## 1 Introduction  
Cloud computing has allowed for the dynamic scaling of computing resources by the end user in a fast, reliable, and simplified way. The end user no longer needs to manage the computing infrastructure they are using and its security [1, 2]. There are many types of services that can be provided by cloud computing, such as laaS, PaaS, SaaS, etc. [3]. These services have allowed cloud computing to become exceedingly important and dominant in the worldwide computing landscape, so much so that the annual cost expended on cloud computing has exceeded USD 595 billion in 2024 and, in 2025, it is expected to increase to over USD 723 billion [4]. In a survey performed on cloud users, most cloud users reported reducing the cost of public cloud usage to be one of their major concerns [5]. All these factors have made reducing the cost of processing workloads in cloud computing an important target.  
Task scheduling is used to optimize the utilization of cloud resources to improve parameters such as cost. Task scheduling for independent tasks can be classified into single machine scheduling and parallel machine scheduling. Single machine scheduling is the simplest form of task scheduling but, nevertheless, has important parallels with scheduling over multiple machines. Parallel machine scheduling concerns the scheduling of independent tasks over multiple machines in parallel. The machines may or may not have identical configurations [6]. Task scheduling has been an active area of research in cloud computing, with many different approaches being proposed [7]. Many task scheduling approaches have also been proposed that use fuzzy logic either by itself or in conjunction with other approaches [8]. As demonstrated in the next section, existing heuristic, metaheuristic, and fuzzy-based works do not try to utilize variable numbers of VMs. It is a fact that automated and fast elasticity is one of the core characteristics of cloud computing [1]. In light of this fact, the incapability of a task scheduling scheme to vary the number of VMs seems to be a gap that has not been given sufficient focus in the existing literature.  
In this study, the problem of task scheduling in a public cloud in a scenario where the tasks have associated priority and deadline constraints is considered. The scenario further involves the requirement that the SLA of high-priority tasks needs to be met, even at elevated costs. To address this scenario, the Fuzzy Priority Deadline (FPD) approach is proposed, which seeks to compulsorily meet the SLA deadline of high-priority tasks. It also seeks to determine the number of VMs needed to process a given task set such that the task scheduling scheme is able to meet the SLA deadline of high-priority tasks. The key contributions of the present study are summarized as follows:  
* A two-stage task scheduling approach called FPD is proposed using a fuzzy controller to process a task set on a public cloud such that the deadline SLA deadline of high-priority tasks is met at any cost.
* The proposed fuzzy controller is able to provide the number of VMs needed and the tasks that are to be mapped to each VM.
* The proposed approach is able to ensure the SLA requirement of high-priority tasks under all input sizes.
* A comparison with established heuristic-based approaches is performed for synthetic and Google cluster-like traces, here the proposed approach demonstrates superior performance over multiple performance measures.  
The rest of this paper is organized as follows: Section 2 covers the related works from the published literature, Section 3 explains the working model of the proposed approach and provides details of experimental setup, and Section 4 presents the threshold adjustment of the proposed fuzzy controller and comparative results obtained with other approaches. Finally, Section 5 provides a discussion about the results of the present study, as well as its limitations and future research directions.