Abstract
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 (Se-DuMi). The suggested model indicated strong bounds in reasonable times, we obtain a definite value of u(0) = 11:876 of the function u(x) over [0; 1] where N = 20 in a time duration less than 60 s and a maximum value of u(0) = 24:987 where N = 50 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.