On Queues with Poisson Arrivals

The system to be studied consists of a service unit and a queue of customers waiting to be served. Service-times of customers are independent, nonnegative variates with the common distribution $B(v)$ having a finite first moment b₁. Customers arrive in a Poisson process (see Feller [4], p. 364) of intensity λ; they form a queue and are served in order of arrival, with no defections from the queue. For previous work on this queueing system see for instance Pollaczek [11], Khintchine [9], Lindley [10], Kendall [7], [8], Smith [12], Bailey [1], and Takacs [14]. Let $W(t)$ be the time a customer would have to wait if he arrived at t. The forward Kolmogorov equation for the distribution of $W(t)$ is solved in principle by the use of Laplace integrals, and Eexp\ - sW(t)\\ is determined in terms of $W(0)$ and the root of a possibly transcendental equation. It is shown that any analytic function of the root can be expanded in Lagrange's series, which provides a way of actually computing the transition probabilities of the process. Let z be the first zero of $W(t)$. A series for Eexp\ - τ z\\ is obtained, and it is proved that pr < ∞\ = 1 if and only if λ b₁ 1. From a functional relation between E(t)\ and pr(t) = 0\ the covariance function R of $W(t)$ is determined. If the service-time distribution $B(v)$ has a finite third moment, then R is absolutely integrable, and the spectral distribution of $W(t)$ is absolutely continuous.