This paper is devoted to the study of a stationary G/G/1 queue in which work input is gradually injected into the system and the work load is processed at unit rate. First, assuming that the input process is stationary, the key formula for the stationary distribution of work in system is derived by appeal to the waiting time of the associated regular G/G/1 queue. The main contribution of this paper is the derivation of Laplace-Stieltjes transforms (LSTs) for the stationary distributions of work in system and related random variables in terms of integrations with respect to a waiting time distribution of the associated regular queue. The results are exemplified by giving explicit formulas for the LST of total work for M/Ek/1 and M/H2/1. The results generalize the results of Pan et al. (1991) for the M/M/1 gradual input queue.
