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There are several commonly occurring situations in which the position of a unit or member of a waiting line is determined by a priority assigned to the unit rather than by its time of arrival in the line. An example is the line formed by messages awaiting transmission over a crowded communication channel in which urgent messages may take precedence over routine ones. With the passage of time a given unit may move forward in the line owing to the servicing of units at the front of the line or may move back owing to the arrival of units holding higher priorities. Though it does not provide a complete description of this process, the average elapsed time between the arrival in the line of a unit of a given priority and its admission to the facility for servicing is useful in evaluating the procedure by which priority assignments are made. Expressions for this quantity are derived for two cases-the single-channel system in which the unit servicing times are arbitrarily distributed (Eq. 3) and the multiple-channel system in which the servicing times are exponentially distributed (Eq. 6). In both cases it is assumed that arrivals occur at random.
Journal of the Operations Research Society of America © 1954 INFORMS