MGM for Aggregated General Distribution Queuing System in discrete time

The general distribution queueing systems (GI/G/s) are stochastic processes that cannot be completely described by Markov chains, but recent research has used the Matrix-Geometric Method (MGM) in discrete time. This paper presents a model that can be calculated by theoretical analysis using a gamma distribution superimposed on a discrete-time GI/G/s queueing model using the Matrix-Geometric and Aggregation methods. A performance evaluation at
$s\leq 2$
showed high-fitting results. This research found that the GI/G/s queueing model can be identified with gamma distribution, when using Aggregation method identified with a hyper-Erlang distribution and a hyper-exponential distribution (hyper-gamma distribution). The adaptive domain of this application is suited for the assessment of delays in the information and telecommunications fields or the performance evaluation in operations research fields of logistics and production processes composed of some agents, jobs, and packets with the same processing capabilities, as well as network models in parallel graph formed.