We consider a deterministic queueing system whose dynamics can be described by a system of ordinary differential equations. The queueing system contains one servicer with two queues 1 and 2. The speed of application reception depends on the number of queue and of the time. Speed of application processing by servicers can be selected from within predetermined limits and are considered control. Speeds of processing are considered control. The problem is to minimize total lengths of queues. The necessary and sufficient conditions are received in the problem of piecewise constant control.