On the Convergence of Workload in Service System to Brownian Motion with Switching Variance
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ON THE CONVERGENCE OF WORKLOAD IN SERVICE SYSTEM TO BROWNIAN MOTION WITH SWITCHING VARIANCE E. S. Garai∗
UDC 519.2
A modification of service system model introduced by I. Kaj and M. S. Taqqu is considered. This model describes the dynamics in time and space of various system workloads created by a set of service processes. In the model under consideration, two types of resource having its own distribution are used. Such a model can be identified with the presence of two operators of the resource. At the time of the active operator failure, one can switch to another operator whose resource has distribution workloads different from the first operator. A limit theorem on the convergence of finite-dimensional distributions of the integral workload process with two types of resource to Brownian motion with switching variance is proved. Bibliography: 7 titles.
1. Introduction In the last decade, mathematical modeling of computer systems based on high-speed connections, such as the Internet, have become actual. These models describe dynamics in time and space of various workloads in a system created by a set of service processes. The mathematical model described below is interesting, because it uses two types of resources each of which has its own distribution. Such a model can be identified with the presence of two resource operators. At the time of the active operator failure one can switch to another operator whose resource has distribution different from the distribution of the first operator. The distribution of the duration of the service processes for these operators may vary. At the initial moment, a particular operator is selected. It is assumed that the consumption of the resource of each type can be interrupted with small probability. It is also assumed that the operator can instantly restore the operation of the system, and hence two operators are enough to provide uninterrupted service. However, more complicated systems using the resource of three or more operators can also be considered. The fundamental results in this topic are obtained by I. Kaj and M. S. Taqqu in [1]. A detailed presentation of the mathematical model can be found in two monographs of M. A. Lifshits [2, 3]. 2. Description of the mathematical model of service system With each service process, we associate random variables characterizing an initial moment, a duration, and a required amount of resources. We assume that these values for different service processes are identically distributed and independent, and the duration and resource within one service process are independent. Assume further that there are two types of resource in the service system, i.e., two sets of i.i.d. positive random variables in each set. It may happen that the consumption of resource of one type or another can be interrupted with small probability. In this case, the system switches to the service with resource of the other type. It is natural to assume that the mechanism of the initial resource failure is determined by i.i.d. Bernoulli’s variables independent of the res
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