Correlations of the Sojourn Times of Subtasks in Fork-Join Queueing Systems with M|M|1-type Subsystems

Abstract

The article analyzes the relationship between the residence times of subtasks in the fork-join subsystems of the queuing system. When arriving the fork-join system, the task is divided into subtasks, each of which is serviced in its own subsystem, the task is considered serviced after the completion of all subtasks that originally comprised it servicing.
There is a dependence between the residence times of subtasks in subsystems, which affects the main performance indicators of the system, for example, the response times, which greatly complicates their analysis. The paper examines the characteristics of the existing dependence. In particular, with the help of generating functions and the Laplace-Stieltjes transformation, exact expressions for the Pearson and Spearman correlation coefficients are obtained.
In addition, using a combination of several methods, including the Nelder-Mead optimization method, the estimation of the Kendall correlation coefficient was obtained, and the model for the response time estimation based on the resulting correlation coefficients was described.
Despite the many works on the study of fork-join queuing systems, there are practically no articles devoted to the correlation analysis of the temporal metrics of the model. Therefore, this article can become one of the first in this area, laying the foundation for the study of the correlation dependence and its influence on the performance parameters of the model, not only for the classical case of M|M|1 subsystems, but also for more complex architectures of this system.