Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T21:33:31.272Z Has data issue: false hasContentIssue false
  • English
  • Français

On counters with random dead time. I

Published online by Cambridge University Press:  24 October 2008

J. M. Hammersley
J. M. Hammersley
Affiliation:
Lecturership in the Design and Analysis or Scientific ExperimentUniversity of Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 4 , October 1953 , pp. 623 - 637
Copyright
Copyright © Cambridge Philosophical Society 1953

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Albert, G. E. and Nelson, L.Contributions to the statistical theory of counter data. Ann. math. Statist. 24 (1953), 9–22.CrossRefGoogle Scholar
(2)Domb, C.The problem of random intervals on a line. Proc. Camb. phil. Soc. 43 (1947), 329–41.CrossRefGoogle Scholar
(3)Domb, C.Some probability distributions connected with recording apparatus. II. Proc. Camb. phil. Soc. 46 (1950), 429–35.CrossRefGoogle Scholar
(4)Domb, C.The statistics of correlated events. I. Phil. Mag. (7), 41 (1950), 969–82.CrossRefGoogle Scholar
(5)Feller, W.On probability problems in the theory of counters. Courant Anniversary Volume (1948), pp. 105–15.Google Scholar
(6)Hammersley, J. M.Tauberian theory for the asymptotic forms of statistical frequency functions. Proc. Camb. phil. Soc. 48 (1952), 592–9.CrossRefGoogle Scholar
(7)Ré;nyi, A.On some problems concerning Poisson distributions. Publ. math., Debrecen, 2 (1951), 66–73.Google Scholar
(8)Rényi, A.On composed Poisson distributions. II. Acta math. hung. 2 (1951), 83–98.CrossRefGoogle Scholar
(9)Stevens, W. L.Solution to a geometrical problem in probability. Ann. Eugen., Land., 9 (1939), 315–20.CrossRefGoogle Scholar
(10)Takács, L.Discussion of phenomena of occurrence and coincidence in case the distribution of the duration of happenings is arbitrary. Magyar Tud. Akad. Mat. Fiz. Oszt. Közle-ményei, 1 (1951), 371–86.Google Scholar
(11)Takács, L.Occurrence and coincidence phenomena in case of happenings with arbitrary distribution law of duration. Acta math. hung. 2 (1951), 275–98.CrossRefGoogle Scholar
(12)Whittaker, E. T. and Watson, G. N.Modern analysis, 4th ed. (Cambridge, 1940).Google Scholar