Ambulance bypass occurs when the emergency department (ED) of a hospital becomes so busy that ambulances are requested to take their patients elsewhere, except in life-threatening cases. It is a major concern for hospitals in Victoria, Australia, and throughout most of the western world, not only from the point of view of patient safety but also financially?hospitals lose substantial performance bonuses if they go on ambulance bypass too often in a given period. We show that the main cause of ambulance bypass is the inability to move patients from the ED to a ward. In order to predict the onset of ambulance bypass, the ED is modelled as a queue for treatment followed by a queue for a ward bed. The queues are assumed to behave as inhomogeneous Poisson arrival processes. We calculate the probability of reaching some designated capacity C within time t, given the current time and number of patients waiting.