Infinite Markings Method in Queueing Systems with the Infinite Variance of Service Time

Possible applications of the developed method with assignment of preemptive priorities determined by infinite markings of the service time semiaxis are analyzed. In queueing systems without priority the stationary average queue length is infinite when the service time has infinite variance even if the utilization is less than one. In case of Poisson arrival flow and Pareto distributed service times with finite mean and infinite variance, the stationary distribution of the queue length is shown to be asymptotically power-law with the exponent less than one; consequently, the stationary average queue length is infinite. But if we use the preemptive priorities in accordance with infinite markings method, the average waiting time in the queue and, hence, its average length become finite. For the case when the request length is proportional to its service time, we introduce an indicator defined as the sum of request lengths in the queue. Generalized Little’s formula for calculating and estimating of average value of the indicator in systems with finite and infinite stationary average queue length is derived. We consider the case when actual service time of arriving requests is unknown. For this situation it is proposed to employ a version of infinite markings method based on dynamically configurable preemptive priorities. In this case, it is found that the infinite markings method provides finite average waiting time.