Many (perhaps most) service systems, such as repair and job shops, computation centers, and transportation networks, experience demand that is non-stationary in time. The paper describes models for situations in which demands made are by a finite number of individuals, who, having been served, do not return until much later. Such a transitory demand or arrival process describes many phenomena, among them being commuter rush hours, and also perhaps the effect on a population of individuals their simultaneous exposure to a dosage of medicine, a disease, or even a pollutant. The paper formulates several models for the service of such demands and describes the manner in which system state may be approximated by Gaussian processes, in particular the Ornstein-Uhlenbeck and Wiener diffusions. (Author)