The goal of "treatment on demand" is to allow all those seeking substance abuse treatment immediate entry into a program. Surprisingly, little is understood regarding the relationship between the demand for treatment, queues, waiting times and treatment admission rates, and treatment capacity. Nor has the increase in treatment capacity required to eliminate drug treatment queues, along with the expected benefits and costs of such an expansion, been studied carefully. In this paper, we present a mathematical model of drug treatment flows for systems where the demand for treatment greatly exceeds available supply. The model produces estimates of queue lengths, waiting times and treatment admission probabilities for any given treatment capacity, and suggests the capacity needed to achieve treatment on demand. The model also enables one to contrast the likely costs and benefits of changes in treatment capacity. We illustrate the model using San Francisco as a case study.