Queuing Systems with a Time Lag

The article discusses various queuing systems (QS) formed by four laws of probability distributions: exponential, hyperexponential, Erlang and hyper-Erlang of the second order. These four laws form sixteen different QS. In contrast to the classical theory, this article considers QS with distribution laws shifted to the right from the zero point. Such QS are of type G/G/1 with arbitrary laws of the distribution of intervals between the requirements of the input flow and the service time. As you know, for such systems it is impossible to obtain solutions for the main characteristic of QS the average waiting time in the general case. Therefore, studies of such systems are important for special cases of distribution laws. The article provides an overview of the author's results for the average waiting time in a queue in a closed form for systems with input distributions shifted to the right from the zero point. To solve this problem, the spectral decomposition method for solving the Lindley integral equation was used. In the course of solving the problem, spectral decompositions of the solution of the Lindley integral equation for eight systems were obtained and with their help calculation formulas were derived for the average waiting time in the queue. It is shown that in systems with delay, the average waiting time is shorter than in conventional systems. The obtained calculation formulas for the average waiting time expand and complement the well-known incomplete formula of the queuing theory for the average waiting time for G/G/1 systems. The proposed approach allows us to calculate the average value and moments of higher orders of waiting time for these systems in mathematical packages for a wide range of changes in traffic parameters. Given the fact that the variation in packet delay (jitter) in the telecommunications standard is defined as the spread of waiting time around its average value, the jitter can be determined through the variance of the waiting time.