By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem in functional analysis we prove that the /G/1 queueing model with vacation times has a unique nonnegative time-dependent solution. 1. Introduction The queueing system when the server become idle is not new. Miller  was the first to study such a model, where the server is unavailable during some random length of time for the M/G/1 queueing system. The M/G/1 queueing models of similar nature have also been reported by a number of authors, since Levy and Yechiali  included several types of generalizations of the classical M/G/1 queueing system. These generalizations are useful in model building in many real life situations such as digital communication, computer network, and production/inventory system [3–5]. At present, however, most studies are devoted to batch arrival queues with vacation because of its interdisciplinary character. Considerable efforts have been devoted to study these models by Baba , Lee and Srinivasan , Lee et al. [8, 9], Borthakur and Choudhury , and Choudhury [11, 12] among others. However, the recent progress of /G/1 type queueing models of this nature has been served by Chae and Lee  and Medhi . In 2002, Choudhury  studied the /G/1 queueing model with vacation times. By using the supplementary variable technique  he established the corresponding queueing model and obtained the queue size distribution at a stationary (random) as well as a departure point of time under multiple vacation policy based on the following hypothesis. “The time-dependent solution of the model converges to a nonzero steady-state solution.” By reading the paper we find that the previous hypothesis, in fact, implies the following two hypothesis. Hypothesis 1. The model has a nonnegative time-dependent solution. Hypothesis 2. The time-dependent solution of the model converges to a nonzero steady-state solution. In this paper we investigate Hypothesis 1. By using the Hille-Yosida theorem, Phillips theorem, and Fattorini theorem we prove that the model has a unique nonnegative time-dependent solution, and therefore we obtain Hypothesis 1. According to Choudhury , the /G/1 queueing system with vacation times can be described by the following system of equations: where ; represents the probability that there is no customer in the system and the server is idle at time ; represents the probability that at time there are customers in the system and the server is on a vacation with elapsed vacation time of the server lying in . represents the probability that
J. Medhi, “Single server queueing system with Poisson input: a review of some recent developments,” in Advances in Combinatorical Method and Applications in Probability and Statistics, N. Balakrishnan, Ed., pp. 317–338, Birkh？user, Boston, Mass, USA, 1997.