On the global values of the queue length in open queueing networks

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. Open queueing networks are of special interest both in theory and in practical applications (packet switch structures, cellular mobile networks, message switching systems, retransmission of video images, assembly lines, etc.). In this paper, we deal with approximations of an open queueing network and present a heavy traffic limit theorem for the global value of the queue length of customers in an open queueing network. A functional limit theorem is proved for the global value of the queue length of customers – an important probability characteristic of the queueing system under conditions of heavy traffic.