On the DO sag curve a minimum concentration occurs at some point, along a stream. If the Streeter–Phelps equation is differentiated with respect to time, and set equal to zero, the time at which the minimum DO occurs is expressed by
To find the value of the critical oxygen deficit, , the Streeter–Phelps equation is combined with the equation above, for the critical time, . Then the minimum dissolved oxygen concentration is
Mathematically it is possible to get a negative value of , even though it is not possible to have a negative amount of DO in reality.
The distance traveled in a river from a given point source pollution or waste discharge downstream to the  (which is the minimum DO) is found by
where  is the flow velocity of the stream. This formula is a good approximation as long as the flow can be regarded as a plug flow (turbulent).