The bureaucracy-hour is a measure of the amount of time that customers or end-users must spend dealing with administrative staff in order to obtain access to a needed service.
Administrative hours "behind the scenes", i.e. not interacting with individual customers, do not count towards this total; time spent in a fixed location (e.g. in a line) waiting for access to an administrative official does count even though interaction is not taking place, since this is time which typically cannot be used for other productive work.
In cases where the usage of the needed service can also be measured in hours, a quick coefficient of bureaucratic frustration (experienced by the customer or end-user) can be expressed in bureaucracy-hours per usage-hour. Bureaucratic frustration in other situations may not be as amenable to being measured with a unitless ratio, however.
To some extent, it is the value of this ratio that determines whether a service has become a bureaucracy.