Internet-type queues with power-tailed interarrival times and computational methods for their analysis

Internet traffic flows have often been characterized as having power-tailed (long-tailed, fat-tailed, heavy-tailed) packet interarrival times or service requirements. In this work, we focus on the development of complete and computationally efficient steady-state solutions of queues with power-tailed interarrival times when the service times are assumed exponential. The classical method for obtaining the steady-state probabilities and delay-time distributions for the G/M/1 (G/M/c) queue requires solving a root-finding problem involving the Laplace–Stieltjes transform of the interarrival-time distribution function. Then the exponential service assumption is combined with the derived geometric arrival-point probabilities to get both the limiting general-time state and delay distributions. However, in situations where there is a power tail, the interarrival transform is typically quite complicated and never analytically tractable. In addition, it is possible that there is only a degenerate steady-state system-size probability distribution. Thus, an alternative approach to obtaining a steady-state solution is typically needed when power-tailed interarrivals are present. We exploit the exponentiality of the steady-state delay distributions for the G/M/1 and G/M/c queues, using level-crossings and a transform approximation method, to develop an alternative root-finding problem when there are power-tailed interarrival times. Extensive computational results are given.