The modern queueing theory is one of the powerful
tools for a quantitative and qualitative analysis of communication systems, computer networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing
systems, arising in the networks theory and communications theory
(called open queueing network). The authors of this research in the
sphere of queueing theory present the theorem about the law of the iterated logarithm (LIL) for the queue length of a customers in open
queueing network and its application to the mathematical model of
the open message switching system.