On Queues with Poisson Arrivals

Abstract

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 $b_1$. Customers arrive in a Poisson process (see Feller [4], p. 364) of intensity $\lambda$; 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 $E\{\exp\{ - sW(t)\}\}$ 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 $E\{\exp\{ - \tau z\}\}$ is obtained, and it is proved that $\mathrm{pr}\{z < \infty\} = 1$ if and only if $\lambda b_1 \leqq 1$. From a functional relation between $E\{W(t)\}$ and $\mathrm{pr}\{W(t) = 0\}$ 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.