The Almost Full Dam with Poisson Input

The Almost Full Dam with Poisson InputAuthor(s): A. M. HasoferSource: Journal of the Royal Statistical Society. Series B (Methodological), Vol. 28, No. 2(1966), pp. 329-335Published by: Wiley for the Royal Statistical SocietyStable URL: http://www.jstor.org/stable/2984376 .

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1966] 329

The Almost Full Dam with Poisson Input

By A. M. HASOFER

The Australian National University, Canberra

[Received August 1965. Revised December 1965]

SUMMARY In this paper, the content of a dam with infinite depth, unit release per unit time and inputs which form a compound Poisson process is investigated by means of an integral equation technique. The Laplace transform of the content distribution for an initially full dam is calculated and inverted. The asymptotic behaviour of the content is described and its mean and variance obtained. Duality relationships with the almost empty dam with Poisson input are pointed out and used to obtain various explicit expressions.

1. INTRODUCTION WE consider a dam with a continuous release of one unit of water per unit time. There are instantaneous inputs at times tl, t2, ..., which constitute a Poisson process with parameter A. The inputs are independent random variables, Xi (i = 0, 1, 2, ...) all having the same distribution B(x). We shall assume that B(O) = 0.

The dam is supposed to have infinite depth, and its content Z(t) at time t before adding the input, if any, is measured from an arbitrary origin. Thus Z(t) can take negative values. When Z(t) = h, the dam is full. This can only happen at the points tn. If the input at any of these points exceeds the deficiency of the dam, overflow occurs instantaneously, so that Z(t) takes the value h, and release resumes immediately from that value.

The initial value of Z(t), at time t = 0, is z. We write

P{Z(t) < x} = W(t, x).

Let us note that Z(t) is closely related to the virtual waiting time of the queue M/G/1 introduced by Takaics (1955). The continuity properties of the corresponding function W(t, x) have been studied by Hasofer (1963) and the arguments used to prove lemmas 1, 2 and 3 in that paper can be applied verbatim to our model. It follows that W(t, x) is Riemann-integrable in t, and that W(t - u, x + u) is continuous in u for u ?min(t,h-x).

2. AN INTEGRAL EQUATION FOR W(t, x) It is clear that for x > h, W(t, x) = 1. Suppose that x < h. Then the event Z(t) < x

can occur in the following ways: (a) There is no input in (0, t), and x > z -t. The probability of this event is

e-At U(x - z + t), where U(x) is the Heaviside unit function, defined by

(1 for x O, U(x) =

~0otherwise.



330 HASOFER - The Almost Full Dam with Poisson Input [No. 2,

(b) The last input before t is at t. = t - u. We consider two possibilities: (i) x > h - u. The event Z(t) < x will then occur with probability one. This con-

dition is equivalent to u > h - x. (ii) x < h-u (i.e. u < h-x), and Z(t-u) + X.-u < x. The probability of this last

event is V(t-u, x + u), where

V(t, x) = W(t, x -y) dB(y).

Thus the probability of event (b) is

rt T A e-uV(t-u,x+u)du if t<h-x,

and rh-x rt f e-Au V(t-u,x+u) du+ A e-Audu if t>h-x.

Finally, we see that W(t, x) satisfies the following equations:

1 for x>h and all t d (la)

e-A U(x -z+ t) + A fe-x V(t-u,x+u)du if t<h-x, (lb) W(t, x) = (t'X) e-At U(x - z + t) + e-A(h-x) -e-At

rh-$x + f e-AtuV(t-u,x+u)du if tkh-x. (Ic)

The existence of all integrals appearing in equations (1) is guaranteed by the properties of W(t, x) given at the end of the preceding section.

3. THE LAPLACE TRANSFORM OF W(t, x) WHEN THE DAM IS INITIALLY FULL

Let us write

(D(p, x) = e-Pt W(t, x) dt, Q(p, x) = fID(p, x -y) dB(y).

It then follows from equations (1) that, for x < h,

1 ~~~~~~A 'D(p, x) = exp-(p+A)(z-x)}+ exp{-(p+ A) (h-x)}

p+A.ep{ p+A z-X+ p(p + A)

rh-$ + A exp{-(p+ A)y} Q(p,x+y) dy. (2)

In the above equation (z - x)+ stand for max (0, z - x). Also, for x ) h, (D(p, x) = p-1. It is shown in the Appendix that equation (2) has a unique solution for Re (p) > 0. This solution has a very simple form when z = h, i.e. when the dam is initially full. In that case it reduces to

h-x (D(p, x) = p-l exp {- (p + A) (h -x)} + A exp {- (p + A) y} Q(p, x +y) dy. (3)



1966] HASOFER - The Almost Full Darn with Poisson Input 331

The solution can be obtained in the form

D(p, x) = C(p) em(P)x.

Replacing in (3), we find that m must satisfy the equation

p-m?+ A{1-O(m)} = O, (4) where

00 0(m) = e-mx dB(x),

and that C(p) = e-mhlp.

We now note that only solutions of equation (4) which have a positive real part can be significant for Re (p) >0, since we must then have Re (p) I (D(p, x) 1I for all x h.

But it has been shown by Beneg (1957) that equation (4) has exactly one root with positive real part for Re (p) > 0. Denoting this root by -q(p), we finally obtain

(D(p, x) = exp {-'-(p) (h-x)}/p.

4. INVERSION OF (D(p,x) The inversion of exp{-q(p)z} has been investigated by Hasofer (1964). Consider a

dam with infinite capacity, having the same input and release rule as the dam con- sidered in the preceding sections. Let 7(t) be the actual content of the dam, i.e. let the dam be empty for Z(t) = 0. Let z be the initial content of the dam, and let f(z) be the time of first emptiness. Let G(t, z) = P{f(z) < t}. It is shown in Hasofer (1964) that

e-Pt dG(t, z) = e-n(P)z,

where -q(p) is as defined in Section 3 of this paper. Thus we have

e-pt W(t, x) dt = p-' e-Pt dG(t, h - x), Re (p) > 0.

Integrating the right-hand side by parts, we obtain

e-pt W(t, x) dt = e-Pt G(t, h - x) dt.

It follows that W(t, x) = G(t, h - x) for all x < h and almost all t. From formula (17) of Hasofer (1964), p. 515, we can deduce that

h -x K(t, t - h + x)

It W(t, x) t hift-xx

W(tn x)= | - 2{uXlO(u, u-h +x)-K(u, u-h +x)} du if t ;3 h-x,

0 otherwise, where

K(t,x)- =EY e ) BnWl n=O n.!




B.(x) is the nth convolution of B(x) with itself, and

a K10(t, x) = t K(t, x).

Also, it follows from the corollary of p. 515 of the same paper that if K(t, x) has continuous derivatives in both t and x at the point (t, t - h + x), where t > h - x, and if we write

EK(t, x) = k(t, x),

then at the point (t, x), W(t, x) has a continuous partial denrvative in t, given by

a ~h-x aW(tq x) = - k(t, t- h +x).

The above conditions are obviously satisfied if B(x) has a continuous derivative b(x). The shape of W(t, x) for fixed x is then as follows:

(a) For t < h - x, W(t, x) vanishes. (b) At t = h - x, there is a jump of magnitude K(h - x, 0) = e-A(h-x). (c) For t> z, W(t, x) is a differentiable curve whose derivative is given by

a h-x h -x ~'I 7, W(t, x)= t k(t, t-h + x)= t Y, e -At) b,(t-h + x), ~~t t t n1 n!

where

b.(x) = dx Bn(X).

In particular

lim 2W(t, x) = k(h-x, 0) = 0, tq (h-x) at

so that the tangent to W(t, x) at t = h - x is horizontal. It follows that we can write

W(t, x) = e-A(-x) +f h x k(u,u-h+x)du, for t >h-x.

Replacing k(t, x) by its value, we find

W(t, x) = e-A(h x) +(h-x) An tf e-Auun4 b (u-h+x) du.

In particular, if b(x) = , e-nx, we find that

W(t, x) = e-A(h-x) + (h - x) 1(A,u) et(hx-) e-(A+/)u I1 [2 j{Ayu(u- h + x)}] du.

Finally, using the results on pp. 516-17 of Hasofer (1964) we conclude that if the inputs Xn take only integral values and if

P E Xr = n} = Pn(t)



1966] HASOFER - The Almost Full Dam with Poisson Input 333

then [t-h+x] h-x

W(t,x)= k=O h-x?nPn(hx+n)

5. THE ASYMPTOTIC BEHAVIOUR OF W(t, x) From the relation W(t, x) = G(t, h - x), we see that W(t, x) is a non-decreasing

function of t. Thus it tends to a limit as t -> oo. This limit can be determined by calculating

lim {p@( p, x)} = lim exp {-(p) (h - x)}. p-o Pmo

Let = lim(p).

P+O

Benes (1957, p. 673) has shown that * exists and satisfies the equation

m Al 1- 0(m)] (5) Let us write o(m) = A[1 - b(m)]. Then we can easily check that cb'(m) >0 and

oi'(m) < 0 for m > 0, o(O) = 0 and lim o(m) = A.

We now consider two cases: (a) on'(0) > 1. Then o(m) > m in the neighbourhood of the origin, and ot(m) < m for

large m. As of'(m) is monotonic decreasing there is exactly one positive root of (5). (b) on'(0) < 1. Then o(m) < 0 for all m > 0, and as o0) = 0 the only non-negative

root of (5) is m = 0. Let us note that the condition on'(0)> 1 is equivalent to - Ab'(0)> 1, and that

- A+'(0) is the mean rate of input per unit time. Thus we have the following Theorem. If the mean rate of input exceeds one, W(t, x) tends to the stationary

exponential distribution

W*(x) = exp{- *(h-x)}, x<h.

If, however, the mean rate of input is smaller than or equal to one, W(t, x) tends to one for all x, i.e.

limP{Z(t) < x} = 1 for all x. t-o00

6. THE MEAN AND VARIANCE OF Z(t) In this section we shall assume for simplicity that h = 0. Obviously this does

not affect the generality of the model, as it only requires a change in the origin of measurement of Z(t).

Let r+0 0

c(t, s) = J esx dx W(t, x) = 1-s esx W(t, x) dx. -00 -00

Then

e-Ptc(t,s)dt= -sj ! e(n+s)xdx o ~ ~~~ P 0 _p An + SY




Let E{Z(t)} = M(t), var{Z(t)} = (J2(t).

Then

e-Pt M(t)dt= --_d _ I

- (6) pds'iq+s s=O 7

f0e-Pt{ur2(t) + M2(t)} dt = p d 2 1 +s 2 (7)

pds2"7?s s=O Jp?7

Thus we obtain another relation between the dam model under investigation, i.e. the almost full dam, and the almost empty dam described at the beginning of Section 4. Let W(t, 0) = P{Z(t) = O}, i.e. the probability of emptiness of the almost empty dam, and let 2(0) = 0. Then it is well known (see, e.g., Benes (1957)) that

Je-Pt W(t, 0) dt= -

Thus we have the relation rt

M(t) =-JW(u, O) du.

From formulae (6) and (7) we easily obtain the mean M* and variance a 2 of the stationary distribution. They are

M*e=

*2 2

7. THE INTEGRO-DIFFERENTIAL EQUATION APPROACH

If we assume that B(x) has a continuous derivative, the arguments used in Hasofer (1963) show that W(t, x) satisfies the integro-differential equation

,a W(t, x) = aE W(t, x) -A W(t, x) + A jwW(t, x -y) b(y) dy.

If we take the Laplace transform with respect to t of this equation, and assume that W(0, x) = 0 for all x < h, we find that

p'D(p, x) = aN Vp, x) - AED(p, x) + A jD (p, x -y) b(y) dy. (8)

It now follows from the general theory of differential-difference equations (see, e.g., Pinney, 1959, p. 21), that the general solution of equation (8) is of the form

(D(p, x) = Yaj(p) exp {mj(p) x},

where the m,(p) are the roots of the characteristic equation

p-m+ AY{1-/(m)}= 0.



1966] HASOFER - The Almost Full Dam with Poisson Input 335

As in Section 3, we argue that only the root with positive real part is relevant, and the arbitrary constant is determined by using the boundary condition D(p, h) =p-1. The result is, of course, identical with that of Section 3.

This method, although simpler than that of Sections 2 and 3, is less general, as it does not apply rigorously, for instance, to inputs of fixed amount.

REFERENCES BENE'S, V. E. (1957), "On queues with Poisson arrivals", Ann. math. Statist., 28, 670-677. HASOFER, A. M. (1963), On the integrability, continuity and differentiability of a family of functions

introduced by L. Takacs", Ann. math. Statist., 34, 1045-1049. (1964), "On the distribution of the time to first emptiness of a store with stochastic input",

J. Aust. math. Soc., 4, 506-17. KOLMOGOROV, A. N. and FOMIN, S. V. (1957), Elements of the Theory of Functions and Functional

Analysis. Rochester, N.Y.: Graylock Press. PINNEY, E. (1959), Ordinary Difference-differential Equations. Berkeley and Los Angeles: Univer-

sity of California Press. TAKAkCS, L. (1955), "Investigation of waiting-time problems by reduction to Markov processes",

Acta Math. Acad. Sci. Hungar., 6, 101-129.

APPENDIX

In this Appendix we prove the existence and uniqueness of the solution of equation (2), using the principle of contraction mappings (see Kolmogorov and Fomin, 1957, Vol. I, p. 43). We take as our basic metric space M the space of all bounded complex-valued functions defined on x < h and integrable in any finite sub-interval of x < h. Our metric will be the uniform metric

p(f,g) = supIg(x)-f(x)l. xSh

We define a mapping A of M into itself by the equation

Aq(x) = exp {-(p + A) (z-x)+} + A) exp (p + ) (h-x)} P+A p(p +A) {() }

Nh-x + A f exp{-(p+ A)y} Q(x+y)dy,

where

Q(x) = f(x-y) dB(y).

We then have h-x

p(Al 01 A02) < sup I 02(x- X1(x) i A exp {(q + A) y} dy x<,h

p(A A ) (01h 0 feA

where q Re (p) > 0. Thus our mapping will be a contraction mapping, and has a unique fixed point,

which is the unique solution of equation (2).



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The Almost Full Dam with Poisson Input