On the law of iterated logarithm for extreme queue length in an open queueing network

The purpose of this research in the field of the open queueing network is to prove the Law of the Iterated Logarithm (LIL) for the extreme value of the queue length of customers in an open queueing network. LIL is proved for the extreme values of the queue length of customers the important probability characteristic of the queueing system under conditions of heavy traffic. Also, we present for extreme queue length of jobs Probability Laws ((theorems on the LIL, Fluid Limits Theorem and Diffusion Limit Theorem) in various conditions of traffic and simulating an open queueing network in Appendices 1 and 2.