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