(1) Introduction A number of beds are reserved each day in most hospitals to accommodate emergency cases arriving during the day. Since the number of emergency cases fluctuates considerably from day to day, it is not immediately obvious how many beds should be set aside for this purpose. If all the beds so reserved are filled during the day, any further emergency cases must either be accommodated in extra beds in the centre of a ward, or referred to another hospital. Neither of these expedients can be regarded as fully satisfactory. The former method involves extra duties for hospital staff, and the latter not only causes extra travelling for emergency cases, but also fails to guarantee a bed. In London, for example, where the transfer of applications for beds is very efficiently organized by the King Edward's Hospital Fund Emergency Bed Service, beds are not immediately available for 10 to 15 per cent, of the cases referred to the Service (Daley, 1953). On the other hand, if a large number of emergency beds is provided to satisfy peak demands, many of them will remain empty on days of low demand. As emergency beds can only be reserved at the expense of the beds available for patients on the waiting list, wastage of this kind should be kept as low as possible. The problem thus resolves itself into balancing two opposing factors :