A queuing network model for minimizing the total makespan of computational grids

Abstract

This paper offers a mathematical solution based on queuing theory and a generalized stochastic Petri net model to minimize the total makespan of the grid computing environments. A grid manager could minimize the total makespan through cautious distribution of subtasks to the grid resources. Subtask arrival rates depend on the arrival rate of the grid tasks submitted to the grid manager, local tasks directly submitted to the grid resources and the processing speed of the resources. Modeling the grid environment using queuing network, the steady state analysis of the network will result in the mean response time of the resources. Therefore, the total makespan could be minimized by minimizing the longest mean response time of the resources. The accuracy of the values obtained for the subtasks arrival rates at each of the grid resources from solving the corresponding queuing network could be further evaluated by steady state analysis of the generalized stochastic Petri net modeling the same grid environment.

Introduction

Grid computing is a technology to build dynamically constructed problem solving environments using geographically and organizationally distributed computational resources connected via communication links [1]. Grid computing provides supercomputing like power on demand, just as a power grid which provides electricity on demand. Typically, when talking about grids and grid environments, the mean is referring to grid computing or computational grids. Based on definition, a computational grid is a large collection of computers linked via the Internet so that their combined processing power can be harnessed to work on difficult or time consuming problems. With the advent of processors that obtain higher and higher performance measures and the emergence of open system operating systems such as Linux, inexpensive clusters of multiple processor systems for medium- and higher-scale computations are more possible and also cost effective. Computational grids allow one to control the extra central processing unit (CPU) cycles available on network and apply those CPU cycles to other, more resource-intensive purposes [2]. The overall motivation for most current grid projects is to enable the resource and human interactions that facilitate large-scale science and engineering such as aerospace systems design, high-energy physics data analysis, climate research, large scale remote instrument operation, collaborative astrophysics and so on [1].

High throughput computing (HTC) is of great concern to grid computing environments, because tasks submitted to a grid manager (GM) are mostly independent and relatively large [3], [4], [5]. HTC is aimed at minimizing the total completion time of all the tasks submitted to a GM in long periods of time, whilst high performance computing (HPC) prefers fast response to individual tasks [6], [7], [8]. To achieve HTC in grid computing environments, GM could keep track of availability and processing power of the grid resources. Thereby, GM could dispatch the subtasks produced by splitting an entire task to the appropriate grid resources considering the total makespan of the grid environment. The makespan of a resource is the time slot between the start and completion of the sequence of tasks assigned to the resource [9], [10], [11]. The total makespan of a computational grid is defined as the largest makespan of the grid resources. Minimizing the total makespan of a grid computing environment, the throughput of the environment can be increased accordingly [6], [11].

Various task scheduling algorithms and dispatching strategies have been proposed in grid computing environments to appropriately dispatch tasks within grid resources to reach a predefined target. This target which is known as a goal of scheduling has a close relationship with quality of service (QoS) parameters in grid environments. Generally, grid scheduling algorithms are presented to optimize at least one of the QoS parameters. The total makespan of the system is one of the most interesting QoS parameters in grid computing environments. Many task scheduling algorithms have been presented to dispatch the grid tasks to grid resources minimizing the total makespan of the environment [4], [6], [8], [9], [12], [13]. Nevertheless, almost none of these algorithms can find a deterministic solution for the scheduling problem, because scheduling tasks between the resources is an NP-complete problem [6], [8], [9], [10], [11], [12], [14]. So, in the best cases, the algorithms find a near optimal solution for the problem and therefore, the estimated value for makespan is not the minimum possible value. In order to overcome this difficulty, some other scheduling and dispatching schemes have been presented to relax the problem into one can be solved with mathematical solutions [15], [16], [17], [18], [19], [20], [21]. Using these approaches, various QoS measures can be taken into account and considered as a goal of scheduling. Furthermore, some realistic assumptions such as local tasks’ arrivals and grid environments’ topologies can be considered in these models in which had been ignored in previously mentioned algorithms.

In this paper, a mathematical approach based on queuing theory is presented to minimize the total makespan of the grid computing environments. To do this, grid resources and tasks should be mapped to the related concepts in queuing theory. Makespan of a grid resource could be considered as the mean response time of the server nodes in steady state analysis of the open queuing network (QN) modeling the grid computing environment [22], [23]. Each of the grid resources can be considered as M/M/1 queuing systems [15], [17], [18] within such QN. In the steady state analysis of M/M/1 queuing systems, the inverse of the difference of task arrival and service rates is equal to the mean response time of the server [24], [25]. To minimize the maximum mean response time among the resources’ mean response times, the subtasks arrival rates to each of the resources has to be adjusted. To achieve this, in our proposed approach, a set of inequalities describing the subtasks arrival constraints and equalities describing the overall tasks arrival rates are built. The major constraint in steady state analysis of QNs is that the subtasks arrival rate has to be less than the service rate in each of the resources; otherwise, there will be an infinite queue of tasks waiting for the resource services. In addition, the overall task arrival rate to the queuing network has to be equal to the sum of the subtask arrival rates to the resources. After finding this equality and inequality system, the Simplex method [26] could be applied to solve it.

In order to provide a formal description of the scheduling scheme in grid environments and graphically represent the grid scheduling workflow, a generalized stochastic Petri net (GSPN) model is offered in this paper. GSPNs are an abstract formal graph model useful for representing concurrency, synchronization and communication. The probabilistic nature of GSPNs allows such operations to be described with a high level of abstraction. Using this extension of Petri nets, one can model time consuming actions of the system (such as task processing in grids) and immediate actions (such as tasks dispatching in GMs) simultaneously. Modeling the grid task scheduling using GSPNs, the results obtained from queuing network solution can be evaluated further. The proposed GSPN models the arrival of grid and local tasks and dispatch of the subtasks in the grid computing environment. In addition, the services provided by the grid resources to execute the submitted tasks are modeled by the presented GSPN. Using steady state analysis of the proposed GSPN and examining various weights for immediate transitions, modeling subtasks distribution, the best dispatching weights for each of the grid resources can be obtained. Applying these weights to the grid tasks arrival rate, the subtasks arrival rates at each of the resources could be estimated. Adjusting the subtasks arrival rates using both the GSPN and mathematical solution of the corresponding QN, the total makespan of the grid environment could be minimized. In Section 5, it is shown that the subtasks arrival rates obtained from steady state analysis of the proposed GSPN are almost the same as the arrival rates obtained by solving the equality and inequality system resulted from corresponding QN.

The remaining parts of the paper are organized as follows: in Section 2, the related works done around the modeling and analysis of the grid environments using QNs and Petri nets are introduced. Sections 3 The proposed QN solution for grid computing environment, 4 The proposed GSPN model for grid computing environment present the proposed approaches for tasks dispatching in grid computing environments using open QNs and GSPNs, respectively. In Section 5, two numerical examples are presented to illustrate the proposed approaches and to show the applicability of them. Finally, Section 6 concludes the paper and presents future work which can be done in this research field.