## 1 Introduction  
The capacity of today's infrastructures and data centers, the ubiquity of network resources, and the low storage cost has led to the emergence of a new field which is the Cloud Computing [1; 2; 3]. This new field has emerged with the directions, in which computation philosophy has shifted from the use of a personal computer or an individual server to a cloud of distributed resources. The main objective is to draw benefits from the underlying infrastructure services to satisfy a Service Level Agreement [4] and a required performance to users [5]. Many scientific and engineering applications present a high complexity; _i.e._, the number of operations generated is very high, which limits the benefits from the performance of a Cloud infrastructure if such an application runs sequentially on one node of the Cloud. If an application is divisible into a number of tasks, then parallelizing it on a number of workers within the Cloud would decrease the execution time or makespan of the application. The producer-consumer pattern [6], also called master-worker, represents a parallelization strategy that is suitable for many of the scientific and engineering applications. For instance, it is used in Google map-reduce programming model [7]; _i.e._, the rationale of why we have chosen the master-worker model as a base model. The master-worker model arises in divisible-load applications, where there is no communication between the workers, such as search for a pattern, compression, join, graph coloring and generic search applications [8], convolution-based applications [9], and image processing applications [10]. Furthermore, as stated in [5], a Cloud is considered as a candidate platform to run heavy-load applications to satisfy performance requirements. Consequently, there is a great interest to run master-worker applications in the Cloud. However, running a divisible load application using a master-worker strategy in a Cloud of nodes face challenges to obtain an optimal performance and analyze the performance of the application for an optimal scheduling of the application's partitions. In the master-worker model, every computing worker processes the load received and transmits the results back to the master worker, which in turn, in a next iteration, performs some data preparations and sends news loads until the application is completed. In this case, the network capacity, the Cloud topology, and the computing performance of individual computing workers must be taken into account to model the system and the tasks' processing. In this work, we use those system parameters and we derive formulas to calculate the blocking probability and the processing time of computing workers, thus providing a dynamic feedback to a scheduler which dynamically alters its resource allocation strategy for optimal performance and better system utilization. By blocking probability, we mean the probability of a task being in a waiting state and blocking all other tasks behind. Our model can also be used as a performance analysis tool to software engineers and Cloud providers to predict the behavior of a divisible load application on a set of Cloud nodes and consequently the system utilization. Although our model and derived formulas can be used for any distributed environment, our model is more useful in a cloud structure where resources are shared in space and time and thus blocking probability is higher as we believe, and an adjustment of scheduling is needed more often. In this paper, we model a Star Cloud topology. However, the mechanisms developed in this paper can be applied to other topologies as well. For instance, the master-worker model can be used in a ring topology; transmissions and computing processing capacities of computing workers should be considered. Several works studied the problem of finding an optimal scheduling algorithm based on system parameters.