(m,n)-Semirings and a Generalized Fault Tolerance Algebra of Systems

We propose a new class of mathematical structures called (m,n)-semirings} (which generalize the usual semirings), and describe their basic properties. We also define partial ordering, and generalize the concepts of congruence, homomorphism, ideals, etc., for (m,n)-semirings. Following earlier work by Rao, we consider a system as made up of several components whose failures may cause it to fail, and represent the set of systems algebraically as an (m,n)-semiring. Based on the characteristics of these components we present a formalism to compare the fault tolerance behaviour of two systems using our framework of a partially ordered (m,n)-semiring.Comment: 26 pages; extension of arXiv:0907.3194v1 [math.GM

Communication-efficient Distributed Multi-resource Allocation

In several smart city applications, multiple resources must be allocated among competing agents that are coupled through such shared resources and are constrained --- either through limitations of communication infrastructure or privacy considerations. We propose a distributed algorithm to solve such distributed multi-resource allocation problems with no direct inter-agent communication. We do so by extending a recently introduced additive-increase multiplicative-decrease (AIMD) algorithm, which only uses very little communication between the system and agents. Namely, a control unit broadcasts a one-bit signal to agents whenever one of the allocated resources exceeds capacity. Agents then respond to this signal in a probabilistic manner. In the proposed algorithm, each agent makes decision of its resource demand locally and an agent is unaware of the resource allocation of other agents. In empirical results, we observe that the average allocations converge over time to optimal allocations.Comment: To appear in IEEE International Smart Cities Conference (ISC2 2018), Kansas City, USA, September, 2018. arXiv admin note: substantial text overlap with arXiv:1711.0197

Derandomized Distributed Multi-resource Allocation with Little Communication Overhead

We study a class of distributed optimization problems for multiple shared resource allocation in Internet-connected devices. We propose a derandomized version of an existing stochastic additive-increase and multiplicative-decrease (AIMD) algorithm. The proposed solution uses one bit feedback signal for each resource between the system and the Internet-connected devices and does not require inter-device communication. Additionally, the Internet-connected devices do not compromise their privacy and the solution does not dependent on the number of participating devices. In the system, each Internet-connected device has private cost functions which are strictly convex, twice continuously differentiable and increasing. We show empirically that the long-term average allocations of multiple shared resources converge to optimal allocations and the system achieves minimum social cost. Furthermore, we show that the proposed derandomized AIMD algorithm converges faster than the stochastic AIMD algorithm and both the approaches provide approximately same solutions

Distributed Algorithms for Internet-of-Things-enabled Prosumer Markets: A Control Theoretic Perspective

Internet-of-Things (IoT) enables the development of sharing economy applications. In many sharing economy scenarios, agents both produce as well as consume a resource; we call them prosumers. A community of prosumers agrees to sell excess resource to another community in a prosumer market. In this chapter, we propose a control theoretic approach to regulate the number of prosumers in a prosumer community, where each prosumer has a cost function that is coupled through its time-averaged production and consumption of the resource. Furthermore, each prosumer runs its distributed algorithm and takes only binary decisions in a probabilistic way, whether to produce one unit of the resource or not and to consume one unit of the resource or not. In the proposed approach, prosumers do not explicitly exchange information with each other due to privacy reasons, but little exchange of information is required for feedback signals, broadcast by a central agency. In the proposed approach, prosumers achieve the optimal values asymptotically. Furthermore, the proposed approach is suitable to implement in an IoT context with minimal demands on infrastructure. We describe two use cases; community-based car sharing and collaborative energy storage for prosumer markets. We also present simulation results to check the efficacy of the algorithms.Comment: To appear as a chapter in "Analytics for the Sharing Economy: Mathematics, Engineering and Business Perspectives", Editors: E. Crisostomi et al., Springer, 2019 (forthcoming book

The Convergence of Finite-Averaging of AIMD for Distributed Heterogeneous Resource Allocations

In several social choice problems, agents collectively make decisions over the allocation of multiple divisible and heterogeneous resources with capacity constraints to maximize utilitarian social welfare. The agents are constrained through computational or communication resources or privacy considerations. In this paper, we analyze the convergence of a recently proposed distributed solution that allocates such resources to agents with minimal communication. It is based on the randomized additive-increase and multiplicative-decrease (AIMD) algorithm. The agents are not required to exchange information with each other, but little with a central agent that keeps track of the aggregate resource allocated at a time. We formulate the time-averaged allocations over finite window size and model the system as a Markov chain with place-dependent probabilities. Furthermore, we show that the time-averaged allocations vector converges to a unique invariant measure, and also, the ergodic property holds

Communication-efficient Distributed Multi-resource Allocation

Distributed resource allocation arises in many application domains such as smart cities, intelligent transportation systems, sharing economy, cloud computing, edge-computing, power systems, etcetera. In several scenarios, the agents such as Internet of Things (IoT) devices may require multiple shared resources to achieve social optimum values; furthermore, they may have heterogeneous resource demands. Such distributed resource allocation problems are challenging to solve, especially when the agents are constrained through communication infrastructure, computational capabilities or do not wish to communicate with other agents in the network due to privacy reasons. Additionally, when the cost functions of agents are non-separable and are coupled through the allocation of multiple resources, in such cases, the single resource allocation algorithms are not efficient and provide suboptimal solutions. In the available distributed solutions for multiple resources, best to my knowledge, agents exchange their information with at least one neighboring agent that may incur communication overhead or compromise agents' privacy. We develop several solutions to solve such problems for multiple divisible and multiple indivisible resources wherein no inter-agent communication is required. Moreover, we assume that each agent has private cost functions coupled with multiple resources; these functions are strictly convex, twice continuously differentiable, and increasing in each variable. Our first contribution is the stochastic distributed algorithm that solves multi-resource allocation problems with no direct agent-to-agent communication for divisible resources; moreover, it achieves social-optimum values. In the algorithms, each agent decides its resource demands locally, and an agent is unaware of the resource allocations of other agents. We solve the divisible multi-resource allocation problem by extending the additive-increase multiplicative-decrease (AIMD) algorithm for single resource allocation by Wirth and co-authors. In the algorithm, the agents keep increasing the demands for a resource linearly until they receive a one-bit signal from a central agency. The central agency broadcasts the signal whenever one of the allocated resources reaches its capacity. Agents then respond to this signal in a probabilistic manner to decrease the resource demand. By doing so, the social optimum is achieved in long-term averages. Our second contribution is a derandomized AIMD algorithm to solve a class of distributed optimization problems for multiple divisible shared resources. The algorithm is a derandomized version of the stochastic additive-increase and multiplicative-decrease (AIMD) algorithm. The developed solution uses a one-bit feedback signal for each resource between the system and the agents, and it does not require inter-agent communication. We show empirically that the long-term average allocations of multiple shared resources converge to optimal allocations, and the system achieves minimum social cost. Furthermore, we show that the derandomized AIMD algorithm converges faster than the stochastic AIMD algorithm, and both approaches provide approximately the same solutions. Our third contribution is the stochastic algorithm for multiple indivisible (unit-demand) resources, inspired by classical stochastic approximation techniques. Each agent's consumption is modeled as a Bernoulli random variable in the solution, and no inter-agent communication is required. Moreover, we provide fundamental guarantees of convergence. Additionally, we present an example illustrating the performance of the algorithm. Finally, we study the development of Internet-of-Things (IoT) enabled sharing economy applications. In many sharing economy scenarios, agents both produce as well as consume a resource; we call them prosumers. A community of prosumers agrees to sell excess resources to another community in a prosumer market. We propose a stochastic algorithm to regulate the number of prosumers in a prosumer community; each prosumer has a cost function coupled through its time-averaged production and consumption of the resource. Furthermore, each prosumer runs its distributed algorithm and takes (binary) decisions in a probabilistic way, whether to produce one unit of the resource or not and to consume one resource unit or not. In the developed approach, prosumers do not explicitly exchange information with each other due to privacy reasons but little with a central agency. Additionally, prosumers achieve the optimal values asymptotically