A two-time-scale model for hospital patient flow

Hospitals are currently facing high incidences of ambulance bypass, long waiting lists for elective procedures and a significant proportion of patients experiencing long stays in the emergency department before moving to a bed in a ward. We observe that the system of wards and the emergency department in essence operates on two time-scales: the wards operate on a time-scale of days, whereas the state of the emergency department evolves on a much faster time-scale. This observation has motivated us to construct a two-time-scale model for a hospital system. The faster time-scale is modelled by a continuous time Markov chain, whereas the slower time-scale (ward occupancy) is modelled by a discrete time Markov chain. Given the nature of the system, the state of the wards has a direct impact on the emergency department.