Waiting Time in a General Active Queue Management Scheme

We derive the waiting time in a queueing scheme, in which an arriving job can be denied service with probability relative to the queue size. Such scheme is a generalization of the tail-drop queue, in which the job is denied service when the buffer (waiting room) is full, and can be found in computer networking, call centers and other everyday life applications of queueing systems. To make the model very general, we use an arrival process which enables shaping arbitrary the job interarrival time distribution and interarrival time autocorrelation, as well as general distribution of the service time and job rejection probabilities. For such model, we prove theorems on the waiting time in the transient case, i.e. as a function of time, as well as in the stationary case. Theoretical results are illustrated via numerical examples, in which the dependence of the behaviour of the system on various parameters is depicted. Among other things, it is demonstrated that the assumed job rejection mechanism may induce rather unexpected waiting times if combined with strong autocorrelation of the arrival process.