A New M/M/1 Queueing System with Bridging Gap Minimization - L. Joey, Farah; June Leong, Wah; Chuei Yee, Chen; Lutfi Bin Othman, Mohammad (Al-Nahrain Journal of Science, 2025)
In queueing systems, the Kendall notation a/b/x/q/y/z is a standard format used to describe six categories: the arrival distribution of clients, the service distribution of servers, servers’ number, the queue capacity, the system capacity, and the queue discipline. In this paper, a new category, referred to as the bridging function with best alpha (Br_∝*) between the arrival and service distributions, is introduced into the Kendall notation. It is placed between the arrival_distribution and the service_distribution, replacing the queue capacity (b), so that the notation becomes (a/ Br_∝*/b/x/y/z). Based on this new notation, the capacity rate of the queueing line can be predicted and represents the gap between the client arrival rate and the client service rate. It is shown that the proposed bridging function with best alpha follows the same distribution of arrival and service, both of which are assumed to be exponentially distributed. The Particle Swarm Optimization (PSO) algorithm is employed to find the optimal rate of best alpha, using the bridging function for the arrival and service distributions as the objective function. Simulation results of M/ Br_∝*/M/1 suggest that the bridging function with best alpha (Br_∝*) can reduce the delay time in the queueing system and converge to the optimal solution when applied to a standard M/M/1 system simulated by MATLAB code.