In this article, the authors use sample-path techniques to derive relations between different performance measures for queues and related stochastic systems. A wider range of topics is reviewed to illustrate the power and versatility of sample-path analysis. They start by reviewing a sample-path version of the renewal-reward theorem (Y=λX). It is then shown how Y=λX can be used to give a simple proof of the sample-path rate-conservation law RCL under more general conditions than previously given. The authors apply Y=λX to obtain simple proofs of relations between continuous-time frequencies of a process defined on a general state space and frequencies at the points of an imbedded point process, including sample-path versions of the stochastic mean-value theorem, the covariance formula, the inverse-rate formula, and necessary and sufficient conditions for ASTA, reversed ASTA, and conditional ASTA. They also show that the RCL implies a sample-path version of the rate-balance formula (equating entrance and exit rates from a given set of states) and give sample-path proofs of relations between forward and backward recurrence-time distributions and their associated equilibrium distributions. Techniques used for deriving relations between (one-dimensional) frequencies are extended to develop a complete sample-path version of the Palm calculus. Strengthen versions of ASTA and PASTA are given, using the theory of martingales. The authors also review and extend sample-path proofs of L=λW and H=λG. Stability of general input-output systems is analyzed, showing that the existence of a limiting frequency distribution implies stability in the sense of equality of input and output rates. They prove the present basic stability result by establishing sufficient conditions that are easily verifiable from conditions on primary processes. Processes with non-countable as well as integer state spaces are considered and studied within one unifying framework. The authors also consider busy-period and idle-period durations and establish their rate stability under the sufficient conditions given previously. Applications to special queueing models and other input-output processes are given. In particular, they give a sample-path proof for rate stability of the workload, queue length, and infinite server queues. Stable versions of Little’s formula are also given.
