In this paper a new mathematical model is proposed for task scheduling and resource allocation in Grid systems. In this novel model, load balancing, starvation prevention and failing strategies are stated as the constraints and the solution is restricted with a predefined quality of service for users with different priorities. These strategies are defined by resource providers based on the amount of submitted jobs to Grid. To solve the proposed model, a modern approximate Auction-based algorithm is developed and it is implemented as a prototype of Grid simulator namely Multi-S-Grid. The results are illustrated on 18 different large-scale Grid systems with different random capabilities and different users. The outcomes reveal the reasonable performance of the proposed Auction-based algorithm to solve Grid system optimization models.