## 3 System Modeling  
As shown in Figure 1, a Cloud consists of multiple Computing Workers and one Master Worker, where these computing workers are independently connected to the master worker in a star topology. Note that the data speed of the link connecting these computing workers and the master worker dictates the speed of communication processing when a task is transmitted. The computing workers could be located at remote distance, and they are operating in an independent heterogeneous mode since each individual computer worker may consist of one or multiple processors, which depend on the built-in capacity in terms of computing power, memory limitation and communication protocols.  
Upon receiving of an application or a user's request, the master worker divides the whole application into a sequence of tasks, which are defined as the minimum data unit to be processed at a computing worker. Since each task contains certain amount of overheads for transmission and computing purposes. To minimize the overheads as well as to achieve an optimum performance efficiency, the master worker includes a scheduler which must take into account of the capacity of the selected computing workers in terms of the capacity of the communication link between the master worker and the selected computing worker, the available memory capacity and the existing computing power in the process of task segmentation, schedule and distribution. On the other hand, the efficiency of scheduling and distributingprocess at the master is certainly affected by the status of operating process at the workers. This paper focuses on the statistics of computing process at the workers under various conditions, including the capacity of receiving computing tasks from the master, the capacity of computing processing at the worker and the capacity of transmitting tasks back to the master. From the best practice point of view, such statistics are able to provide valuable information for the master to dynamically control the process of scheduling the task, selecting the suitable workers and distributing the tasks to the selected workers using a scheduling algorithm [23, 25]. Note that the master worker can assign tasks to itself; _i.e._, the master itself can be a computing worker. The master worker collects partial results from the computing workers and combines them to constitute the final application's result.  
In the following analysis, a Cloud is considered to have \(R\) heterogeneous computing workers. A computing worker \(r\), \(r\in\{1,\ldots,R\}\), in the Cloud is modeled by a tandem processing system consisting of 3 components in series. The first component represents the task receiving processing capacity of \(\mu_{1,r}\) tasks per second, \(r\) is the index of computing worker. The second component represents the computing capacity of \(\mu_{2,r}\) tasks per second, and the third component represents the task transmission capacity of \(\mu_{3,r}\). It is assumed that all the computing workers can run in parallel, but our model does not impose this and computing workers may run at different stages during application execution.  
An application can be either dynamically submitted to the Cloud to run or it is statically there but a dynamic request of execution is submitted by the user. Applications' or execution requests' arrival is assumed to be a Poisson process with a mean rate \(\lambda\). Poisson is often an assumption made for the arrival of tasks to a master scheduler ([12, 26]). When an application arrives at the Cloud, the master worker segments the application into \(R\) tasks to be assigned for each individual computing worker. The task assigned to a computing worker \(r\) is \(\alpha_{r}\lambda\), which is also a Poisson process. Note that \(\alpha_{r}\) is the weight of a task load assigned to a computing worker \(r\) and \(\alpha_{r}=\sum_{r=1}^{r=R}\alpha_{r}=1\) is the whole application load.  
Figure 1: A Star-Topology Computing Platform Model.