TWO PAIRS OF DUAL QUEUEING SYSTEMS WITH CONVENTIONAL AND SHIFTED DISTRIBUTION LAWS
Keywords:Erlang and exponential distribution laws, Lindley integral equation, spectral expansion solution method, Laplace transform
Context. The relevance of studies of G/G/1 systems is associated with the fact that they are in demand for modeling data transmission systems for various purposes, as well as with the fact that for them there is no final solution in the general case. We consider the problem of deriving a solution for the average delay of requests in a queue in a closed form for ordinary systems with Erlang and exponential input distributions and for the same systems with distributions shifted to the right.
Objective. Obtaining a solution for the main characteristic of the system – the average delay of requests in a queue for two pairs of queuing systems with ordinary and shifted Erlang and exponential input distributions, as well as comparing the results for systems with normalized Erlang distributions.
Methods. To solve the problem posed, the method of spectral solution of the Lindley integral equation was used, which allows one to obtain a solution for the average delay for the systems under consideration in a closed form. For the practical application of the results obtained, the method of moments of the theory of probability was used.
Results. Spectral solutions of the Lindley integral equation for two pairs of systems are obtained, with the help of which calculation formulas are derived for the average delay of requests in the queue in a closed form. Comparison of the results obtained with the data for systems with normalized Erlang distributions confirms their identity.
Conclusions. The introduction of the time shift parameter into the distribution laws of the input flow and service time for the systems under consideration transforms them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the value of the variation coefficients of the intervals between the arrivals of claims and their service time, and as is known from the queuing theory, the average delay of requests is related to these variation coefficients by a quadratic dependence. If a system with Erlang and exponential input distributions works only for one fixed pair of values of the coefficients of variation of the intervals between arrivals and their service time, then the same system with shifted distributions allows operating with interval values of the coefficients of variations, which expands the scope of these systems. The situation is similar with shifted exponential distributions. In addition, the shifted exponential distribution contains two parameters and allows one to approximate arbitrary distribution laws using the first two moments. This approach makes it possible to calculate the average latency and higher-order moments for the specified systems in mathematical packets for a wide range of changes in traffic parameters. The method of spectral solution of the Lindley integral equation for the systems under consideration has made it possible to obtain a solution in closed form, and these obtained solutions are published for the first time.
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