Performance Evaluation of an $M / G /1$ Queueing System with Interrupted Vacation Governed by a Double-Threshold Randomized $(p, N_{1}, N_{2})$– Strategy

This study investigates an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$M / G / 1$</tex> queueing system featuring a dual-threshold randomized <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(p,N_{1},N_{2})$</tex>–strategy and interruptible vacation. By applying the total probability in probability theory and an integral transform, the Laplace(L) transform, the transient probability distribution of the queue length is analyzed. Recursive expressions for the equilibrium queue length distribution are obtained, along with essential performance indicators like the probability generating function (PGF) and the expectation of queue length. Numerical examples are provided to explore system capacity optimization.