The common queueing problem has always focused on domestic and industrialized activities. Various improved models of queueing theory are widely used to solve problems. The aim of this paper is to deduce the exact solution of Bogoyavlenskii equation via direct extended modified algebraic method. In addition, we apply it to determine the upper and lower bounds through the semidefinite optimization packages software (SeDuMi). The suggested model indicated strong bounds in reasonable times, we obtain a definite value of [Formula: see text] of the function [Formula: see text] over [Formula: see text] where [Formula: see text] in a time duration less than 60 s and a maximum value of [Formula: see text] where [Formula: see text] in a time frame of approximately 7 min. This study enriches the theoretical optimization queueing network and provides an analysis and decision making method for perfecting the theory.

中文翻译：

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Bogoyavlenskii方程上下界直接扩展修正代数法在队列管理与优化中的应用

常见的排队问题一直集中在国内和工业化活动上。排队论的各种改进模型被广泛用于解决问题。本文的目的是通过直接扩展修正代数法推导出 Bogoyavlenskii 方程的精确解。此外，我们通过半定优化包软件（SeDuMi）应用它来确定上限和下限。建议的模型在合理的时间内显示出很强的界限，我们获得了函数 [公式：参见文本] 的 [公式：参见文本] 的确定值，超过 [公式：参见文本] 其中 [公式：参见文本] 在持续时间更少超过 60 秒，最大值为 [公式：见文本]，其中 [公式：见文本] 在大约 7 分钟的时间范围内。