Queuing systems with finite buffers are reasonable models for many manufacturing, telecommunication, and healthcare systems. Although some approximations exist, the exact analysis of multi-server and finite-buffer queueswith general service time distribution is unknown. However, the phase-type assumption for service time is a frequently used approach. Because the Cox distribution, a kind of phase-type distribution, provides a good representation of data with great variability, it has a vast area of application in modeling service times.
The research focus is twofold. First, a theoretical structure of a multi-server and finite-buffer queuing system in which the service time is modeled by the two-phase Cox distribution is studied. It is focused on finding an efficient solution to the stationary probabilities using the matrix-geometric method. It is shown that the stationary probability vector can be obtained with the matrix-geometric method by using level-dependent rate matrices, and the mean queue length is computed. Second, an empirical analysis of the model is presented. The proposed methodology is applied in a case study concerning the geriatric patients. Some numerical calculations and optimizations are performed by using geriatric data.