Balls into Bins: strict Capacities and Edge Weights

We explore a novel theoretical model for studying the performance of distributed storage management systems where the data-centers have limited capacities (as compared to storage space requested by the users). Prior schemes such as Balls-into-bins (used for load balancing) neither consider bin (consumer) capacities (multiple balls into a bin) nor the future performance of the system after, balls (producer requests) are allocated to bins and restrict number of balls as a function of the number of bins. Our problem consists of finding an optimal assignment of the online producer requests to consumers (via weighted edges) in a complete bipartite graph while ensuring that the total size of request assigned on a consumer is limited by its capacity. The metric used to measure the performance in this model is the (minimization of) weighted sum of the requests assigned on the edges (loads) and their corresponding weights. We first explore the optimal offline algorithms followed by the analysis of different online techniques (by comparing their performance against the optimal offline solution). LP and Primal-Dual algorithms are used for calculating the optimal offline solution in O(r · n) time (where r and n are the number of requests and consumers respectively) while randomized algorithms are used for the online case. We propose randomized online algorithms in which the consumers are selected based on edge probabilities (that can change with consumer failures; due to capacity exhaustion) and evaluate the performance of these randomized schemes using probabilistic analysis. The performance of the online algorithms is measured using competitive analysis assuming an oblivious adversary who knows the randomized algorithm but not the results produced. For the simplified model with equal consumer capacities an average-case competitive ratio (which compares the average cost of the output produced by the online algorithm and the minimum cost of the optimal offline solution) of ( d mini, j di, j ) (where d is the edge weight / distance) is achieved using an algorithm that has equal probability for selecting any of the available edges with a running time of O(r). In the extending the model to arbitrary consumer capacities we show an average case competitive ratio of ( d·c c·mini, j di, j ) . This theoretical model gives insights to a (storage) cloud system designer about, how the different attributes (producer requests, edge weights and consumer capacities) effect the overall (read / write) performance of a distributed storage management system over a period of time.