Static assignment of complex stochastic tasks using stochastic majorization

We consider the problem of statically assigning many tasks to a (smaller) system of homogeneous processors, where a task's structure is modeled as a branching process, and all tasks are assumed to have identical behavior. We show how the theory of majorization can be used to obtain a partial order among possible task assignments. Our results show that if the vector of numbers of tasks assigned to each processor under one mapping is majorized by that of another mapping, then the former mapping is better than the latter with respect to a large number of objective functions. In particular, we show how measurements of finishing time, resource utilization, and reliability are all captured by the theory. We also show how the theory may be applied to the problem of partitioning a pool of processors for distribution among parallelizable tasks

Universality of Load Balancing Schemes on Diffusion Scale

We consider a system of $N$ parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among $d$ randomly selected servers otherwise $(1 \leq d \leq N)$. This load balancing scheme subsumes the so-called Join-the-Idle Queue (JIQ) policy $(d = 1)$ and the celebrated Join-the-Shortest Queue (JSQ) policy $(d = N)$ as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin-Whitt heavy-traffic regime, and establish that it does not depend on the value of $d$, implying that assigning tasks to idle servers is sufficient for diffusion level optimality

Asymptotically Optimal Load Balancing Topologies

We consider a system of $N$ servers inter-connected by some underlying graph topology $G_N$. Tasks arrive at the various servers as independent Poisson processes of rate $\lambda$. Each incoming task is irrevocably assigned to whichever server has the smallest number of tasks among the one where it appears and its neighbors in $G_N$. Tasks have unit-mean exponential service times and leave the system upon service completion. The above model has been extensively investigated in the case $G_N$ is a clique. Since the servers are exchangeable in that case, the queue length process is quite tractable, and it has been proved that for any $\lambda < 1$, the fraction of servers with two or more tasks vanishes in the limit as $N \to \infty$. For an arbitrary graph $G_N$, the lack of exchangeability severely complicates the analysis, and the queue length process tends to be worse than for a clique. Accordingly, a graph $G_N$ is said to be $N$-optimal or $\sqrt{N}$-optimal when the occupancy process on $G_N$ is equivalent to that on a clique on an $N$-scale or $\sqrt{N}$-scale, respectively. We prove that if $G_N$ is an Erd\H{o}s-R\'enyi random graph with average degree $d(N)$, then it is with high probability $N$-optimal and $\sqrt{N}$-optimal if $d(N) \to \infty$ and $d(N) / (\sqrt{N} \log(N)) \to \infty$ as $N \to \infty$, respectively. This demonstrates that optimality can be maintained at $N$-scale and $\sqrt{N}$-scale while reducing the number of connections by nearly a factor $N$ and $\sqrt{N} / \log(N)$ compared to a clique, provided the topology is suitably random. It is further shown that if $G_N$ contains $\Theta(N)$ bounded-degree nodes, then it cannot be $N$-optimal. In addition, we establish that an arbitrary graph $G_N$ is $N$-optimal when its minimum degree is $N - o(N)$, and may not be $N$-optimal even when its minimum degree is $c N + o(N)$ for any $0 < c < 1/2$.Comment: A few relevant results from arXiv:1612.00723 are included for convenienc

Tight Load Balancing via Randomized Local Search

We consider the following balls-into-bins process with $n$ bins and $m$ balls: each ball is equipped with a mutually independent exponential clock of rate 1. Whenever a ball's clock rings, the ball samples a random bin and moves there if the number of balls in the sampled bin is smaller than in its current bin. This simple process models a typical load balancing problem where users (balls) seek a selfish improvement of their assignment to resources (bins). From a game theoretic perspective, this is a randomized approach to the well-known Koutsoupias-Papadimitriou model, while it is known as randomized local search (RLS) in load balancing literature. Up to now, the best bound on the expected time to reach perfect balance was $O\left({(\ln n)}^2+\ln(n)\cdot n^2/m\right)$ due to Ganesh, Lilienthal, Manjunath, Proutiere, and Simatos (Load balancing via random local search in closed and open systems, Queueing Systems, 2012). We improve this to an asymptotically tight $O\left(\ln(n)+n^2/m\right)$. Our analysis is based on the crucial observation that performing "destructive moves" (reversals of RLS moves) cannot decrease the balancing time. This allows us to simplify problem instances and to ignore "inconvenient moves" in the analysis.Comment: 24 pages, 3 figures, preliminary version appeared in proceedings of 2017 IEEE International Parallel and Distributed Processing Symposium (IPDPS'17

Supermarket Model on Graphs

We consider a variation of the supermarket model in which the servers can communicate with their neighbors and where the neighborhood relationships are described in terms of a suitable graph. Tasks with unit-exponential service time distributions arrive at each vertex as independent Poisson processes with rate $\lambda$, and each task is irrevocably assigned to the shortest queue among the one it first appears and its $d-1$ randomly selected neighbors. This model has been extensively studied when the underlying graph is a clique in which case it reduces to the well known power-of-$d$ scheme. In particular, results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the size of the clique gets large, the occupancy process associated with the queue-lengths at the various servers converges to a deterministic limit described by an infinite system of ordinary differential equations (ODE). In this work, we consider settings where the underlying graph need not be a clique and is allowed to be suitably sparse. We show that if the minimum degree approaches infinity (however slowly) as the number of servers $N$ approaches infinity, and the ratio between the maximum degree and the minimum degree in each connected component approaches 1 uniformly, the occupancy process converges to the same system of ODE as the classical supermarket model. In particular, the asymptotic behavior of the occupancy process is insensitive to the precise network topology. We also study the case where the graph sequence is random, with the $N$-th graph given as an Erd\H{o}s-R\'enyi random graph on $N$ vertices with average degree $c(N)$. Annealed convergence of the occupancy process to the same deterministic limit is established under the condition $c(N)\to\infty$, and under a stronger condition $c(N)/\ln N\to\infty$, convergence (in probability) is shown for almost every realization of the random graph.Comment: 32 page

Scalable Load Balancing Algorithms in Networked Systems

A fundamental challenge in large-scale networked systems viz., data centers and cloud networks is to distribute tasks to a pool of servers, using minimal instantaneous state information, while providing excellent delay performance. In this thesis we design and analyze load balancing algorithms that aim to achieve a highly efficient distribution of tasks, optimize server utilization, and minimize communication overhead.Comment: Ph.D. thesi

Communication Patterns for Randomized Algorithms

Examples of large scale networks include the Internet, peer-to-peer networks, parallel computing systems, cloud computing systems, sensor networks, and social networks. Efficient dissemination of information in large networks such as these is a funda- mental problem. In many scenarios the gathering of information by a centralised controller can be impractical. When designing and analysing distributed algorithms we must consider the limitations imposed by the heterogeneity of devices in the networks. Devices may have limited computational ability or space. This makes randomised algorithms attractive solutions. Randomised algorithms can often be simpler and easier to implement than their deterministic counterparts. This thesis analyses the effect of communication patterns on the performance of distributed randomised algorithms. We study randomized algorithms with application to three different areas. Firstly, we study a generalization of the balls-into-bins game. Balls into bins games have been used to analyse randomised load balancing. Under the Greedy[d] allocation scheme each ball queries the load of d random bins and is then allocated to the least loaded of them. We consider an infinite, parallel setting where expectedly λn balls are allocated in parallel according to the Greedy[d] allocation scheme in to n bins and subsequently each non-empty bin removes a ball. Our results show that for d = 1,2, the Greedy[d] allocation scheme is self-stabilizing and that in any round the maximum system load for high arrival rates is exponentially smaller for d = 2 compared to d = 1 (w.h.p). Secondly, we introduce protocols that solve the plurality consensus problem on arbitrary graphs for arbitrarily small bias. Typically, protocols depend heavily on the employed communication mechanism. Our protocols are based on an interest- ing relationship between plurality consensus and distributed load balancing. This relationship allows us to design protocols that are both time and space efficient and generalize the state of the art for a large range of problem parameters. Finally, we investigate the effect of restricting the communication of the classical PULL algorithm for randomised rumour spreading. Rumour spreading (broadcast) is a fundamental task in distributed computing. Under the classical PULL algo- rithm, a node with the rumour that receives multiple requests is able to respond to all of them in a given round. Our model restricts nodes such that they can re- spond to at most one request per round. Our results show that the restricted PULL algorithm is optimal for several graph classes such as complete graphs, expanders, random graphs and several Cayley graphs

Adaptive Algorithms for Coverage Control and Space Partitioning in Mobile Robotic Networks

We consider deployment problems where a mobile robotic network must optimize its configuration in a distributed way in order to minimize a steady-state cost function that depends on the spatial distribution of certain probabilistic events of interest. Three classes of problems are discussed in detail: coverage control problems, spatial partitioning problems, and dynamic vehicle routing problems. Moreover, we assume that the event distribution is a priori unknown, and can only be progressively inferred from the observation of the location of the actual event occurrences. For each problem we present distributed stochastic gradient algorithms that optimize the performance objective. The stochastic gradient view simplifies and generalizes previously proposed solutions, and is applicable to new complex scenarios, for example adaptive coverage involving heterogeneous agents. Finally, our algorithms often take the form of simple distributed rules that could be implemented on resource-limited platforms

Resource Allocation in Distributed Service Networks

The past few years have witnessed significant growth in the use of distributed network analytics involving agile code, data and computational resources. In many such networked systems, for example, Internet of Things (IoT), a large number of smart devices, sensors, processing and storage resources are widely distributed in a geographic region. These devices and resources distributed over a physical space are collectively called a distributed service network. Efficient resource allocation in such high performance service networks remains one of the most critical problems. In this thesis, we model and optimize the allocation of resources in a distributed service network. This thesis contributes to two different types of service networks: caching, and spatial networks; and develops new techniques that optimize the overall performance of these services. First, we propose a new method to compute an upper bound on hit probability for all non-anticipative caching policies in a distributed caching system. We find our bound to be tighter than state-of-the-art upper bounds for a variety of content request arrival processes. We then develop a utility based framework for content placement in a cache network for efficient and fair allocation of caching resources. We develop provably optimal distributed algorithms that operate at each network cache to maximize the overall network utility. Next, we develop and evaluate assignment policies that allocate resources to users with a goal to minimize the expected distance traveled by a user request, where both resources and users are located on a line. Lastly, we design and evaluate resource proximity aware user-request allocation policies with a goal to reduce the implementation cost associated with moving a request/job to/from its allocated resource while balancing the number of requests allocated to a resource. Depending on the topology, our proposed policies achieve a 8% - 99% decrease in implementation cost as compared to the state-of-the-art