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counter theory; Laplace transform; generating function; dead time
In the paper the authors investigate the explicit form of the joint Laplace transform of the distances between two subsequent moments f particle registrations by the Type II counter (the counter with prolonged dead time), in the general case, and the generating function of the number of particles arriving during the dead time. They give explicit solutions to the complicated integral equations obtained by L. Takács and R. Pyke, respectively. Moreover, they study the geometric behaviour of the distribution of the latter above mentioned random variable, and make some remarks on the Type II counter and the case of registration of $m$ types of particles.
[1] Л. Г. Афанасьева И. В. Михайлова: Предельное распределение периода занятости в системах $G /D /\infty$ и $M /G /\infty$ в условиях большой загруски. в. кн. Материалы Всесоюзного симпозиума по статистике случайных процессов, изд. Киевского гос. ун-та, Киев (1973), 12-15.
[2] Л. Г. Афанасьева И. В. Михайлова: О восстановлениии характеристик некоторых систем массового обслуживания по выходящему потоку. в кн. Труды мат. фак. ВГУ, вып. 9, изд. Воронежского гос. ун-та, Воронеж (1973), 132-138. MR 0436667 | Zbl 1221.53041
[3] Л. Г. Афанасьева И. В. Михайлова: О числе требований, обслуженных за период занятости. Изв. АН СССР, Тех. Кибернетика, (1978), 88-96. Zbl 1234.93001
[4] G. E. Albert L. Nelson: Contribution to the statistical theory of counter data. Ann. Math. Stat., 24 (1953), 9-22. MR 0053447
[5] R. Barlow: Applications of semi-Markov processes to counter problems. in: Studies in applied probability and management science, Stanford Univ. Press, Stanford, Calif., (1962), 34-62. MR 0139205
[6] M. Berman: The covariance of two type II counters. J. Appl. Probab., 18 (1981), 782-787. MR 0621246 | Zbl 0467.60053
[7] А. Цвуреченский, др.: Об оценке плотности следов в трековых камерах. ОИЯИ, 5-81-362, Дубна (1981) 1-14. Zbl 1167.00300
[8] А. Двуреченский, др.: О применении систем массового обслуживания с бесконечным числом каналов к некоторым задачам физике энергий. ОИЯИ, P 5-82-682 (1982). 1 - 4. Zbl 1164.20358
[9] A. Dvurečenskij G. A. Ososkov: On a busy period of discretized $GI /GI /\infty$ queue. JINR, E5-82-855, Dubna (1982).
[10] В. Феллер: Введение в теорию вероятностей и ее приложения. T. I., ,,Мир", Москва (1967). Zbl 1103.35360
[11] В. И. Голъданский А. В. Куцешо, M И. Подгорецкий: Статистика отсчетов при регистрации ядерных частиц. ,Физматгиз", Москва (1959). Zbl 1234.81002
[12] R. L. Glückstern: Determination of bubble density. Nuclear. Instr. Meth. 45 (1966), 166-172.
[13] А. И. Маркушевич: Краткий курс теории аналитических функций. ,Наука", Москва (1978). Zbl 1130.91322
[14] F. Pollaczek: Sur la théorie stochastique des compteurs électroniques. C. R. Acad. Sci. Paris, 238 (1954), 322-324. MR 0059508 | Zbl 0055.12601
[15] R. Pyke: On renewal processes related to Type I and Type II counter models. Ann. Math. Stat., 29 (1958), 737-754. DOI 10.1214/aoms/1177706533 | MR 0099089 | Zbl 0086.33702
[16] R. Pyke: Markov renewal processes of zero order and their application to counter theory. in: Studies in Applied Probability and Management Science, Stanford Univ. Press, Stanford, Calif., (1962), 173-183. MR 0133889 | Zbl 0116.36402
[17] G. Sankaranarayanan: Limit distributions in the theory of counters. Ann. Math. Stat., 32 (1961), 1271-1285. Correction, ibid 33 (1962), 1466. DOI 10.1214/aoms/1177704866 | MR 0137175 | Zbl 0111.33103
[18] G. Sankaranarayanan: Theory of particle counters. Math. Student, 32 (1964), 29-38. MR 0184307
[19] G. Sankaranarayanan C. Suyambulingom: Distribution of the maximum of the number of impulses at any instant in a type II counter in a given interval of time. Metrika, 18 (1971/72) 227-233. Correction, ibid 20 (1973), 245. DOI 10.1007/BF02614253 | MR 0413307
[20] R. M. Sekkapan: A problem in type II counter. Calcutta Statis. Assoc. Bull., 15 (1966), 169-174. DOI 10.1177/0008068319660405 | MR 0217904
[21] W. L. Smith: Renewal theory and its ramifications. J. Roy. Stat. Soc. B, 20 (1958), 243 - 284. MR 0099090 | Zbl 0091.30101
[22] L. Takács: On processes of happenings generated by means of a Poisson process. Acta. Math. Acad. Sci. Hungar., 6 (1955), 81-99. DOI 10.1007/BF02021269 | MR 0070887
[23] L. Takács: On a probability problem arising in the theory of counters. Proc. Cambr. Phil. Soc., 52 (1956), 488-498. DOI 10.1017/S0305004100031480 | MR 0081585
[24] L. Takács: On the sequence of events, selected by a counter from a recurrent process of events. Teor. Verojat. i. Prim. 1 (1956), 90-102. MR 0084219
[25] L. Takács: On some probability problems concerning the theory of counters. Acta Math. Sci. Acad. Hungar., 8 (1957), 127-138. DOI 10.1007/BF02025237 | MR 0090168
[26] L. Takács: On a probability problem in the theory of counters. Ann. Math. Stat., 29 (1958), 1257-1263. DOI 10.1214/aoms/1177706457 | MR 0099717
[27] L. Takács: On a coincidence problem concerning particle counters. Ann. Math. Stat., 32 (1961), 739-756. DOI 10.1214/aoms/1177704969 | MR 0133893
[28] L. Takács: Introduction to the theory of queues. Oxford Univ. Prass 1962. MR 0133880
[29] L. Takács: Queues with infinitely many servers. RAIRO Recherche Opérat., 14 (1980), 109-113. MR 0575658
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