This paper presents a simple procedure for investigating the distributions of queue length at post-departure and random epochs as well as the waiting time of a random customer in an \(M/G^{a,b}/1\) queueing system. A set of differential-difference equations generated with the remaining service time as the supplementary variable constitutes the foundation of this study. The approach suggested in this investigation does not require the formulation of a transition probability matrix, which is a traditional procedure for analyzing the queue length distribution at post-departure epoch. By applying the residue theorem and partial fraction technique, we are able to get closed-form expressions for the distributions of queue length at post-departure and random epochs as well as the waiting time of a random customer. We also present some numerical results to validate the accuracy of our findings and to support the validity of the analytical process.
