## 1 Introduction  
Recent technological advancements and the widespread adoption of smart devices have given rise to a plethora of innovative applications, including augmented reality (AR) [1], virtual reality (VR) [2], automatic driving [3], real-time video analytics [4], and mobile online gaming [5], which are complex, computationally intensive, and highly energy demanding [6]. Despite being equipped with advanced computation and communication technology, smart devices or mobile devices (MDs) have limitations, in terms of energy resources, computational power, and memory [7]. These constraints constitute a significant challenge for processing computation-intensive and time-sensitive applications in MDs [8]. An effective strategy is to transfer computation-intensive tasks from these resource-constrained devices to devices with a more powerful computational capacity, i.e., mobile edge computing (MEC) servers that are situated at the network's edge [9; 10; 11]. This strategy enables faster processing of data. By transferring the computation-intensive tasks to MEC servers, MDs can reduce the burden of handling resource-intensive computations. They can instead concentrate on carrying out lightweight operations. As a result, this strategy not only increases overall system capability while consuming less energy but also extends the battery life of the MDs [12]. Even though MEC offers several advantages, it is imperative to consider the delay introduced by the physical distance between MDs and MEC servers [13; 14], especially in time-sensitive applications (e.g., AR and VR). In the context of AR and VR applications, even minor delays can impede the seamless integration of the virtual and physical worlds, ultimately affecting user engagement and overall experience. Moreover, during crowded events or festivals, offloading tasks to MEC servers becomes challenging, due to increased network congestion and reduced resource capacity. It is also difficult to establish a connection that is reliable between MDs and MEC servers because of the increased user activity. Therefore, unmanned aerial vehicles (UAVs) equipped with MEC (UAV-MEC) servers can be employed as a potential solution, to provide communication and computation services to the MDs [15; 16; 17; 18]. Deploying UAVs near MDs significantly reduces data processing time, making them suitable for time-sensitive applications that require real-time analysis. UAVs can adapt their position and coverage based on user density and location, ensuring better service quality. Moreover, they establish direct communication links with MDs. The performance of the MEC system is, therefore, expected to be significantly enhanced by the deployment of UAVs. A lot of research has been conducted on UAV location optimization, focusing on two types of deployment scenarios: static deployment [19; 20; 21; 22] and mobile deployment [23; 24; 25; 26; 27] in 2D or 3D environments. UAVs serve fixed or predefined areas for specific purposes, using static deployment. Mobile UAV deployment involves UAVs with the ability to change locations and relocate as necessary. However, existing research problems have not explored the potential advantages of using UAVs with ellipsoidal movement, which allows dynamic adjustments in position, trajectory, and shape within a predefined area. This adaptability is essential for scenarios with varying user densities, allowing UAVs to increase coverage, serve more MDs, reduce data transmission distances, and expedite task execution. In contrast to task-offloading studies [28; 29; 30], which often simplify user scenarios by assigning a single task per user, our research addresses the task offloading request rate, considering scenarios where MDs may have multiple tasks or no tasks. When MDs offload tasks, task transmission consumes their energy resources, which is further influenced by the relative positions of the UAVs and MDs. The increased energy consumption can significantly reduce the MD's battery life, potentially impacting the network's overall longevity.