Hama, Hiromitsu

Sein, Myint Myint

Osaka City University Repository

Mem. Fac. Eng., Osaka City Univ., Vol. 36, pp.61-65.(1995)
Dam in Continuous Time
by
Myint Myint SEIN*, Hiromitsu HAMA**.
(Received September 29, 1995)
Synopsis
This paper discribes a dam model in continuous time. On the assumption that the inflow
into the dam has an exponential distribution, we derive the stochastic distribution for the water
level. First, we study the fundamental equation of a dam. Next, we consider the dam with
infinite capacity which has exponential inflow and a constant (unit) outflow. Finally, we
investigate the dam problem under the outflow rule suggested by Holdaway.
Keywords: Moran Dam model, Holdaway's release rule, Stochastic processes
1. Introduction
We will consinder a particular type of storage system which corresponds to the kind of
situation which occurs in a dam build to store water l). This system consists of three
components namely the input, output and the content of a dam. The input is the amount of
water that flows into the dam, which depends on some natural phenomena, such as rainfall,
melting ·of snow, stream flows. And these vary from time to time and have a probability
distribution 2) in a random manner at every moment, day by day and from year to year. The
output is the amount of water to be released from the dam during a certain interval. This
component may be deterministic or random. It also may depend on the amount of water in
the dam and on the "release rule" which prescribes how and when the water is released 3) •
The purpose' of the dam is to make the amount released (the output) more uniform then the
inflow in some statistical sense; in this connection, it is relevant to study the underlying
stochastic process. For a given input and release rule, a special interest to a designer is the
varing pattern of contents of the dam. The content of the dam by which we mean the
amount of water stored in the dam. Now, we shell study the fundamental equation of a dam
modell, 2, 4)
2. Problem formulation
We consider a dam which can hold at most K units of water which we call the capacity
of the dam. Suppose that during a time interval (t,t+ 1),(t=0,1,2,3,...) X t units of water flow
••
Research Student,Department of Information and Computer Engineering.
Professor,Department of Information and Computer Engineering.
61
62 Myint MYint SEIN~ Hiromitsu HAMA.
into the dam. Let Z t units be the amount of water in the dam at the monent 1 then, it may
happen that Z t +X t is greater than K and the overflow of amount Z t + X t - K is caused.
Clearly the distribution of this overflow can be obtained directly from the distribution of X t
and Z t.
At the end of time interval 1 the water is released in according to some definite rule.
As we are trying to construct the simplest possible model, it is supposed that we release a
definite quantity M < K if there is at least amount in the dam, or the amount Z t + X t, if
Z t + X t < M, thus leaving finally Z t+ 1. Specifically, Moran's dam model as follows:
Suppose that during the time interval (/,1+ 1), (/=0,1,2, ...) X t units of water flow into
the dam, any overflow being lost. Provided that the dam is not dry, M units of water flowed
at the end of each time interval. Let Z t denote the amount of water left in the dam just
before the input X t occurs, then Moran formulated the dam equation as follows :
Zt+1 = min{Zt+Xt,K} - min{Zt+Xt,M}, 0 < M < K (2.1)
The above Eq.(2.1) can be understand as follow:
If Z t + X t > K then min{ Z t + X t, K} = K, otherwise min{ Z t + X t, K} = Z t + X t,
If Z t + X t > M then min{ Z t + X t, M} = M, otherwise min{ Z t + X t, M} = Z t + X t,
Zt+1= K-M, Zt+Xt> K
= Z t + X t - M, M < Z t + X t < K (2.2)
=0 Zt+Xt<M
Thus we can easily see that Z t, lies in the interval (0, K- MJ. Now we consider the
stationary distribution of dam content for continuous case. By assuming that X t has a
continuous distribution with a probability density which we denote by!(x), U t = Z t + X t also
has a continuous probability density which we denote by g(x).
Assuming !(x)=0 if x < 0, we have
M K
g(x) = f(x)fg(u)du + fg(u)f(x +M -u)du + f(x+ K - M)fg(u)du (2.3)
o M K
The three terms arising according as
o ~ U t ~ M (When the dam runs dry ),
M < Ut < K (When there is no overflow nor does the dam run dry) , and
K < Ut (When there is an overflow of amount U t - K).
3. Dam With Exponential Inflow
In this section we will consider a dam of infinite capacity with negative exponent inputs
and unit release. Let X t denote by the inflows during the time interval (/,1+ 1),
(/=0,1,2,3, ...), and the amountof water after the release, then we have
Zt+1 = Zt+Xt - min{ Zt+Xt,l } (3.1)
Let us write U t = Z t + X t and define the probability density function of U t by g(x) .
and the probability density function of X t by!(x), then from equation (3.1) the following
integral equation is obtain :
Dam in Continuous Time
M M+x
g(x) = f(x)fg(t)dt + ff(M +x - t)g(t)dt
o M
63
..........(3.2)
In this case it is more convennient to consider the cumulative distribution of X t and U t
Let us write
u
F(u) = f f( t)dt and
o
u
G(u) = fg(t)dt
o
Then we obtain from equation (3.2)
u
G(u) = fG(M + u - t)dF(t)
o
for M = 1, we have
u
G(u) = fG(1 + u - t)dF(t)
o
which we can write as follow:
u
H(u -1) = f H(u - t)dF(t)
o
.........(3.3)
..........(3.4)
........•.(3.5)
Now we will consider the case where the input distribution has the form by Pearson's
type III
a n+1
dF(t) = dF(t) =_tne-at ••••••••••(3.6)
n!
with n a non -negative integer. Solving the equation (3.2) , we have obtain the possible
solution
n+l
H(u) = 1+~ cjeZ;u
i-O
where is possibly complex. We have
..........(3.7)
n+l
1+~ cjeZ;u =
i=O
c .an+1e z(u+l) n (u + 1) n-s n+l C.1+ 1 _an+le-a(u+l)~ ~ 1
(a + Z. )n+l LJ (n - s)I 4 (a + Z. )S+11 s-O· 1-0 1
......(3.8)
which implies,
n+l
~ Cj 01---- =0, for s= , ,.... .,n ,i-o (a + Zi )-*1 ..........(3.9)
64 Myint Myint SEIN, Hiromitsu HAMA.
which are n + 1 equations for the constants Cl, •••••••• , Cn+ 1 and H(u) can be shown to be a
distribution function and is therefore the required solution.
4. Holdaway's Dam Model in Continuous Time.
Now, we will consider a dam model in which the release rule is governed by the
Holdaway's scheme. This kind of dam model is known as Holdaway's dam model. In the
following, we investigate this dam model by assuming that the input distribution is negative
exponential. Let us Z t denoted by the content of the dam at a time t just after the release,
then the release scheme of Holaday is as follows :
For D < M < L < K
(1) If Z t + X t ~ K + M, a quantity M is released through the dam and aquantity
Z t + X t - K - M goes the waste, so that Z t + 1 = K.
(2) If L +M~ Z t + X t < K+ M, quantity M is released and the final state is :
Zt+l= Zt+Xt-M
(3) If L + D~ Z t + X t < L + M, a quantity Z t + X t - L is released and the final state is
Zt+l=L.
(4) If D < Z t + X t < L + D, a quantity D is released and the final state is :
Zt+l=Zt+Xt-D
(5) If Zt+Xt < D,Zt+Xt is released and Zt+l= 0
This release rule can be rewritten as follow :
If Wt de note the amount of water to be flow out at the time t, then from the above scheme
we have,
w=t
Zt + Xt, whenZt +Xt < D
D,whenD < Zt + Xt < L +D
Zt + Xt, whenL+ D < Zt +Xt < L- M
M, whenL + M < Zt + Xt < K + M
M, whenZt +Xt > K +M,with
overflow
..........(4.1)
By using the release rule , we obtain the following solution system is
(1) for Y < L
D+Y
G(y) = fG(t)f(Y + D- t)dt,
D
..........(4.2)
Dam in Continuaus Time
(2) for L < Y < K
65
D+L M+Y
G(y) = f G(t)f(Y + D - t)dt + fG(t)f(Y + M - t)dt,
D L+M
(3) for Y > K
..........(4.3)
D+L K+M
G(y) = f G(t)f(Y + D - t)dt + fG(t)f(Y + M - t)dt + F(Y - K),
D L+M
Where G(y) = pr(Zt + X t S Y), dF(x) = f(x)dx,F(x) = pr(Xt S x).
Conclusion
.....(4.4)
In this paper, we have considered some problems of dams which have been recently
drawn attention by research workers in the theory of stC?rage. Specifically, we have paid our
attention on a problem in which release rule is made under Holidaway scheme. By assuming
that the input distribution is negative exponential, we have obtained some explicit result for
stationary distribution of the dam content. One of the ways to extend this model is to
consider the cases in which the input distribution forms a general process such as Person's
type III, general independent process and Markovian-type process and so on.
Reference
(1) Moran, P.A.P. : "The Theory of Storage", Methuen, London,(1959).
(2) Feller, W : "An Introduction to Probability Theory and Its application", VOL.1(2nd Ed.) Wiley,
NewYork,(1967).
(3) Ghoshal, A. : "On the Continuous Analogue of Holdaway's problem for the finite dam",
Aust.J.Appl.Sci.l0:365 -70,(1959)
(4) Pyke Tin & Phatarfod, R.M : "On Infinite Dams with Inputs Forming a Stationary rocess",
J.Appl.Prob.11,(1974).


Dam in Continuous Time

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Dam in Continuous Time

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