A Two-Phase Service Access Beside Retrial Queues, Orbit Mechanism, Working Vacation Policies for Single Server Environments

This paper analyzes the M/G (P1, P2)/1 queue using a single server that provides essential services in both the first phase (FPS) and the second phase (SPS). It was assumed that the server undertakes unreliability due to starting failure (SF) during the beginning of the first phase with probability (α) or service begins by the idle server successfully with probability (1-α) for the arriving or retrial customer. Then immediately took the failed server for repair and launched the customer into orbit. Also, the server operates at varying rates instead of fully ceasing service during its vacation. After returning from vacation, the server resumes normal operations for the customers. It also discussed the system, orbit size, and other significant metrics, utilizing the supplementary variable technique (SVT) and the probability generating function (PGF). Applications for this retrial queueing system incorporating the model and a general decomposition law were discussed. Numerical analysis was established to examine the impact of different factors on the system’s efficiency. MATLAB software was used to establish several impacts of the system’s behavioral measures.