This paper introduces a new methodology for obtaining the stationary waiting time distribution in single-server queues with Poisson arrivals. The basis of the method is the observation that the stationary density of the virtual waiting time can be interpreted as the long-run average rate of downcrossings of a level in a stochastic point process. Equating the total long-run average rates of downcrossings and upcrossings of a level then yields an integral equation for the waiting time density function, which is usually both a linear Volterra and a renewal-type integral equation. A technique for deriving and solving such equations is illustrated by means of detailed examples.

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The purpose of this paper is to describe the new System Point method for analyzing queues. It considers the stationary probability distribution of the waiting time in variations of the M/M/R queue with first come first served discipline for a large class of service mechanisms. It is shown that the stationary probability density function of the waiting time evaluated at w > 0 can be interpreted as the long run average of the number of times that the virtual wait becomes less than w, per unit time. Theorems are presented which establish this interpretation of the probability density function of the virtual waiting time in terms of point processes generated by level crossings in the state space. These theorems, in combination with the principle of stationary set balance, generate a system of model equations that can be written directly. In addition, the forms of these equations are often linear Volterra integral equations of the second kind with parameter, which yield direct analytical solutions. An analogous theorem is proved for a variant of M/G/1. Two illustrative examples are presented.

In many modern applications of queueing theory, the classical assumption of exponentially decaying service distributions does not apply. In particular, Internet and insurance risk problems may involve heavy-tailed distributions. A difficulty with heavy-tailed distributions is that they may not have closed-form, analytic Laplace transforms. This makes numerical methods, which use the Laplace transform, challenging. In this paper, we develop a method for approximating Laplace transforms. Using the approximation, we give algorithms to compute the steady state probability distribution of the waiting time of an M/G/1 queue to a desired accuracy. We give several numerical examples, and we validate the approximation with known results where possible or with simulations otherwise. We also give convergence proofs for the methods.