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A queueing system model is considered, consisting of m (m ≥ 1) servicing devices and a maximum number of tasks with n (n ≥ 1) in the waiting queue. Each task is characterized by three random parameters (ν, β, ω), where ν is the number of servicing devices required to perform the task, β is the maximum time required to complete the task and ω is the possible time that the task can wait before assigning to run, after which it leaves the system without service. Tasks are accepted for service in the order of their entry into the system, i.e., FIFO (First-In-First-Out) discipline is used. In paper the equations are obtained for the state probabilities of the system in the stationary mode, which can serve as an assessment for real multiprocessor systems using MPI and OpenMP technologies.

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Cluster computing is becoming increasingly practical for high performance computing research and development. A computer cluster is a set of connected computers that work together so that, they can be viewed as a single system. Clusters offer a scalable means of linking computers together to provide an expansive environment for hosting enterprise applications. As the number of nodes in cluster configurations grows, the cluster administration becomes more challenging. We need to study the challenges of cluster management and to provide a solution. To have an effective cluster management we need to have an effective task scheduling algorithm. With the explosive growth of information, the demand on computing is sharply increasing. Due to a large number of computing tasks, the scheduling algorithm is an important part of cluster computing and has a great influence on the quality of claster service. In cluster computing, some large tasks may occupy too many resources and some small tasks may wait for a long time based on First-In-First-Out (FIFO) scheduling algorithm. This paper provides an overview of an improved scheduling algorithm that shortens the execution time of tasks and increases the resource utilization.

The queue state in multiprocessor computing systems is an actual problem for the process of optimal scheduling of tasks. In this paper, a system of equations is obtained describing the distribution of the queue for the system in a steady state. The resulting linear system of equations is solved using conventional numerical methods and can be used in schedulers.