions m= # movie types April 2013

Conclusions  Algorithms for resource placement and assignment  Geared for distributed network settings  Arbitrary demand pattern (e.g., highly-variable, correlated)  Joint placement-assignment problem  Multi-dimensional stochastic demand  New solution techniques 25 Rochman, Levy, Brosh April 2013

Questions? 26Rochman, Levy, Brosh April 2013

 An alternative allocation: Proportional mean  Allocate movies proportion to mean of distribution How good are the results? 27 Rochman, Levy, Brosh April 2013

 Two resource-types.  Single region, capacity n  Proportional Mean:  Expected profit= 2*n/(k+1)  Optimal allocation: n replicas to red.  Expected profit=n. Proportional Mean Not optimal n Pr(N=x) 1-1/k 1/k nk 2 x= Rochman, Levy, Brosh April 2013 demand

Reduction to single region S t.... Capacity, Weight Movie typeMovie type, # replicas 29Rochman, Levy, Brosh April 2013 Flow value= s

 Convert max to min Correctness 30Rochman, Levy, Brosh Original New If solution is April 2013

Correctness S t.... Capacity, Weight Movie type, # replicas 31Rochman, Levy, Brosh April 2013 < < Concavity! <

Reduction to multi region 32Rochman, Levy, Brosh  Convert max to min: Global Local April 2013 Semi-Separability!