We analyze a queueing system in which customers can call in to request service. A proportion, say 1p of them on their arrival test the availability of the server. If the server is free the customer enters service immediately. Otherwise, if the service system is occupied, the customer joins a source of unsatisfied customers called the orbit. The remaining p proportion of the initial customers enter the orbit directly, without examining the state of the server. We consider two models characterized by the discipline governing the order of re-requests for service from the orbit. First, all the customers from the orbit apply at a fixed rate. Secondly, customers from the orbit are discouraged and reduce their rate of demand as more customers join the orbit. The arrival at and the demands from the orbit are both assumed to be according to the Poisson Process. However, the service times for both primary customers and customers from the orbit are assumed to have a general distribution. We calculate several characteristic quantities of these queueing systems.