Queueing models are used to model applications in various areas, including computer networks, population dynamics, chemical kinetics, stock markets, and actuarial science. These models help provide more insight about such systems and predict certain random behaviors.

Consider a single-server steady-state queueing model {Q(t, t ≥ 0}. Let {W(t, t ≥ 0} denote the workload and τ denote the busy period of the server. The integral function of the form appears when queueing models are used to model systems such as asynchronous transfer mode (ATM) computer networks, insurance systems, risk, and percolation. Hence, it is pertinent to study the behavior of this function, in particular, its tail behavior, which is the focus of this paper. In the integral, f is a deterministic function and X=Q or W and T= τ or some finite or infinite constant.

The authors provide a detailed discussion of the tail behavior of the integral function for a queueing model in which the server serves the customers according to a subexponential distribution, and for a model in which the service distribution is light-tailed. The study further extends to a time-dependent GI/G/1 queuing model for heavy- and light-tailed cases; to a risk insurance system with T being a finite deterministic quantity; to a multivariate insurance system with T being a generic interarrival time; and finally to regulated processes.

The authors carry out the mathematical analysis elegantly.