This article addresses the Edge User Allocation (EUA) problem in edge computing, where the appropriate mapping from users to edge servers (ESs) is crucial for optimizing performance metrics. Game theory is one of the powerful tools used in edge computing for user allocation to edge resources and task offloading. However, this approach takes longer to converge to Pure Nash Equilibrium (PNE), which is called a stable optimal solution. In this article, we propose a grouping technique for ESs, enabling parallel execution of game-theoretic approaches to achieve faster convergence at the PNE. Our contributions include the grouping method, the introduction of parallel Best Response (BR) dynamics for rapid convergence, and proof that the parallel BR dynamics will eventually halt at PNE. We also provide empirical evidence demonstrating its efficiency compared to traditional BR dynamics. This research enhances the scalability and effectiveness of game-theoretic approaches in resolving the EUA problem, offering practical solutions for edge computing scenarios.

In edge computing, the app vendors hire resources from edge servers and allocate them to app users to overcome the challenge of the limited computing capacities of their IoT devices. An app vendor intends to provide app services to the maximum number of users with the least number of edge servers in order to make efficient use of edge resources while reducing overall system costs. However, when an edge server has to serve more app users than its capacity, the Quality of Service (QoS) deteriorates. Thus, establishing a tradeoff between cost and QoS is a critical challenge in the process of allocating edge computing resources to users. It is referred to as the app user allocation (AUA) problem. To solve the AUA problem, we propose a distributed game-theoretic approach that finds a pure Nash equilibrium (PNE) as the optimal stable solution. We first model the AUA problem as a constrained optimization problem and then introduce a user allocation game (UAGame) to solve it. This UAGame employs a distributed edge server allocation (ESA) algorithm to reach PNE. The time complexity of the ESA algorithm is reduced by the edge server clustering. It has also been shown that the UAGame is a potential game, and therefore the ESA algorithm is guaranteed to converge at PNE. The performance of the ESA algorithm has also been studied theoretically and validated numerically.

Internet of Things (IoT) devices have enabled communications in resource-limited computing environments. Sensor nodes from the multiple IoT devices collectively work for many applications, such as disaster management, border security management, smart farming, smart cities, etc. In such applications, the data from a single source node is often destined for multiple nodes. Multicast communication is preferred over unicast or broadcast communication for such applications, as multicast uses fewer resources. Efficient construction of the multicast tree leads to cost-effective multicast transmission. This article introduces a path selection game (PSGame), a game-theoretic approach that formulates the construction problem of the least-cost multicast tree as a potential game. Our proposed path selection algorithm (PSA) quickly converges to the pure Nash equilibrium (PNE), bringing the least cost multicast tree. Our findings show that the overhead incurred in terms of energy consumption and delay is minimal in the proposed algorithm compared to other mechanisms. The theoretical analysis proves that the proposed algorithm quickly converges to PNE in$O(n.r_{\max })$steps. It also proves that the cost ratio between the proposed solution and the centralized optimum will be bounded by$\log (n)$. The numerical analysis substantiates the theoretical analysis.

The research paper [1] proposes a differential privacy algorithm in the context of Federated Learning and provides its performance analysis, mainly focusing on proving a convergence bound for the loss function. In this paper, we show that some of the mathematical derivations given in [1] are not valid. Thus the bounds they prove in the paper do not hold for all loss functions. In this work, we give the correct derivation of the best possible local sensitivity bound, which is valid for all loss functions. We also state the modifications in the bounds for global sensitivity and the standard deviation of the Gaussian noise added, both before and after aggregation.