On the law of the iterated logarithm in hybrid multiphase queueing systems

Whitby in his book about artificial intelligence [21] states that the human brain consists of 100 networks of networks (NoN). One of the examples of NoN is a hybrid multiphase queueing system (HMQS). At first, we present the summary of works dedicated to a particular case of HMQS and multiphase queueing system (MQS, see Fig. 1). One can apply limit theorems for a waiting time of a customer and a queue length of customers to get probabilistic characteristics of MQS under various conditions of heavy traffic [2, 3]. The most fundamental example (a single-phase case, where the time intervals in between the arrivals of customers to MQS are independent identically distributed random variables and there is a single device, working independently of the output in heavy traffic) has been completely investigated by several authors [2, 8]. Iglehart [5] carefully investigated a single-device case and obtained laws of the iterated logarithm (LIL) for this case. It is surprising to note that the fundamental results of Iglehart on the queueing systems, working in heavy traffic are rarely used [4–8]. There are only a few papers on the theory of MQS in heavy traffic [10, 12, 13] with, however, no proof of LIL for the probabilistic characteristics of MQS in heavy traffic. LIL for a cumulative waiting time