On counters with random dead time. I

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Abstract
The following work was undertaken in connexion with a device for counting blood cells electronically, which has been developed in the Clinical Pathology Department of the Radcliffe Infirmary with financial assistance from the Medical Research Council and the Nuffield Foundation. Although the results will apply to other types of counter, it will help to fix the ideas if we consider the problem for this specific device. A large number of blood cells, contained in a shallow chamber, are scanned by a photoelectric cell. The depth of the chamber and the concentration of blood cells in solution therein allow blood cells (supposed distributed at random throughout the chamber) to overlap when viewed from above by the scanner. The field of view of the scanner at any instant is somewhat larger than the size of a blood cell, but is, nevertheless, of much the same order of magnitude. With passage of time the chamber moves underneath the photocell so that the field of view traces out a long narrow path not crossing or overlapping itself and only embracing a portion of. the whole chamber. The blood cells have no motion relative to the chamber. As each blood cell comes under the photocell it produces an electrical impulse, whose duration depends upon the size and shape and orientation of the blood cell. These impulses go to a counter, which counts them except that it will not count any impulse which is overlapped by a previous impulse. The problem is to determine the number of blood cells in the chamber from a knowledge of the recorded count and the distribution of the lengths of individual impulses. A further complication arises because it is inadvisable to count two impulses separated only by a very short interval of time, and therefore the counter itself generates some self-paralysing impulses additional to the blood-cell impulses and in a manner which is correlated with them. Until § 3, however, we shall neglect paralysis, because a proper understanding of its effects cannot be reached unless we first know how the system behaves in the absence of paralysis.

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