We consider distribution systems with a single depot and many retailers each of which faces external demands for a single item that occurs at a specific deterministic demand rate. All stock enters the systems through the depot where it can be stored and then picked up and distributed to the retailers by a fleet of vehicles, combining deliveries into efficient routes. We extend earlier methods for obtaining low complexity lower bounds and heuristics for systems without central stock. We show under mild probabilistic assumptions that the generated solutions and bounds come asymptotically within a few percentage points of optimality (within the considered class of strategies). A numerical study exhibits the performance of these heuristics and bounds for problems of moderate size.