A state‐dependent queue alternating between arrivals and services

A queueing system usually involves an arrival process, service time distribution, service discipline, system capacity, etc. and steady‐state performance measures extensively studied. However, time‐dependent probabilities are important in applications and they do not involve convergence conditions. Further, several different models can have the same steady‐state probabilities. We consider a novel state‐dependent queueing system which alternating between arrival and service‐states. We assume that the arrival, service and transfer rates depend on the number present in the system. Time‐dependent system size probabilities and the duration of the busy period are obtained in a closed form for state‐independent rates. We present numerical illustrations when the parameters are state‐dependent. We have employed Continued Fractions effectively to achieve these transient results for this complex system.