RESEARCH OF TWO SYSTEMS E2/H2/1 WITH ORDINARY AND SHIFTED DISTRIBUTIONS BY THE SPECTRAL DECOMPOSITION METHOD
Keywords:Delayed system, E2/H2/1 system, the average waiting time, Laplace transform, the spectral decomposition method.
ABSTRACT Context. In the queueing theory, the studies of G/G/1 systems are relevant because it is impossible to obtain solutions for the waiting time in the final form in the general case with arbitrary laws of distributions of the input flow and of the service time. Therefore, the study of such systems for particular cases of input distributions is important. The problem of deriving a solution for the average waiting time in a queue in closed form for a pair of systems with ordinary and with shifted Erlang and hyperexponential input distributions is considered.
Objective. Obtaining a solution for the main system characteristic – the average waiting time in queue for two queuing systems of type G/G/1 with conventional and with shifted second-order Erlang and Hyperexponential input distributions.
Method. To solve this problem, we used the classical spectral decomposition method for solving the Lindley integral equation, which plays an important role in the theory of G/G/1 systems. This method allows obtaining a solution for the average waiting time for the considered systems in a closed form. For the practical application of the obtained results, the well-known probability theory moments method is used.
Results. For the first time, spectral expansions of the solution of the Lindley integral equation are obtained for two systems, with the help of which the formulas for the average waiting time in the queue are derived in closed form.
Conclusions. Spectral expansions of the solution of the Lindley integral equation for the systems under consideration are obtained and their complete coincidence is proved. Consequently, the formulas for the average waiting time in the queue for these systems are the same, but with modified parameters. It is shown that in the system with a delay in time, the average waiting time is less than in a conventional system. The resulting for waiting time formulas expand and supplement the known queuing theory incomplete formula for the average waiting time for G/G/1 systems with arbitrary laws distributions of input flow and service time. This approach allows us to calculate the average latency for these systems in mathematical packages for a wide range of traffic parameters. All other characteristics of the systems are derived from the waiting time. In addition to the average waiting time, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time. The results are published for the first time.
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