Exact Solution for One Type of Lindley's Equation for Queueing Theory and Network Calculus

TL;DR

This paper introduces a novel method to solve a specific type of Lindley's equation, enabling exact solutions for certain queueing waiting-time distributions and providing insights into backlog distribution approximations in network calculus.

Contribution

The paper develops a new approach to solve Lindley's equation with finite negative real roots, deriving exact distributions for multiple queueing models and comparing with existing network calculus models.

Findings

Exact waiting-time distributions for M/M/1, M/H2/1, M/E2/1, and D/M/1 queues.

Comparison of the new method with effective bandwidth and capacity models.

Backlog distribution can be exactly or approximately obtained depending on the model.

Abstract

Lindley's equation is an important relation in queueing theory and network calculus. In this paper, we develop a new method to solve one type of Lindley's equation, i.e., the equation V(s)T(-s)-1=0 only has finite negative real roots. V(s) and T(-s) are the Laplace transforms of service time's probability density function (PDF) and interarrival time's PDF (evaluated at -s). For queueing theory, we use this method to derive the exact M/M/1, M/H2/1 and M/E2/1 waiting-time distributions, and for the first time find the exact D/M/1 waiting-time distribution. For network calculus, we use two examples to compare our method with the effective bandwidth model and its dual, the effective capacity model, respectively. We observe that the distribution function of backlog size in the first example can be obtained exactly by our method and partially by the effective bandwidth model; however, such a distribution function in the second example cannot be obtained by our method but can be approximated by the effective capacity model.