Abstract

Multiplication effects in point processes are important in a number of areas of electrical engineering and physics. We examine the properties and applications of a point process that arises when each event of a primary Poisson process generates a random number of subsidiary events with a given time course. The multiplication factor is assumed to obey the Poisson probability law, and the dynamics of the time delay are associated with a linear filter of arbitrary impulse response function; special attention is devoted to the rectangular and exponential case. Primary events are included in the final point process, which is expected to have applications in pulse, particle, and photon detection. We refer to this as the Thomas point process since the counting distribution reduces to the Thomas distribution in the limit of long counting times. Explicit results are obtained for the singlefold and multifold counting statistics (distribution of the number of events registered in a fixed counting time), the time statistics (forward recurrence time and interevent probability densities), and the counting correlation function (noise properties). These statistics can provide substantial insight into the underlying physical mechanisms generating the process. An example of the applicability of the model is provided by betaluminescence photons produced in a glass photomultiplier tube, when Cherenkov events are also present.

© 1983 Optical Society of America

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References

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  1. E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).
  2. D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).
  3. D. R. Cox, J. R. Stat. Soc. B 17, 129 (1955).
  4. M. S. Bartlett, Biometrika 51, 299 (1964).
  5. A. J. Lawrance, “Some Models for Stationary Series of Univariate Events,” in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, Ed. (Wiley-Interscience, New York, 1972), pp. 199–256.
  6. M. C. Teich, B. E. A. Saleh, Phys. Rev. A 24, 1651 (1981).
    [CrossRef]
  7. M. C. Teich, B. E. A. Saleh, J. Opt. Soc. Am. 71, 771 (1981).
    [CrossRef]
  8. B. E. A. Saleh, M. C. Teich, Proc. IEEE 70, 229 (1982).
    [CrossRef]
  9. J. Neyman, Ann. Math. Stat. 10, 35 (1939).
    [CrossRef]
  10. J. Neyman, E. L. Scott, J. R. Stat. Soc. B 20, 1 (1958).
  11. M. C. Teich, Appl. Opt. 20, 2457 (1981).
    [CrossRef] [PubMed]
  12. B. E. A. Saleh, M. C. Teich, “Statistical properties of a non-stationary Neyman-Scott cluster process,” IEEE Trans. Inf. Theory, to be published (November1983).
    [CrossRef]
  13. G. Vannucci, M. C. Teich, J. Opt. Soc. Am. 71, 164 (1981).
    [CrossRef]
  14. B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
    [CrossRef]
  15. B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
    [CrossRef]
  16. K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
    [CrossRef]
  17. M. Thomas, Biometrika 36, 18 (1949).
    [PubMed]
  18. R. E. Burgess, Discuss. Faraday Soc. 28, 151 (1959).
    [CrossRef]
  19. L. Mandel, Br. J. Appl. Phys. 10, 233 (1959).
    [CrossRef]
  20. M. C. Teich, B. E. A. Saleh, Opt. Lett. 7, 365 (1982).
    [CrossRef] [PubMed]
  21. A. van der Ziel, Noise in Measurements (Wiley-Interscience, New York, 1976).
  22. B. E. A. Saleh, Photoelectron Statistics (Springer, New York, 1978).
  23. D. Gross, C. M. Harris, Fundamentals of Queuing Theory (Wiley, New York, 1974).
  24. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  25. The analogous result for the SNDP, represented in Eq. (49) of Ref. 8, is incorrect. The correct result is P2SNDP(τ)=1α{μ∫-∞∞h(t) exp[-hτ(t)]dt·∫-∞∞h(t+τ)× exp[-hτ(t)]dt+ ∫-∞∞h(t)h(t+τ) exp[-hτ(t)]dt}·exp(μ ∫-∞∞{exp[-hτ(t)]-1}dt). The graphical result for the SNDP, presented in Fig. 11 of Ref. 8, is correct, however.
  26. J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, Elmsford, N.Y., 1964).
  27. W. Viehmann, A. G. Eubanks, “Noise Limitations of Multiplier Phototubes in the Radiation Environment of Space,” NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Md., Mar.1976).
  28. W. Viehmann, A. G. Eubanks, F. G. Pieper, J. H. Bredekamp, Appl. Opt. 14, 2104 (1975).
    [CrossRef] [PubMed]
  29. E. N. Gilbert, H. O. Pollak, Bell Syst. Tech. J. 39, 333 (1960).
  30. K. Matsuo, M. C. Teich, B. E. A. Saleh, “Poisson branching point processes,” in preparation.

1983 (1)

B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
[CrossRef]

1982 (3)

K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Proc. IEEE 70, 229 (1982).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Opt. Lett. 7, 365 (1982).
[CrossRef] [PubMed]

1981 (5)

M. C. Teich, Appl. Opt. 20, 2457 (1981).
[CrossRef] [PubMed]

G. Vannucci, M. C. Teich, J. Opt. Soc. Am. 71, 164 (1981).
[CrossRef]

M. C. Teich, B. E. A. Saleh, J. Opt. Soc. Am. 71, 771 (1981).
[CrossRef]

B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Phys. Rev. A 24, 1651 (1981).
[CrossRef]

1975 (1)

1964 (1)

M. S. Bartlett, Biometrika 51, 299 (1964).

1960 (1)

E. N. Gilbert, H. O. Pollak, Bell Syst. Tech. J. 39, 333 (1960).

1959 (2)

R. E. Burgess, Discuss. Faraday Soc. 28, 151 (1959).
[CrossRef]

L. Mandel, Br. J. Appl. Phys. 10, 233 (1959).
[CrossRef]

1958 (1)

J. Neyman, E. L. Scott, J. R. Stat. Soc. B 20, 1 (1958).

1955 (1)

D. R. Cox, J. R. Stat. Soc. B 17, 129 (1955).

1949 (1)

M. Thomas, Biometrika 36, 18 (1949).
[PubMed]

1939 (1)

J. Neyman, Ann. Math. Stat. 10, 35 (1939).
[CrossRef]

Bartlett, M. S.

M. S. Bartlett, Biometrika 51, 299 (1964).

Birks, J. B.

J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, Elmsford, N.Y., 1964).

Bredekamp, J. H.

Burgess, R. E.

R. E. Burgess, Discuss. Faraday Soc. 28, 151 (1959).
[CrossRef]

Cox, D. R.

D. R. Cox, J. R. Stat. Soc. B 17, 129 (1955).

Eubanks, A. G.

W. Viehmann, A. G. Eubanks, F. G. Pieper, J. H. Bredekamp, Appl. Opt. 14, 2104 (1975).
[CrossRef] [PubMed]

W. Viehmann, A. G. Eubanks, “Noise Limitations of Multiplier Phototubes in the Radiation Environment of Space,” NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Md., Mar.1976).

Gilbert, E. N.

E. N. Gilbert, H. O. Pollak, Bell Syst. Tech. J. 39, 333 (1960).

Gross, D.

D. Gross, C. M. Harris, Fundamentals of Queuing Theory (Wiley, New York, 1974).

Harris, C. M.

D. Gross, C. M. Harris, Fundamentals of Queuing Theory (Wiley, New York, 1974).

Lawrance, A. J.

A. J. Lawrance, “Some Models for Stationary Series of Univariate Events,” in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, Ed. (Wiley-Interscience, New York, 1972), pp. 199–256.

Mandel, L.

L. Mandel, Br. J. Appl. Phys. 10, 233 (1959).
[CrossRef]

Matsuo, K.

K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
[CrossRef]

K. Matsuo, M. C. Teich, B. E. A. Saleh, “Poisson branching point processes,” in preparation.

Neyman, J.

J. Neyman, E. L. Scott, J. R. Stat. Soc. B 20, 1 (1958).

J. Neyman, Ann. Math. Stat. 10, 35 (1939).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).

Pieper, F. G.

Pollak, H. O.

E. N. Gilbert, H. O. Pollak, Bell Syst. Tech. J. 39, 333 (1960).

Saleh, B. E. A.

B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Proc. IEEE 70, 229 (1982).
[CrossRef]

K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Opt. Lett. 7, 365 (1982).
[CrossRef] [PubMed]

M. C. Teich, B. E. A. Saleh, J. Opt. Soc. Am. 71, 771 (1981).
[CrossRef]

B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Phys. Rev. A 24, 1651 (1981).
[CrossRef]

K. Matsuo, M. C. Teich, B. E. A. Saleh, “Poisson branching point processes,” in preparation.

B. E. A. Saleh, M. C. Teich, “Statistical properties of a non-stationary Neyman-Scott cluster process,” IEEE Trans. Inf. Theory, to be published (November1983).
[CrossRef]

B. E. A. Saleh, Photoelectron Statistics (Springer, New York, 1978).

Scott, E. L.

J. Neyman, E. L. Scott, J. R. Stat. Soc. B 20, 1 (1958).

Snyder, D. L.

D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

Stoler, D.

B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
[CrossRef]

Tavolacci, J. T.

B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
[CrossRef]

Teich, M. C.

B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
[CrossRef]

B. E. A. Saleh, M. C. Teich, Proc. IEEE 70, 229 (1982).
[CrossRef]

K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Opt. Lett. 7, 365 (1982).
[CrossRef] [PubMed]

B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
[CrossRef]

M. C. Teich, B. E. A. Saleh, J. Opt. Soc. Am. 71, 771 (1981).
[CrossRef]

M. C. Teich, Appl. Opt. 20, 2457 (1981).
[CrossRef] [PubMed]

G. Vannucci, M. C. Teich, J. Opt. Soc. Am. 71, 164 (1981).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Phys. Rev. A 24, 1651 (1981).
[CrossRef]

B. E. A. Saleh, M. C. Teich, “Statistical properties of a non-stationary Neyman-Scott cluster process,” IEEE Trans. Inf. Theory, to be published (November1983).
[CrossRef]

K. Matsuo, M. C. Teich, B. E. A. Saleh, “Poisson branching point processes,” in preparation.

Thomas, M.

M. Thomas, Biometrika 36, 18 (1949).
[PubMed]

van der Ziel, A.

A. van der Ziel, Noise in Measurements (Wiley-Interscience, New York, 1976).

Vannucci, G.

Viehmann, W.

W. Viehmann, A. G. Eubanks, F. G. Pieper, J. H. Bredekamp, Appl. Opt. 14, 2104 (1975).
[CrossRef] [PubMed]

W. Viehmann, A. G. Eubanks, “Noise Limitations of Multiplier Phototubes in the Radiation Environment of Space,” NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Md., Mar.1976).

Ann. Math. Stat. (1)

J. Neyman, Ann. Math. Stat. 10, 35 (1939).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

E. N. Gilbert, H. O. Pollak, Bell Syst. Tech. J. 39, 333 (1960).

Biometrika (2)

M. Thomas, Biometrika 36, 18 (1949).
[PubMed]

M. S. Bartlett, Biometrika 51, 299 (1964).

Br. J. Appl. Phys. (1)

L. Mandel, Br. J. Appl. Phys. 10, 233 (1959).
[CrossRef]

Discuss. Faraday Soc. (1)

R. E. Burgess, Discuss. Faraday Soc. 28, 151 (1959).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. E. A. Saleh, J. T. Tavolacci, M. C. Teich, IEEE J. Quantum Electron. QE-17, 2341 (1981).
[CrossRef]

J. Math. Phys. (1)

K. Matsuo, B. E. A. Saleh, M. C. Teich, J. Math. Phys. 23, 2353 (1982).
[CrossRef]

J. Opt. Soc. Am. (2)

J. R. Stat. Soc. B (2)

D. R. Cox, J. R. Stat. Soc. B 17, 129 (1955).

J. Neyman, E. L. Scott, J. R. Stat. Soc. B 20, 1 (1958).

Opt. Lett. (1)

Phys. Rev. A (2)

B. E. A. Saleh, D. Stoler, M. C. Teich, Phys. Rev. A 27, 360 (1983).
[CrossRef]

M. C. Teich, B. E. A. Saleh, Phys. Rev. A 24, 1651 (1981).
[CrossRef]

Proc. IEEE (1)

B. E. A. Saleh, M. C. Teich, Proc. IEEE 70, 229 (1982).
[CrossRef]

Other (12)

A. J. Lawrance, “Some Models for Stationary Series of Univariate Events,” in Stochastic Point Processes: Statistical Analysis, Theory, and Applications, P. A. W. Lewis, Ed. (Wiley-Interscience, New York, 1972), pp. 199–256.

E. Parzen, Stochastic Processes (Holden-Day, San Francisco, 1962).

D. L. Snyder, Random Point Processes (Wiley-Interscience, New York, 1975).

A. van der Ziel, Noise in Measurements (Wiley-Interscience, New York, 1976).

B. E. A. Saleh, Photoelectron Statistics (Springer, New York, 1978).

D. Gross, C. M. Harris, Fundamentals of Queuing Theory (Wiley, New York, 1974).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

The analogous result for the SNDP, represented in Eq. (49) of Ref. 8, is incorrect. The correct result is P2SNDP(τ)=1α{μ∫-∞∞h(t) exp[-hτ(t)]dt·∫-∞∞h(t+τ)× exp[-hτ(t)]dt+ ∫-∞∞h(t)h(t+τ) exp[-hτ(t)]dt}·exp(μ ∫-∞∞{exp[-hτ(t)]-1}dt). The graphical result for the SNDP, presented in Fig. 11 of Ref. 8, is correct, however.

J. B. Birks, The Theory and Practice of Scintillation Counting (Pergamon, Elmsford, N.Y., 1964).

W. Viehmann, A. G. Eubanks, “Noise Limitations of Multiplier Phototubes in the Radiation Environment of Space,” NASA Tech. Note D-8147 (Goddard Space Flight Center, Greenbelt, Md., Mar.1976).

B. E. A. Saleh, M. C. Teich, “Statistical properties of a non-stationary Neyman-Scott cluster process,” IEEE Trans. Inf. Theory, to be published (November1983).
[CrossRef]

K. Matsuo, M. C. Teich, B. E. A. Saleh, “Poisson branching point processes,” in preparation.

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Figures (9)

Fig. 1
Fig. 1

Random multiplication of events: (a) primary events; (b) subsidiary events; (c) primary events plus randomly delayed subsidiary events.

Fig. 2
Fig. 2

Production of Thomas events: (a) primary Poisson events; (b) filtered Poisson events (shot noise); (c) primary events plus subsidiary doubly stochastic Poisson events whose rate is shot noise.

Fig. 3
Fig. 3

Model for generation of the Thomas point process.

Fig. 4
Fig. 4

Dependence of the degrees-of-freedom parameters and ′ on the ratio β = T/τp. exp and M exp are for the exponential case, whereas rect and M rect are for the rectangular case. Observe that exp = rect = τp/T and M exp = M rect = 3 for T/τp ≪ 1, whereas M exp = M rect = M exp = M rect = 1 for T/τp ≫ 1.

Fig. 5
Fig. 5

Counting distribution p(n) vs count number n, with 2T/τp as a parameter, for the Thomas process. The impulse response function for the filter is exponential with time constant τp/2. In all cases the mean number of events in the counting time T is 〈N(T)〉 = 10. (a) multiplication parameter α = 0.5; (b) multiplication parameter α = 2.0.

Fig. 6
Fig. 6

Thomas and SNDP counting distributions, p(n) vs count number n, in the limit T/τp ≫ 1. In this limit, the SNDP reduces to the Neyman type-A distribution. (a) μT = 5 and α = 1 for both cases, so that 〈N(T)〉SNDP = 5 and 〈N(T)〉Thomas = 10; (b) α = 2 and 〈N(T)〉SNDP = 〈N(T)〉Thomas = 10 for both cases.

Fig. 7
Fig. 7

Forward recurrence time probability density function P1(τ) for the Thomas process. The impulse response function for the filter is exponential with time constant τp/2. In all cases the mean forward recurrence time is 〈τ〉 = 1. (a) Filter time constant τp/2 = 1, multiplication parameter α = 0.1,1,10; (b) α = 1, τp/2 = 0.1,1,10. P1(0) is always equal to 1/〈τ〉.

Fig. 8
Fig. 8

Interevent (pulse-interval) probability density function P2(τ) for the Thomas process. The impulse response function for the filter is exponential with time constant τp/2. In all cases the mean interevent time is 〈τ〉 = 1. (a) Filter time constant τp/2 = 1, multiplication parameter α = 0.1,1,10. (b) α = 1, τp/2 = 0.1,1,10.

Fig. 9
Fig. 9

Model for the photon point process generated by betaluminescence photons plus single Cherenkov photon bursts. Observe the relation to Fig. 3, which is the mathematical model studied here.

Equations (95)

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N = k = 1 M A k + M ,
Q N ( s ) = Q M [ s - ln Q A ( s ) ] .
N = M ( 1 + A ) ,
N ( N - 1 ) = ( 1 + A ) 2 M ( M - 1 ) + M A ( A - 1 ) ,
var ( N ) = ( 1 + A ) 2 var ( M ) + M var ( A ) .
Q M ( s ) = exp { M [ exp ( - s ) - 1 ] } ,
Q N ( s ) = exp { M [ exp ( - s ) Q A ( s ) - 1 ] } .
var ( N ) = ( 1 + c 1 ) N ,
c 1 = A + var ( A ) 1 + A .
Q N ( s ) = exp [ M ( exp { - s + A [ exp ( - s ) - 1 ] } - 1 ) ] ,
N m + 1 = M k = 0 m ( m k ) N m - k d k + 1 ,             N 0 = 1 ,
d k + 1 = ( 1 + A ) d k + A r = 1 k ( k r ) d k - r , d 0 = 1.
N = M ( 1 + A ) , N 2 = M [ ( 1 + M ) ( 1 + A ) 2 + A ] , N 3 = M [ ( 1 + 3 M + M 2 ) ( 1 + A ) 2 + 3 A ( 1 + M ) ( 1 + A ) + A ] .
var ( N ) = M ( 1 + 3 A + A 2 ) ,
Q N f ( s ) = ( 1 - s ) N ,
Q N f ( s ) = Q N ( s ) s = - ln ( 1 - s ) .
Q N f ( s ) = exp { M [ ( 1 - s ) exp ( - A s ) - 1 ] } .
F m = ( - 1 ) m m s m Q N f ( s ) | s = 0 ,
F m + 1 = N k = 0 m ( m k ) F m - k b k + 1 , F 0 = 1 ,
b k = A k - 1 1 + A { A + k } .
F 1 = N , F 2 = N { N + A 2 + A 1 + A } , F 3 = N { N 2 + 3 A N 2 + A 1 + A + A 2 3 + A 1 + A } .
F T = var ( N ) N = 1 + 3 A + A 2 1 + A .
p ( n ) = 1 n ! n z n G N ( z ) | z = 0 ,
G N ( z ) = z N .
G N ( z ) = Q N ( s ) s = - ln z .
G N ( z ) = exp ( M { z exp [ A ( z - 1 ) ] - 1 } ) .
( n + 1 ) p ( n + 1 ) = N k = 0 n c ( k ) p ( n - k ) ,             n = 0 , 1 , ,
p ( 0 ) = exp ( - M ) ,
c ( k ) = 1 + k 1 + A A k exp ( - A ) k ! ,
p ( n + 1 ) = exp ( - M ) k = 1 n + 1 ( k A ) n + 1 - k exp ( - A k ) ( n + 1 - k ) ! M k k ! .
L Z ( s ) = exp [ - - Z ( t ) s ( t ) d t ] .
L Z ( s ) = exp ( - - Z p ( t ) ( s ( t ) - - h ( τ - t ) { exp [ - s ( τ ) ] - 1 } d τ ) d t ) = L Z p ( s ( t ) - - h ( τ - t ) { exp [ - s ( τ ) ] - 1 } d τ ) ,
L Z p ( s ) = exp ( μ - { exp [ - s ( t ) ] - 1 } d t ) .
L Z ( s ) = exp { μ - [ exp ( - s ( t ) + - h ( τ - t ) × { exp [ - s ( τ ) ] - 1 } d τ ) - 1 ] d t } .
Q N ( s ) = L Z ( s ) s ( t ) = s v * ,
N = ( N 1 , N 2 , , N L ) , s = ( s 1 , s 2 , , s L ) , v ( t ) = [ v 1 ( t ) , v 2 ( t ) , , v L ( t ) ] , v j ( t ) = u ( t - t j ) - u ( t - t j - T j ) .
Q N ( s ) = exp { μ - [ exp ( - j = 1 L s j v j + - h ( τ - t ) × { exp [ - j = 1 L s j v j ( τ ) ] - 1 } d τ ) - 1 ] d t } .
Q N ( s ) = exp [ μ - T 0 ( exp { - s + [ exp ( - s ) - 1 ] h T ( t ) } - 1 ) d t + μ 0 ( exp { [ exp ( - s ) - 1 ] h T ( t ) } - 1 ) d t ] ,
h T ( t ) = 0 T h ( t + t ) d t .
N m + 1 ( T ) = N ( T ) k = 0 m ( m k ) N m - k ( T ) D k ,             N 0 ( T ) = 1 ,
D k = 1 1 + α 1 T [ - T 0 H k + 1 ( t ) d t + 0 E k + 1 ( t ) d t ] ,
H k + 1 ( t ) = [ 1 + h T ( t ) ] H k ( t ) + h T ( t ) r = 1 k ( k r ) H k - r ( t ) ,             H 0 = 1 ,
E k + 1 ( t ) = h T ( t ) r = 0 k ( k r ) E k - r ( t ) ,             E 0 = 1.
N ( T ) = μ T ( 1 + α ) ,
var [ N ( T ) ] = μ T ( 1 + 3 α M + α 2 M ) ,
M - 1 = 1 α 2 T - h T 2 ( t ) d t ,
M - 1 = 1 α T - T 0 h T ( t ) d t + 1 3 α T 0 h T ( t ) d t ,
α = 0 h ( t ) d t .
F T = 1 1 + α ( 1 + 3 α M + α 2 M ) .
F SNDP = 1 + ( α / M ) ,
h ( t ) = 2 α τ p exp ( - 2 t / τ p ) u ( t ) ,
M = 2 β / [ exp ( - 2 β ) + 2 β - 1 ] ,
M = 3 β / [ exp ( - 2 β ) + 3 β - 1 ] ,
h ( t ) = { α / τ p 0 t τ p , 0 otherwise ,
M = { 1 / ( β - β 2 / 3 ) β 1 , β / ( β - 1 / 3 ) β 1 ,
M = { 3 / ( 1 + β ) β 1 , β / ( β - 1 / 3 ) β 1.
F T = ( 1 + 3 α + α 2 ) / ( 1 + α ) ,
Q N f ( s ) = exp ( μ - T 0 { ( 1 - s ) exp [ - s h T ( t ) ] - 1 } d t + μ 0 { exp [ - s h T ( t ) ] - 1 } d t ) .
F m + 1 = N k = 0 m ( m k ) F m - k B k + 1 ,
F 0 = 1 ,
B k = 1 T ( 1 + α ) { - T 0 [ h T ( t ) + k ] [ h T ( t ) ] k - 1 d t + 0 [ h T ( t ) ] k d t } .
p ( n ) = ( - 1 ) n ( n / s n ) Q N f ( s ) s = 1 .
( n + 1 ) p ( n + 1 ) = N k = 0 n C ( k ) p ( n - k ) ,
p ( 0 ) = exp [ N 1 + α ( 1 T 0 { exp [ - h T ( t ) ] - 1 } d t - 1 ) ] ,
C ( k ) = 1 k ! 1 ( 1 + α ) T [ ( k + 1 ) - T 0 J k ( t ) d t + 0 J k + 1 ( t ) d t ] , J k ( t ) = [ h T ( t ) ] k exp [ - h T ( t ) ] .
R ( t 1 , t 2 ) = [ N ( t 1 + T ) - N ( t 1 ) ] [ N ( t 2 + T ) - N ( t 2 ) ] .
R ( t 1 , t 2 ) = s 1 s 2 Q N ( s ) | s = 0 .
R ( τ ) = N T 2 + N ( T ) ϕ ( τ ) ,
ϕ ( τ ) = ( 1 - τ T ) u ( T - τ ) + 1 1 + α { 1 T - h T ( x ) h T ( x - τ ) d x + - T T ( 1 - x T ) [ h ( x - τ ) + h ( τ - x ) ] d x } , τ 0 ,
ϕ ( 0 ) = 1 1 + α ( 1 + 3 α M + α 2 M ) ,
τ = t 2 - t 1 ,             u ( x ) = { 1 x 0 , 0 x < 0 ,
ϕ ( τ ) = { 1 + 3 α + α 2 1 + α ( 1 - τ T ) + α ( 2 + α ) ( 1 + α ) ( 1 2 β ) × [ exp ( - 2 β ) cosh ( 2 τ / τ p ) - exp ( - 2 τ / τ p ) ] , τ T , α 2 + α 1 + α b exp ( - 2 τ / τ p ) τ T ,
b = cosh ( 2 β ) - 1 2 β ,
β = T / τ p .
P 1 ( τ ) = lim Δ τ 0 1 Δ τ Prob [ 0 events in ( t , t + τ ) , 1 event in ( t + τ , t + τ + Δ τ ) ] .
L = 2 , v 1 ( t ) = u ( t ) - u ( t - τ ) , v 2 ( t ) = u ( t - τ ) - u ( t - τ - Δ τ ) ,
Q N ( s ) = exp { μ - [ exp ( - j = 1 2 s j v j ( t ) + - h ( σ - τ ) × { exp [ - j = 1 2 s j v j ( σ ) ] - 1 } d σ ) - 1 ] d t } .
G N ( z ) = Q N ( s ) s = - lnz ,
z = ( z 1 , z 2 ) , lnz = ( ln z 1 , ln z 2 ) .
Prob [ 0 events in ( 0 , τ ) , 1 event in ( τ , τ + Δ τ ) ] = z 2 G N ( z ) | z = 0 .
P 1 ( τ ) = μ { 1 + 0 exp [ - h τ ( t ) ] h ( τ + t ) d t } × exp [ - μ ( τ - 0 { exp [ - h τ ( t ) ] - 1 } d t ) ] ,
h τ ( t ) = 0 τ h ( t + t ) d t .
P 2 ( τ ) = lim Δ τ 1 0 Δ τ 2 0 1 Δ τ 1 Δ τ 2 Prob [ 1 event in ( t , t + Δ τ 1 ) , 0 events in ( t + Δ τ 1 , t + τ + Δ τ 1 ) , 1 event in ( t + τ + Δ τ 1 , t + τ + Δ τ 1 + Δ τ 2 ) ] lim Δ τ 1 0 1 Δ τ 1 Prob [ 1 event in ( t , t + Δ τ 1 ) ]
L = 3 , v 1 ( t ) = u ( t ) - u ( t - Δ τ 1 ) , v 2 ( t ) = u ( t - Δ τ 1 ) - u ( t - Δ τ 1 - τ ) , v 3 ( t ) = u ( t - Δ τ 1 - τ ) - u ( t - Δ τ 1 - τ - Δ τ 2 ) ,
Q N ( s ) = exp { μ - [ exp ( - j = 1 3 s j v j ( t ) + - h ( σ - t ) × { exp [ - j = 1 3 s j v j ( σ ) ] - 1 } d σ ) - 1 ] d t } .
Prob [ 1 event in ( 0 , Δ τ 1 ) , 0 events in ( Δ τ 1 , τ + Δ τ 1 ) , 1 event in ( τ + Δ τ 1 , τ + Δ τ 1 + Δ τ 2 ) ] = z 1 z 3 G N ( z ) | z = 0 .
Prob [ 1 event in ( 0 , Δ τ 1 ) ] = z 1 G N ( z ) | z = 0 .
P 2 ( τ ) = 1 1 + α exp ( μ τ - μ 0 { exp [ - h τ ( t ) ] - 1 } d t ) × [ h ( τ ) exp [ - h τ ( 0 ) ] + h ( - τ ) + 0 h ( t ) h ( t + τ ) exp [ - h τ ( t ) ] d t + μ { 1 + 0 h ( t + τ ) exp [ - h τ ( t ) ] d t } × { exp [ - h τ ( 0 ) ] + 0 h ( t ) exp [ - h τ ( t ) ] d t } ] ,
h τ ( t ) = 0 τ h ( t + t ) d t .
L Z ( s ) = exp [ - - Z ( t ) s ( t ) d t ] .
L Z ( s ) = exp [ - - Z p ( t ) s ( t ) d t ] exp [ - - U ( t ) s ( t ) d t ] | F Z p ,
exp [ - - U ( t ) s ( t ) d t ] | F Z p = exp ( - Z p ( t ) - h ( τ - t ) × { exp [ - s ( τ ) ] - 1 } d τ d t ) .
L Z ( s ) = exp [ - - Z p ( t ) ( s ( t ) - - h ( τ - t ) × { exp [ - s ( τ ) ] - 1 } d τ ) d t ] = L Z p ( s ( t ) - - h ( τ - t ) { exp [ - s ( τ ) ] - 1 } d τ ) ,
L Z p ( s ) = exp ( μ - { exp [ - s ( t ) ] - 1 } d t ) .
P2SNDP(τ)=1α{μ-h(t)exp[-hτ(t)]dt·-h(t+τ)×exp[-hτ(t)]dt+-h(t)h(t+τ)exp[-hτ(t)]dt}·exp(μ-{exp[-hτ(t)]-1}dt).

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