Aging Properties of Actual and Virtual Waiting Times in the GI|G|1|∞ Queuing Model

Abstract

The article considers a single-channel non-Poisson queuing model GI|G|1|∞ with a FIFO “first-in-first-out” service discipline in stationary conditions and system load intensity less than one. One of the important concepts of the mathematical theory of reliability is the property of increasing the hazard rate (IHR) of homogeneous elements forming reliability systems, otherwise called the aging property. This problem is also of importent for the queuing theory. Two independent in the aggregate and independent of each other sequences: sequence of the waiting times before the start of servicing of actual calls and sequence of the waiting times of virtual calls, starting at time t, or, more precisely, sequence of the durations of the time intervals starting at time t and ending at the moment when the system is free from calls received into the system until time t, are considered. Using the properties of ladder points and ladder heights, as well as applying formulas of Takac’s, Cohen’s and Hook’s, it proved that the widely used in the theory of random walks two above-mentioned sequences of random variables are IHR.