The study of queueing phenomena is an important branch of applied probability theory. Queueing models have wide-spread use in engineering and management. Among all the queueing models, those with a single server play an important role, since they not only have many direct applications but also are basic components of more sophisticated models. In this talk, we present a novel technique for solving general queueing models with a single server in equilibrium. The models that can be solved include the G/G/1 queue, the G/D/1 queue, and the queue fed by a general, two-state fluid. The storage capacities of the models are not necessarily infinite.

Despite of much previous effort in solving general single server queueing models such as the G/G/1 queue, available results in the existing literature are still quite limited. Since explicit, closed-form solutions to the general queueing models are usually very difficult or even impossible to determine, approximation and numerical techniques are necessary. The major drawback of the conventional techniques is that they cannot provide approximate or numerical o the real solutions of the general queueing models, since they typically need to change the original problems to some other problems tractable for the mathematical tools used.

The solution technique presented in this talk is based on the Fredholm integral equations of the second kind, derived for the general single server queueing models. Once the integral equation is solved, the solution to the corresponding queueing model follows. Solving the Fredholm equation is much easier than solving the corresponding queueing model with the existing methods. Therefore, our technique can solve queueing models that are very difficult or even impossible to solve with the traditional techniques. Actually, as long as a unique solution exists for a single server queueing model, the solution can then be obtained at least numerically with our technique. In addition, compared to the traditional approaches, our technique is easy to use in practice. To illustrate the technique, we have applied it to various queueing models with a single server. The solutions obtained include explicit closed-form solutions, analytical approximations, and numerical solutions. The analytical approximations and the numerical solutions can be arbitrarily close to the real solutions.