Uncovering the evolution from finite to infinite high-priority capacity in a priority queue

Infinite capacity queues are often used as approximation for their finite real-world counterparts as they are mathematically tractable. It is generally known that tail probabilities of low-priority system content in a two-class priority queue with infinite capacity for customers of both priority classes can be non-exponential, even if the interarrival time and service time distributions are exponentially decaying. In contrast, when the capacity for the high-priority customers is finite, tail probabilities of low-priority system content are always exponentially decaying. Therefore, using the results for one as an (accurate) approximation for the other is not obvious. From an analytical point of view, the non-exponentiality in the infinite case is caused by the arisal of an implicitly defined function, a root of the kernel, in the probability generating function for the low-priority system content. However, up till now, it has been unclear how this non-exponentiality suddenly emerges when taking the limit from to the finite to the infinite case. Our main contribution is that, under the restriction of a maximum of two arrivals per slot, a recurrence relation in the high-priority capacity is constructed resulting in an explicit expression for the corresponding generating function for the finite case. Amazingly, this expression contains all roots of the kernel in the infinite case. Taking the limit of this expression leads to the well-known behavior for the infinite case as the root inside the complex unit circle dominates the other roots uncovering the evolution from the finite to the infinite case. Furthermore, we investigate under which circumstances the standard tail characterizations are inaccurate