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On a first-passage problem for a cumulative process with exponential decay
Stochastic Procmses and their Applications 0 North-Holland
kilra
hunji
Faculty of Engineering, Himshim Utiiversity,~i~~s~~i~ 730, Japan
Received 24 April 1974 Revised 1 July 1973
fiist-paassuge problem fof a clrmulatilre process is intresti
80
A. Tsunri,S. Owki / A first-passageproblem for a cumulativeprocess
the steady sta;ee* is, however, of FJeat importance to study first-pasIS.For instance, materials and stucsage problems fo can be described as such a secondary tures fail by degra enomena are also process with expo giroblem will occur in a neuron taken into accoun eilson and Ross [ 5 ] firing model if inhibitory effects are neglected. prodiscussed the passage-time distributions for rnstein-Uhlenbeck cesses as an approximation of such models. In this paper we discuss a first-passage problem for a cumulative pros cumul’ative process is assumed to be generated by a lpcGsson cess. recess and the Iamplitutle generated by an event is assumed to (~Lecayexponentially. First, making use of ‘I’akacs’s result we derive a~: explicit expression of the transition probability, assuming that the ampllitudes generatell by tthe events are distributed exponentially. Next we give an integral equation for the probability density function (p&f.) of the first passage time until the total amnlitude exceeds a pre-specified threshold Ilying the explicit exi:-essioni for the transition probability, we derive thr;:,Laplace transform of the p.id.f. of the first-passage time. Finally we pre:ient the exact f’orms of the mean first-passage times anId plot these cur~:s versus the threshold level,
G nen tiai
decay
Let Sfi (;Y= 1,2, ...) denote a r~-&~~rnvariable that exy>resfiesthr: If’occurrence of the rzth even . N(t) i,c defined as a ranciic)m emoting the number of cverlts occurred MYto tlrnp if. In partiassume that the process (IV(t), t 2 0) is a Poisrson process t us introduce a cumullative process
{Z(t),
t >
0 :}
genuatetZ by
A. Tsurui, 5’~dsaki / k first-passage problem _&lra cumulatiw process
81
function
Further, a constant cyis a decay parameter. Thus Z(t) denotes the total amplitude at time t. Of interest is the time T that the total amplitude exceeds a pre-specified time-independent th.reshold level K for the first time (see Fig. 1). Before discussing this pr&lem, however, we will derive the transition probability r(x, t IJ+ dxdy =
[x~Z(s+t)~x+dxIEi
which will be used in wh.at follows. Takacs [9] derived a characteristic [ii?+@) G x]
function of the distribution
:
(bz(t,(U)
=
exP --A/
0
(2.2)
(1 -$H(~e-CYU))
du
1,
(2.3
where q+&) is a characteristic function of H(x). Generally, it is difficult to obtain the inverse. Fourier transCorm of eq. (2.3) in a closed foArmfor an arbitrary distribution H(x). For the special but most interesting case, i.e., H(x) = 1 - exp(-x/o), &owever, we are able to car out the inverse Fourier transform in a closed form. Namely, in that case, since
various density
fhxtians
sh
(‘r(x -ye- a!#
d&fly)=
I 0
otherwise,
problem that the total t is of inkrest to discuss the first-pass ant threshold level a pre-specified co 0) = 0. So we shall seek the p.d.f. of the random enote the probability that the total amplitude exceeds K) at time t, e and falls into an int en it must rst-passage time is easily obtained. from d it with respect to the variable z over the
adding the two parts, we have r(x, t
I 0) dx
= q(.rc, t) dx
Eliminating the quantity 4(x, t) from eqs. (3.3) a integral equation (3.1). Thus the proof is comple Solving the integral equati
(3.1) with Ithe a range (K, 03)) w
.l)S
(t)] =Lqe- t
x
[I+
alcwlatedand eq. (
X
-4
A. T&rui, S. Osaki /A first-passage problem for a cumulative process
86. se1
function of order zero.
d if ar tends to infinity, wleget
(t) = eWKe-“-Kt be
)
(4.8)
expected.
It is of spec:al importance to investigate the mean first-passage time fter some <:/umbersomebut strai.ghtforward calcuMon$, -we find
ET = -_ e-K lim d&J) p-+o dp :>
I! eK+K =-ii 1 +
z&
W-~O,,l (n+a,
mL$~fl+m--n(111 _.---.
pq+l
m= 1 (W2,1 m ! 1
+,G
I
1m-
21-q
k_“n+n
i )2n+2
m !
pj
d -I_-
m=O
l-m m! 1
n!
Km
1
m-!l -m I 2
(5.1)
a
pecially, if! is a small integer, ET has a simple form; for 2= 1, ET =
eK-1
+
A? 1 mt:ln+ 1)Z
(5.2)
’
for I = 2, =
(e
+
i2 (8n+1)(m+2)(m+3) m
!
c-1
A. Tswui, S. Osaki /A f&t-passage problmz for CIcumdat~ve prscc YS
Fig. 2. Curves of the mean first-passage times versus the threshold level.
curves we WI make crude estimations of the above problem.
for the mean first-passage time
87
88
A. Tsurui, S. Gsaki / .4 first-passage problem for a cumulative process
ough we assumed th d to the case that L slight modifications. Sine applicable to various real plications in physics, engi would be laborious and s Siinc:ewe assumed tha successfully 0 passage time. tegral equatio tion is found except numerically. It is of interest to discuss models with more general decay typ [2! and [$I). owever, it seems very difficult to obtain expli pressions for t mean first-passage time, since in these cases tion probability does not have a simple structure.
eferences [ 1] D.R. Cox, Renewal theory (Methuen, London, 1962). [ 2) D.J. Iglehart, Weak convergence of compot nd stochastic process I, Stochastic Proce:.Ges Appl. l(l973) 11-31. [ 3 ] P.I.M. Johannesma, Diffusion models for stochastic activity of neurons, in: E.R. Caia licllo, ed., Neutral Networks (Springer, Berlin, 1968). [4] J. Keilson and N.D. Mermin, The second-order distribution of integrated shot noise, Trans. Inform. Theory IT-S (1959) 75-77. [ 5 ] J. Keilson and H.F. Ross, Passage-time distributions for gaussian Markov (OrnsteinUhlenbeck) statistical processes, in: Selected Tables in Mathematical Statistics (publication sponsored by the Inst. of Math. Statist.), to appear. [6] R.C. Morey, Some stochastic properties of a compound-renewal damage modeI, 0 tions Res. 14 (1966) 902--908. [7] V.K. Murthy and B.P. Lientz, On cumulative damage and reliability of components, Document ARL 68=-0180, Aerospace Res. Lab., Wri t-Patterson Air For= astic Processes (Holdcnry processes generat Acad. Sei. Hung. 5
kilra
hunji
Faculty of Engineering, Himshim Utiiversity,~i~~s~~i~ 730, Japan
Received 24 April 1974 Revised 1 July 1973
fiist-paassuge problem fof a clrmulatilre process is intresti
80
A. Tsunri,S. Owki / A first-passageproblem for a cumulativeprocess
the steady sta;ee* is, however, of FJeat importance to study first-pasIS.For instance, materials and stucsage problems fo can be described as such a secondary tures fail by degra enomena are also process with expo giroblem will occur in a neuron taken into accoun eilson and Ross [ 5 ] firing model if inhibitory effects are neglected. prodiscussed the passage-time distributions for rnstein-Uhlenbeck cesses as an approximation of such models. In this paper we discuss a first-passage problem for a cumulative pros cumul’ative process is assumed to be generated by a lpcGsson cess. recess and the Iamplitutle generated by an event is assumed to (~Lecayexponentially. First, making use of ‘I’akacs’s result we derive a~: explicit expression of the transition probability, assuming that the ampllitudes generatell by tthe events are distributed exponentially. Next we give an integral equation for the probability density function (p&f.) of the first passage time until the total amnlitude exceeds a pre-specified threshold Ilying the explicit exi:-essioni for the transition probability, we derive thr;:,Laplace transform of the p.id.f. of the first-passage time. Finally we pre:ient the exact f’orms of the mean first-passage times anId plot these cur~:s versus the threshold level,
G nen tiai
decay
Let Sfi (;Y= 1,2, ...) denote a r~-&~~rnvariable that exy>resfiesthr: If’occurrence of the rzth even . N(t) i,c defined as a ranciic)m emoting the number of cverlts occurred MYto tlrnp if. In partiassume that the process (IV(t), t 2 0) is a Poisrson process t us introduce a cumullative process
{Z(t),
t >
0 :}
genuatetZ by
A. Tsurui, 5’~dsaki / k first-passage problem _&lra cumulatiw process
81
function
Further, a constant cyis a decay parameter. Thus Z(t) denotes the total amplitude at time t. Of interest is the time T that the total amplitude exceeds a pre-specified time-independent th.reshold level K for the first time (see Fig. 1). Before discussing this pr&lem, however, we will derive the transition probability r(x, t IJ+ dxdy =
[x~Z(s+t)~x+dxIEi
which will be used in wh.at follows. Takacs [9] derived a characteristic [ii?+@) G x]
function of the distribution
:
(bz(t,(U)
=
exP --A/
0
(2.2)
(1 -$H(~e-CYU))
du
1,
(2.3
where q+&) is a characteristic function of H(x). Generally, it is difficult to obtain the inverse. Fourier transCorm of eq. (2.3) in a closed foArmfor an arbitrary distribution H(x). For the special but most interesting case, i.e., H(x) = 1 - exp(-x/o), &owever, we are able to car out the inverse Fourier transform in a closed form. Namely, in that case, since
various density
fhxtians
sh
(‘r(x -ye- a!#
d&fly)=
I 0
otherwise,
problem that the total t is of inkrest to discuss the first-pass ant threshold level a pre-specified co 0) = 0. So we shall seek the p.d.f. of the random enote the probability that the total amplitude exceeds K) at time t, e and falls into an int en it must rst-passage time is easily obtained. from d it with respect to the variable z over the
adding the two parts, we have r(x, t
I 0) dx
= q(.rc, t) dx
Eliminating the quantity 4(x, t) from eqs. (3.3) a integral equation (3.1). Thus the proof is comple Solving the integral equati
(3.1) with Ithe a range (K, 03)) w
.l)S
(t)] =Lqe- t
x
[I+
alcwlatedand eq. (
X
-4
A. T&rui, S. Osaki /A first-passage problem for a cumulative process
86. se1
function of order zero.
d if ar tends to infinity, wleget
(t) = eWKe-“-Kt be
)
(4.8)
expected.
It is of spec:al importance to investigate the mean first-passage time fter some <:/umbersomebut strai.ghtforward calcuMon$, -we find
ET = -_ e-K lim d&J) p-+o dp :>
I! eK+K =-ii 1 +
z&
W-~O,,l (n+a,
mL$~fl+m--n(111 _.---.
pq+l
m= 1 (W2,1 m ! 1
+,G
I
1m-
21-q
k_“n+n
i )2n+2
m !
pj
d -I_-
m=O
l-m m! 1
n!
Km
1
m-!l -m I 2
(5.1)
a
pecially, if! is a small integer, ET has a simple form; for 2= 1, ET =
eK-1
+
A? 1 mt:ln+ 1)Z
(5.2)
’
for I = 2, =
(e
+
i2 (8n+1)(m+2)(m+3) m
!
c-1
A. Tswui, S. Osaki /A f&t-passage problmz for CIcumdat~ve prscc YS
Fig. 2. Curves of the mean first-passage times versus the threshold level.
curves we WI make crude estimations of the above problem.
for the mean first-passage time
87
88
A. Tsurui, S. Gsaki / .4 first-passage problem for a cumulative process
ough we assumed th d to the case that L slight modifications. Sine applicable to various real plications in physics, engi would be laborious and s Siinc:ewe assumed tha successfully 0 passage time. tegral equatio tion is found except numerically. It is of interest to discuss models with more general decay typ [2! and [$I). owever, it seems very difficult to obtain expli pressions for t mean first-passage time, since in these cases tion probability does not have a simple structure.
eferences [ 1] D.R. Cox, Renewal theory (Methuen, London, 1962). [ 2) D.J. Iglehart, Weak convergence of compot nd stochastic process I, Stochastic Proce:.Ges Appl. l(l973) 11-31. [ 3 ] P.I.M. Johannesma, Diffusion models for stochastic activity of neurons, in: E.R. Caia licllo, ed., Neutral Networks (Springer, Berlin, 1968). [4] J. Keilson and N.D. Mermin, The second-order distribution of integrated shot noise, Trans. Inform. Theory IT-S (1959) 75-77. [ 5 ] J. Keilson and H.F. Ross, Passage-time distributions for gaussian Markov (OrnsteinUhlenbeck) statistical processes, in: Selected Tables in Mathematical Statistics (publication sponsored by the Inst. of Math. Statist.), to appear. [6] R.C. Morey, Some stochastic properties of a compound-renewal damage modeI, 0 tions Res. 14 (1966) 902--908. [7] V.K. Murthy and B.P. Lientz, On cumulative damage and reliability of components, Document ARL 68=-0180, Aerospace Res. Lab., Wri t-Patterson Air For= astic Processes (Holdcnry processes generat Acad. Sei. Hung. 5