In queueing theory Wiener-Hopf techniques were first used in a non-probabilistic context by W. L. Smith, and later in a probabilistic context by F. Spitzer. The specific problem considered by these authors was the solution of the Lindley integral equation for the limit d.f. of waiting times. The more general role played by these techniques in the theory of random walks (sums of independent and identically distributed random variables) is now well known; it has led to considerable simplification of queueing theory.
Now the queueing phenomenon is essentially one that occurs
in continuous time, and the basic processes that it gives rise to are continuous time processes, specifically those with stationary independent increments. It is therefore natural to seek an extension of some of the Wiener-Hopf techniques in continuous time. Recent work has yielded several results which can be used to simplify existing theory and generate new results. In the present paper we illustrate this by considering the special case of the compound Poisson process X(t) with a countable state-space. Using the Wiener-Hopf factorization for this process, we shall derive in a simple manner all of the properties of a single-server queue for which X(t) represents the net input.