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ON ^ BIBLIOTHEEK TU Delft P 1945 3384A S T U D Y OF T H E T I M E D E P E N D E N T , STOCHASTIC B E H A V I O U R OF STORAGE SYSTEMS WITH A D D I T I V E INPUT
PROEFSCHRIFT
TER V E R K R I J G I N G V A N DE G R A A D V A N DOCTOR IN DE TECHNISCHE WETENSCHAPPEN A A N DE TECH-NISCHE HOGESCHOOL D E L F T , OP G E Z A G V A N DE RECTOR M A G N I F I C U S IR. H.R. V A N N A U T A L E M K E , H O O G L E R A A R IN DE A F D E L I N G DER E L E K T R O -T E C H N I E K , VOOR EEN COMMISSIE U I -T DE S E N A A -T -TE V E R D E D I G E N OP WOENSDAG 26 MEI 1971 TE 14 UUR
DOOR
/q(/s 32(Py
PIET BERNARD MARIE ROES
wiskundig ingenieur GEBOREN TE R O O S E N D A A L EN NISPEN
m/LlOTHEEK
DrR
TECHNiSCHt HOGESCHOOL
DELFT
1.1. A mathematical idealisation of a dam I (discrete case) 3
1.2. A mathematical idealisation of a dam II (continuous case) 8
1.3« Level crossings by certain stochastic processes 17
1.4. Level crossings by the content process v(t) 23
1.5. Review of recent research on related dam processes 25
1.6. Some notational and analytical detail 37
Chapter II Discrete Dams with Additive input 41
2.1 o The transient behaviour and first entrance times for the
infinit.e dam 41
2.2. First surpassage times for the infinite dam 46
2.3. First entrance times and transient behaviour for the
finite dam 51
Chapter III The infinitely high and deep dam 55
3.1. Introduction 55
3.2. The transient behaviour of the unrestricted dam 57
3.3. Calculation of the first entrance times ö ^0
3.4. Calculation of the taboo first entrance times 0 c2 -u;xy
3.5. Calculation of the first skip times ir 65
^ -xy
3.6. Calculation of the taboo first skip times 7t ó8 -u;xy
4.2. Calculation of the first passage times g~ 72
4.3. The transient behaviour of the infinitely deep dam 73
Chapter V The infinitely high dam 77
5.1. Introduction 77
5.2. The transient behaviour of an initially dry dam 79
5.3. The transient behaviour of the infinite dam 82
-I- 86
5.4. Calculation of the first entrance times ^
5.5- Calculation of the taboo first entrance times é 89 ^ ; x y
5.6. Calculation of the first skip times T 93 -xy
5o7. Calculation of the taboo first skip times 7r 96 -u;xy
Chapter VI The finite dam 101
5.1. Introduction 101
6.2. Calculation of the first entrance times é 102 6.3. The transient behaviour of the finite dam 104
References 111
The phenomena to be studied in this thesis are described
easiest in relation to water storage situations. It should be
recognized though that the methods we develop in the sequel
may be applied to certain Markovian time homogeneous Random
Walks, such as occur in the theories of Queues and Inventory
Gontrol. The particular feature investigated is the effect of
the imposition of boundaries on unrestricted or partially
res-tricted processes of this kind.
With a view to indicate the type of problems we intend to
examine, imagine a basin into which water is admitted from time
to time. The water is stored in the basin and released as
re-quired according to some rules. It is often of interest to Ido*
e.g. how much water is on hand at any given time, the maximum
content of the basin in a certain time interval and whether a
more or less continuous supply of water can be guaranteed.
The vital importance of the answer to such questions is
exem-plified by the use to which water in a reservoir is commonly
put and the reason for which dams are sometimes build viz.
drinking water, irrigation, generation of electricity, cooling
naturally depends to a large extend on the way in which water
is admitted, the input. Great fluctuations of the input will
cause a similar behaviour of the content. Further, the capacity
of the reservoir is an important factor. If the capacity is
small a lot of water is likely to go to waste. Finally, the
release rules are of importance. Thrift, if applied wisely,
may ensure a relatively long if meagre supply during draught.
Of necessity these remarks are vague. It is obvious that
the three aspects mentioned i.e. input, capacity and release
do influence the behaviour of the content of the dam in time,
but how and to what extend depends on the three factors
joint-ly. A fairly constant input and release may imply that a big
capacity of the reservoir is unnecessary. A near critically
small excess of average input over average use may nevertheless
necessitate a huge capacity. In view of these considerations
it is therefore pertinent to give a precise description of the
dajn models to be studied. This description will be given in
sections i and 2.
The approach throughout this study of Markovian Dam
distributions from the timedependent behaviour of the chain.
Alternatively, given the return time distributions of a Markov
Chain, the transient behaviour of same may be found. Often
return times for one Markov Chain can be found from same of a
related Markov Chain. Indeed, the return times of the finite
and semi infinite dams considered are found from those of
re-lated unrestricted processes. For a particular Markov Chain
this procedure is illustrated in Chapter II and it will be seen
that it relies heavily on Feller's Theory of Recurrent Events.
The application of a si.-nilar approach to certain Strong Markov
ProcesG?;s requires a relation between the transient behaviour
oF such processes and the renewal functions of the Renewal
Pro-ces.ses imbedded at any level. This relation is found in the
fairly basic theorem of section 3 regarding level crossings in
certain morr.entarily skipfree Stochastic Processes. In section 5
tae apprOtiCh presented is compared with other methods.
1.1. A n:?.thematical idealisation of a dam I (discrete time).
Consider a dam built to store water and thus forming a
contribu-tory rivers originating in the catchment area. The rainfall in
this area is a random process and therefore the inflow into the
reservoir is subject to random fluctuations; it is natural to
assume that this inflow is never negative.
Following Moran (1954), we consider the storage system only
at equidistant moments n = 0, 1, 2, ..., i.e. after equal
(year-lyt say) intervals. Denote by z* the content of the reservoir
at time n, while z^ = z is the initial nonnegative integer
con-tent. Let the positive integer K be the capacity of the
reser-voir. A dam for which K is infinite will be refered to as an
infinite dam; quantities pertaining to the latter will be
unstarred, e.g. z is the content of an infinite dam at time n.
The release of water occurs during the first half of the unit
time intervals (n,n-i-l), n = 0, 1, 2, ... . The amount of water
released during (n,n+l) is either a positive integer M units
or the total content at the start of the interval z* whichever —n
is less.
The input process of the cam is described by the total
amounts that become available in unit time intervals. Let x —n
the content of the reservoir in the second half of this
inter-val. Excess water overflows if the content z at time n minus —n
the release (min(il'i,z*)) plus the input x exceeds K. In this —n —n case an amount z -i- x - K - min(M,z ) goes to waste.
—n —n —n
The choice of the time at which the inflows become
availa-ble is fairly arbitrary. We are forced to make such a choice
due to the discretisation of the problem.
With the above assumptions we can express the
relation-ship between the content process 1^ , n = 0 , 1, •••[ and the
input process | x , n = 0 , 1, 2, ...I in the following recurrence
relation for n = 0, 1, 2, ... , z* ., = z* -I- X - M if z* > M, z* -(• X - M i K , (1.1) —n-i-1 — n — n — n * ' — n — n ' ^ - n if z** < M , X 4 K, —n ~ —n * = K if z*4 M, x^ j: K, = K if z* + X - M ? K . —n —n
Writing x = max(0,x), x = min(0,x), this can be condensed to
z* , = K + ((z* - M)"^ -I- x - K ) " . (1.2) —n+1 —n —n
As values of M exceeding unity lead to complications, we v/ill
take M = 1. This simplifies the formulae considerably and thus
gives a better insight in the structure of the results. The
loss of generality seems justified by the gain in clarity.
With a view to get as complete a picture as possible of the
stochastic behaviour of the dam as time proceeds, we further
define two accumulative processes. The total input X(n) during
the interval (0,n), clearly
n-1
X(n) = y~ X for n = 1, 2, ... ; (1.3)
— < — n 1^0
the total dry time d_(n) of the dam in the interval (0,n). This
dry time is the sum of the number of intervals in (0,n) during
which the reservoir was empty initially. Note that not
with-standing an inflow during the second half of the interval
(n,n+l) it is counted as a dry interval if z = 0 .
Writing (A) for the indicator function of the event A, that is
(A) = 1 if A occurs and zero otherwise, (1.4)
n-1
d*(n) = 1 ^ (z* = 0) for n = 1, 2, ... . (1.5) i=o
Alternatively, we have for the total input •jx(n) , n = O, 1, 2,..f
and total dry time |d^(n), n = O, 1, 2, ...r processes the
following recurrence relations for n = O, 1, 2, ... ^
and
X(n+1) = X(n) + X , (1.6)
d|(n+l) = d(n) + (z* = 0 ) ,
while we define X(0) = d(0) = 0.
Finelly, we will have to be more specific about
The inout process.
The inflows x of water during the caoosen disjoint unit
time intervals (n,n+l), n - 0, 1, 2, ... are independent and
identically distributed random variables with a discrete
dis-tribution on the nonnegative integers.
Let P(p) be the probability generating function of x t
that is
and let a be the expected value of the input during the
choosen unit intervals, i.e.
a = EJxjjj = P'(l) ^ <». (1.8)
The value of a relative to unity is naturally of particular
importance when dealing with the (semi) unrestricted darrs.
1.2. A mathematical idealisation of a dam II (continuous time).
The description of a finite dam in continuous time is a
rather more intricate than for the discrete case. v;e will
restrict ourselves in the first instance to the somewhat
pecu-liar class of input processes whose sample functions are
nonde-creasing step-functions vanishing at t = 0 . Subsequently, we will
limit the type of input processes even further to processes with
independent increments and almost all whose sample functions have
a finite number of jumps, that is to compound Poisson input. The
reason for the latter restriction is discussed at the end of
section 1.4.
Let X(t) be the input of the dam during the time interval
(0,t3 and let the input process jx't), t ^ o\ be a separable stochastic process whose sample functions are nondecreasing
step-functions vanishing at t = 0. That is, X(t) is the amount
of water that flows into the reservoir in the interval from
0 to t. The release of water will be assumed to be at unit rate.
Possibly the simplest model for a dam is one where the
reservoir is so large that it can never overflow and contains
so much water that it never runs dry. This is what we will call
an infinitely high and deep dam or briefly an unrestricted dam.
Let v^(t) be the water level for such a dam at time t, while
v(0) = V is the initial level. We will acsume that the level
of water increases or decreases by one unit if one unit of
water is added or withdrawn respectively. Alternatively we may
refer to v(t) as the content of such a dam at time t, it
be-ing understood that v(t) may assume any real value. Unlike
common usage of the word content, it is measured here relative
to some arbitrarily fixed reference level. Since X(t) is the
total input in the interval (0,t], while no water is lost and
release is continuous, we may write
v(t) = v(0) -1- X(t) - t. (2.1)
The content v(t) at any time t is therefore completely
been received since.
Next, we consider the infinitely deep dam, (cf. Hasofer
(1966)) Again we assume that the reservoir contains so much water
that it can never become empty, but the water level can not
exceed the positive value K at any time. Let v (t) be the water
level for the infinitely deep dam or by the same abuse of
langua-ge as before the content of this dam at time t. The process
v_ (t), t > Oj be;iaves like that of the unrestricted dam as long as the latter does not exceed K. As soon as an input occurs
that would cause the content v_ (t) to exceed K, the excess water overflows and the process is restarted with v (t) = K.
We will now derive (cf. Gani and Pyke (I960)) an explicit
expression for the content v (t) of the infinitely deep dam.
Firstly, if no overflow occurs in the interval (0,t], the
con-tent V (t) at t equals the right hand side (R.H.S.) of (2.1),
with v(0) replaced by v (0). On the other hand, if the dam
over-flows in (0,t] define o and T, the time of first overflow and the last time such happened in (0,t],respectively. The content
at time t when an overflow has ocoured in (0,t] equals K plus
the input from the moment T_ the dam overflowed last miniB the
v"(t) = K -I- X(t) - X(r) - t -I- T . (2.2)
Now consider some moment s in the interval (t ,tj . Since
the dam did not overflow during (X>U we have v_ (s) 4 K, thus
from (2.2) it follows that X(s) - X(T) - s + t ^ 0, so that for
any se(t,tl
X(t) - X(l) - t + T 4 x(t) - X(s) - t + s, (2.3)
Similarly, let s be any time in the interval ( 0 , ^ .
Since the dam overflows at t_, the total input in the interval
(S,T1 has exceeded the length of this interval, so that
X ( T ) - X(s);j: 1_ - s and again (2.3) holds in this case for S€(0,T1 . It follows that if the dam overflows in the interval
(O.tJ at least once, the content at time t equals the H.H.S.
of (2.2) for which we may write
v~(t) = K + inf |x(t) - X(x) - t + T I . (2.4) OiTitI- J
Further, since v~(0) -t- X(s) - s ^ K for all se(0,tl if the
dam does not overflow in this interval it follows that
v~(0) -I- X(t) - t does not exceed the R.H.S. of (2.4) in this
v~(0) + X(w) - 2 ;j K so that the R.H.S. of (2.4) does not
exceed v (0) + X(t) - t then. We conclude that we have for the content of the infinitely deep dam that
v"(t) = minrv"(0)
+
X(t) - t, K •^ inf }x(t) - X(t) - t -^ t|] . (2.5)
We now turn our attention to the infinitely high dam or
infinite dam as it is commonly known. The content process
\z. (*)f * ^ Oj for this dam behaves as that of the unrestricted dam as long as the zero level is not reached. The content may
assume any nonnegative value. When this dam runs dry, release
ceases and the content remains zero until water flows into the
reservoir. As soon as this happens release at unit rate is
re-sumed and the behaviour is again as for the unrestricted dam
with restart content equal to the amount of water received. An
explicit expression may be derived for the content at any time
t in the same spirit as before for the infinitely deep dam as,
v'^(t) = max[v'^(0) + X(t) - t, sup |x(t) - X ( T ) - t -^t|| . (2.6)
We come now to the description of the finite dam. Let the
v^ (t) the content of the finite dam at time t, while
V (0) = v 4 K is the initial nonnegative content. The process
•jv (t), t > Of behaves like that for the infinite dam as long as
the value K is not exceeded. If an input occurs so as to take
the content v (t) of the infinite dam beyond the level K, the
process is now stopped, excess water overflows and the v (t)
process is restarted again with restart content v^ (t) = K.
This procedure is repeated indefinitely. This way we obtain the
behaviour of the finite dam as a random walk between two
re-flecting barriers 0 and K. If this process does not reach the
boundaries during a certain time interval, its behaviour is
like that of the unrestricted dam; if only the lower boundary
is reached, like the behaviour of the infinite dajn, while if
during some time interval only the upper boundary is attained
it bahaves like the infinitely deep dam in this interval. Note
that if the upper level K is reached, the lower level cannot
be reached subsequently until at least a time interval K has
elapsed, since decrease of v (t) can only occur at unit rate.
In order to find the content v (t) of the finite dam of
capacity K at time t we choose a positive quantity h < K and
((n-l)h,nh], (nh,tj, where n is the largest integer not
exceeding r. For each of these subintervals the final content
is now determined successively. Consider the interval ( S , S + A ]
with A <K and suppose that the initial content v (s) is known,
then using (2.6) one can determine the content v (u) at any
time ue(s,s-i-Al if the dam did not have finite capacity. This
enables one to determine whether an overflow for the finite
dam occurs. Then the final content at time s + A is found from
(2.6) if no overflow occurs as
V * ( S + A) = max[v*(s) -i- X ( S + A ) - X ( S ) - A , (2.7) sup fx(s + A) - X(T) - s - A + T|| .
Te(s,s+A)
If an overflow does occur, i.e. if
sup |max(v*(s) + X(o) - X(s) - o + s, (2.8)
oé(s,s+A)
sup fx(<^) - X(-c) - CT + t]U > K,
TÉ(s,ff)^ JJJ
then we have from (2.5) that
v*(s-HA) = K + inf (X(S-HA) - X(r) - s - A -I-T|, (2.9)
Here use is made of the fact that A <K so that, since the
re-lease is at unit rate, the dam cannot have become empty after
the overflow occured. This procedure enables us to determine
the content of the finite dam at any time t, through succesive
calculation of the final content in each of the subintervals
introduced above. It hardly needs mention that criterion (2.8)
is rather prohibitive in nature, so that the whole procedure
seems fairly impractical. Nevertheless, it can and has been
used to determine the transient behaviour of the finite dam by
Roes (1970a).
As in the discrete case, we further define the process r ^ 1 "'
Id (t), t :^ OJJ where d, is the total dry time in the interval . That is d^ (t) is the total time during which the dam is
empty in the interval from 0 to t. Note that (cf. (1.4))
d (t) = (v (u) = 0)du, (2.10) o
Also, for the infinite dam we have if v (0) = v for the total
dry time é (t) in the interval (0,tj that
We proceed with a description of the input process in
more detail.
The input process.
Prom here onwards we assume, when dealing with dams in
continuous time, that ]x(t), t > Oj is a separable stochastic
process with independent stationary increments. We stated
al-ready that almost all sample functions Of the input process
are nondecreasing right-continuous step-functions with X(0) = 0.
The Laplace-Stieltjes transform of the input process X(t)
has the form (cf. Takacs (1967))
E[e-?i^*)] = e - * ^ ( ^ \ (2.12)
where t ^ 0, Re J ^ 0 and
OB
?(9) =
J
(l-e"^*)dN(x), (2.13)
0 +
and N(x), 0 < x < C Ö , is a nondecreasing, right continuous
function for which lim N(x) = 0 and / xdN(x) <oo , We note
X-.OD J
Define (cf. (1.8)) a = / xdN(x) 4: 00 , the expected input f °'^]
per unit time interval; EjX(t)l = at. The expected nximber of
jumps in an inteiTral of length t exceeding x is -tN(x). Define
A = -iy(O).^ 00 ; thus ^ t is the expected number of jumps in an
interval of length t. If \<oothe input process is a compound
Poisson process. This latter type of process will be the input
process for the dams considered in this section from here
onwards.
1.3» Level crossings b.y certain stochastic processes.
In the present section the stage is set on which most of
the analysis in subsequent chapters is based. A relation is
derived between the timedependent distribution of a certain
type of Random Walk and the expected number of level crossings
of same. This relation affords us to link the behaviour of
restricted dams with that of the unrestricted one.
Consider a stochastic processwhich increases and
decrea-ses by simple jumps as well as smoothly. The rate of smooth
increase and decrease with time is a function of the state of
the process. The process is not constant in time except,
relation between the expected number of true crossings of a
level X, say, in an interval of time and the time-dependent
distribution. By a true crossing of the level x at time t we
mean that the process increases or decreases smoothly at t,
as opposed to a skipping which happens when the state x is
bypassed due to a jump from a state below x to a state
excee-ding X or from above to below x. The relationship is found
from a close examination of the sample paths.
The approach appears to be a powerful tool if the original
process is a strong, time homogeneous Markov process, since
then the expected number of crossings of a level x is the
re-newal function of an imbedded rere-newal process at that level.
Consequently, this enables one to find the first entrance
times of the Markov process. Moreover, the approach is
reversi-ble; the first entrance time distributions may be used to
ob-tain the timedependent distribution of the process.
Let lz.(t), t > Or be a stochastic process on the
nonnega-tive real line, whose sample functions z{t,u), u(Sl, are right continuous. Let T(w) = }t:z(t,i)) > 0| and assume that (i) the
right hand sample derivative exists for every tfcT(w), (ii) its
(iii) the number of jumps is finite in every finite interval.
A typical sample path is shown in figure 1, to which we refer
in the next paragraph.
z(t ,6))
Figure 1 A general sample path
In the sequel we will need to distinguish between
cros-sings and skippings of a level. The level x is skipped at t = t
if z(t--,w) < X < z(t ,0)) or z(t--,u) > x > z(t. ,(.>)• If on the
other hand z(t-,(ü) = x = z(t,<o), the level x is crossed; this
may be done from above as at tp or below as at t-. For
complete-ness' sake we may define the situation z{t.-,cj) = x < z(t.,Cij) as a skipping at t ; while if, as at t_, the sample derivative
changes sign from negative to a positive value, and z(tj.-,<J) =
= z(t_,(u) = y, then a crossing from above occurs. Similar
de-serve no attention as they play no vital role in the sequel.
It will be more convenient to deal with processes for
which r(x) = 1 and we therefore transform the range through
division by r(x). We now consider a typical sample path as
shown in figure 2 and define N(T,X,Ü>) as the number of times
the path actually crosses the line z(t,Ci>) = x > 0 in the
inter-val (0,TJ. We observe the following identity
X T
ƒ N(T,x,(o)dx =
where the L.H.S. is the sum of lengths of the projections
on the vertical axis of the oblique parts of z(t,CLi)
under-neath level X and between 0 and T; the R.H.S. the sum of
lengths of projections of same on the t - axis (cf. figure 2 ) .
Indeed, the identity follows from the one to one correspondence
between line segments. Note that this argument only applies if
the number of discontinuities and points of sign change of
the sample derivative (kink^ between 0 and T is finite. We will
assume that M(T), the expected value of this number, is finite
(cf. end of next section).
Let N(T,x) be the number of level x-crossings of z^(t) in
z{t,CJ)
Figure 2 A sample path for r(x)= 1
(O.TJ. Since ^(t) has a finite number of discontinuities and
kinks with probability one in every finite interval (as
M(T) < co), it follows that N(T,x) is constant almost everywhere
and its number of discontinuities is finite with probability
one. Hence N(T,x), xe(-asoo) is a measurable stochastic process.
Let P (t,x) = P z^(t) ^ xj^j, where (p stands for the initial condition.- Further let m^^^(T) = E N(T,x))^ be the expected
number of level x-crossings of ^(t) in (0,T . Prom the
finite-ness of M(T) it follows that m^^^(T) is finite and integrable P
(3.1), applying a theorem of Doob (1953) p. 62, we may
inter-change the order of integration and obtain that
X T
ƒ m^^^(T)dx = / [Pp(t,X) - P^(t,0)|dt. (3.2) 0 0
Finally, on taking transforms we have the follov/ing
Theorem
Let £(t), tero,ao) be a stochastic process on the
nonnega-tive reals, whose sample functions possess a right hand
derivative, the absolute value of which equals unity except
possibly when z^(t) = 0. Let F (t,x) be the distribution func-(x)
tion of z(t) and m (t) the expected number of level x cros-— P
sings in (0,tj. If the number of jumps and points of sign
change of the sample derivative is finite, we have for
Re s > 0, Re s> ^ 0 that
oo o
r r
e"^*d^m^J^t)dx = ƒ e"""^ / e"^''d^P^( t ,x)dt. (3.3) 0 0 0 0 +
The interest of the present theorem is exemplified by the (x)
fact that m . (t) may be a renewal function. Indeed, if the
original process z^(t) is a strong Markov process, the average (x) T
a renewal process imbedded at level x if distinction is made
between up-crossings and down-crossings. That is, define
P'*'(t,x) = pfz(t) ^ X, z'(t) = l|^| and m'*'^^^(T) as the expected
nvunber of level x uo-crossings of z^(t) in (0,T] , then (3.3) + (x)
holds for these functions and m , (T) is a renewal function.
<P
An analoguous relation applies for down-crossings.
1.4. Level crossings by the content process v(t).
As announced we assume that X, the average number of jumps
per time unit of the input process X(t) is finite. With this
assumption, only a slight modification in the proof of the
theorem of the previous section is required to obtain as the
equivalent of (3.3) for the content process v(t) for
Re s .^ 0, Re 5> = 0 that
CO
e"'^^ ƒ e-^^d^m^^^t)dx = j e"^* j f."'''d^P^(t,x)dt, (4.1) 5 0 0
where P^(t,x) = p{v(t) ^ x(v(0) = v , m^^^(t) = E|N(T,X)|_V = v|,
N(T,x) is the number of level x cro.-ïsings of v(t) in (0,T] and
assumed character of the input X(t) it follows that there
exists an s > t such that v(u) < x for ue(t,s) if v(t) = x.
Hence the level x crossings of the content process v(t) are all
down-crossings.
From the time..dependent behaviour of the process v.(t)t
determined by (3.2.3), it follows that the transition functions
of this process are continuous in the initial condition
v(0) = V. Further, the process is a right continuous,
time-ho-mogeneous Markov process. Therefore v^(t) is a Feller process,
as defined by Dynkin, and hence a strong Markov process (cf.
Dynkin (1961) p. 114). It follows that m^^^(t) is the renewal
function of the renewal process imbedded at level x in the
strong Markovian process v(t).
The relation (3.2) underlies all derivations and results
in the subsequent chapters concerning the continuous time
para-meter dam process. The only restriction on the validity of
(3.2) for these processes is the condition that A should be
finite. All attempts to give a rigorous deduction of its
vali-dity for infinite A have not been succesful as yet. It is
strong-ly conjectured, however, that (3.2) and even (3.1) (for almost
It should be mentioned that an approach as given by the
author in his paper Roes (1970a) leads to a rigorous analysis
of the transient behaviour of the finite dam with input X(t)
for which A = co(although a rather weak condition on ^(f) (cf.
(2.13)) still has to be imposed). In the latter approach,
how-ever, the various Markov times that emerge in the present
stu-dy, do not play any role and the same difficulties would arise
if one wanted to find them.
1.5. Review of recent research on related dam processes.
The early research on dams was reviewed by Gani (1957) and
Moran (1959). Subsequently, Prabhu (1964) gave a comprehensive
survey of the timedependent results obtained. We give a brief
summary of the work finalised since then and a discussion of
three methods which have been suggested recently for dealing
with the problem on hand i.e. the calculation of the
timedepen-dent distribution of the content of the finite dam.
Hasofer (1964) gave a rigorous derivation of the
distri-bution of the residual wet period for an infinite dam. This
to first emptiness of an infinite dam with prescribed initial
content. In Hasofer (1956) the model for the infinitely deep
dam with compound Poisson input was develloped with a view to
investigate the transient behaviour of a finite dam which is
nearly full. He derives an integral equation for the
timedepen-dent distribution of the content, shows that it has a unique
solution and produces same. Hasofer (1956) also gives the
limiting distribution of content for increasingly large values
of time and the distribution of the time to first overflow for
this model.
Phatarfod (1959) uses a birth and death technique to
find for the finite dam with Poisson input the transform of the
time to first emptiness before overflow and of the time to first
overflow before emptiness.
In Prabhu (1968) useful asymptotic results are compiled
for the infinite dam, for the residual wet period if the
ini-tial content is large, for the part of the interval (0,t]]
during which the dam is dry d (t), for the nett input and for
the dam content v (t) for large values of t.
In Roes (1970a) we were able to find the transient
characterising function ^(p) (cf. 2.13)) had the property
^(?) = 0(y), o< < 1, as |j>( ^ oo , Re J> > 0. This was achieved by
enclosing the content process v (t) between two processes and
applying Cohen's (1969) analytic method to the latter. The
method subject of Roes (1970b, 1970c) is the one which has been
stated more precisely in the present study.
As a means of comparison with the present approach as
summarised at the end of the first section, we discuss three
alternative methods.
Green's Function Method.
Keilson (1965) introduces a method to deal with bounded
processes. This method is in one respect not unlike the one we
give here in that he too first considers the unrestricted
pro-cess and derives the behaviour of the restricted propro-cess from
it. In order to give an idea of Keilson's method we quote a
simple example.
Consider the process |w , n = 0, 1, 2, ...y defined by
5n = 0 if E^_i + ün « 0 , (5.1)
= w , -I- u if 0<w . + u < K, —n-1 —n —n-1 —n ^ '
where |u ,n = 1, 2, ...> is a sequence of independent random
variables with common distribution A(.). Let A(K,y) =
= pjw i yl w .. = xj be the single step transition distribution
of w . It follows for 0 $ X ^ K that
A(x,y) = 0 if y < 0 , (5.2)
= A(y-x) if 0 ^ y < K,
= 1 if y ^ K .
Let P (x) = Pjw < x| be the distribution of w , which is
pres-n I—a * J -n' ^
cribed for n = 0. We have for n = 1, 2, ... that
P^(y) = ƒ A(x,y)dF^_^(x), (5.3)
which may be written as
Pji^y) = /A(y-x)dP^_^(x) + C^(y), (5.4)
where the compensation function
modifies (5,4) in the sense that (5.4) would be the equation
for the vuirestricted process but for the presence of C (y).
Defining C (y) = P (y) we have from (5.4) that
n
^J^^ ' Y i \
A^''"''^*(y-x)dCj^(x), (5.6)
k=othe star indicating the convolution. Prom (5.5) and (5.6) one
obtains
C^^l(y) = f[A(x,y) - A(y-x)] fi Uy''-^^(x-z)dC^(z), (5.7)
from which the C (y) may be obtained recursively.
A similar treatment is possible for a compound Poisson
process, the analogue of (5.7) being an integral equation for
the compensation function rather than a difference equation.
For the transient behaviour of the finite dam with
com-poiind Poisson input Keilson (1965) proposes to apply a limiting
procedure j-> oo to the sequence of processes v^.(t) = v -H X(t) +
J
-I- Y.(t). The process X(t) is the compound Poisson input, Y.(t)
J J
is a compound Poisson process with jump frequency j and jump
limit-ing procedure should be executed, taklimit-ing into account at the
same time compensation as sketched above.
It appears that much remains to be worked out before this
method can be fully evaluated. The results so far indicate
suitability of the method for obtaining the limiting behaviour
of v(t) as t -»• oo, but the usefulness of the approach for the
transient behaviour is not quite so clear as yet.
Combinatorial Method.
Takacs (1967) gave, after studying generalisations of the
classical ballot problem, the following theorem for processes
with cyclically interchangeable increments.
"If |x(u), 0 .^ u i tj is a real-valued seperable stochastic
process with cyclically interchangeable increments almost all
of whose sample functions are nondecreasing stepfunctions
vanishing at u = 0, then
Pfx(u) < u for 0 < u $ t | x ( t ) = y] = ( t - y ) / t i f O ^ y ^ t , (5.8)
and zero otherwise, where the conditional probability is
defined up to an equivalence".
section 2 is particularly noticeable in view of the formulae
(2.4) and (2.6). Rather than inquiring into the proof of this
theorem, which involves a thorough examination of the sample
paths, or into the way Takacs (1967) applies his result to dam
processes, we take the opportunity to illustrate the usefulness
of the theorem by applying same to part of the area of
conjec-ture of section 4. More specifically, we show that N(T,x), the
number of crossings of level x by the process v(t) in (0,T]] has
finite moments of all orders, irrespective whether A = oo or not.
To start with, it is easily seen that the jumps of v(t)
exceeding a positive value u, occuring in a Poisson stream (cf.
discussion following (2.13)) with mean N(u), cannot attribute
to the value of N(T,x) to the extent of causing it to become
infinite. Indeed their contribution is less than 1 -i- Tu . Now
consider the process v (t) v/ith an input from which the jumps
exceeding u are suppressed; that is v (t) is a content process
v_{t), for which the input characterising function ^(.) equals
u
o-i-^ „ ( 9 ) = I (l-e"o-i-^'')dN(x). (5.9)
u a, = ƒ xdN(x) < 1. (5.10) 0 + u
'. = ƒ
(5.11) 1This is always possible by virtue of the fact that / xdN(x)< co
0 +
Next, we appeal to Takacs' ((1967), p. 37,38) theorem,
based on the cyclical interchangeability of the increments of
the input, restated in terms of the content process v(t) for
0 :$ y ^ t as
P [ V ( T ) < 0 for 0 < T ^ t|v(0) = 0, v(t) = -y] = 2,
from which it is concluded that (cf. convention introduced
immediately preceding (1.2))
pfv(T) < 0 for 0 < T ^ t | v ( 0 ) = o| = E | [ - ^ ^ J |V(0) = o| , (5.12)
Choosing t = 1 , we obtain in particular that
P [ V ( T ) < 0 for 0 < T 4 l|v(0) = oj = (5.13)
1
= 1 - P [x(l) > IJ - I xd^pjx(l) $ x ( ^ o
oo
> 1 - / xd^pfxd) é x| = 1 - a.
Application of this inequality to the process v (t) with
trun-cated input, yields that
P [ V ^ ( T ) < 0 for 0 < T ^ l|v^(0) = o| ^ 1 - a^ > 0. (5.14)
This implies that there is a positive probability for the time
between two successive crossings of a level to exceed unity.
A standard argument (cf. Prabhu (1965) p. 155) yields for
k = 0, 1, 2, ... that
E [(a(T,0))*'|v(0) = o| < 00. (5.15)
Analytic Method.
The transient behaviour of the finite dam with compound
Poisson input was first found by Cohen and is given in his book
Cohen (1969), Actually, he finds the transient behaviour of
input has occured. Prom either of these processes the content
at any time t may be found in the same way as Cohen (1969,
p.294) does for the G/G/1 queue. The advantage of dealing with
imbedded processes is the possibility this gives to show that
the solution of the integral equation, which has to be solved
in the course of the derivation of the transient behaviour,
has an unique solution. Prom the transient behaviour of the
im-bedded processes Cohen (1969) derives the distributions of
several Markov times.
We give an outline of our paper Roes (1970a), which is
very similar to Cohen's (1959) treatment of the finite dam.
First it is shown that the distribution of the process
ju(t), t€|0,OQ)V converges to that of v (t), where u(t) is
de-fined for t = nA by (cf. convention introduced just before
(1.2))
u(0) = V,
u ( t + A) =
[K- A + [ u ( t ) + X(t + 4) - A(t) - K[~y ,
u ( t + r ) = [ j i ( t ) - T]"^ f o r O < t < A .
integral equation for U(5>,s) = / e ^]e — ]dt
e i-x 1 y -'nx ~ 2TTW (P-(p -ii)n e d-ri where 0 < Re'ri < Re ? , (5.17)
-?x
e ^ " 2ÏÏÏ -/ "(yT^'^^'di^ "here [Re fj"" < Re 1^ ,
and the contour B is along the vertical line Re T| = c from
c - ioo to c + ioo. This is achieved as follows. From (5.16) and
(5.17) we have for 0 < Re 7J < Re e < Re? that
-?u(t+A) _
j
^
r_y -^(k-A)
® 2ni J (.9-'Tin 15.IB;
[T _ _ i _ f _l!^_p-e(u(t)-i-x(t-i-A)-x(t)-K) ]
A similar identity is obtained from u(t) = [K+|u(t)-KJ~J . On
subtraction, division by A, taking the limit A->-0 (which
in-volves some nontrivial interchanges of limit operations and
for which we make use of the additional assumption ^(i) = = 0(f>°'),o<< 1, \if\ -> oo , Re ? > 0) and Laplace transforms with respect to t it is found that U(5>,s) satisfies for Re s ^ 0,
(s-f-H^(y))U(?,s)-e"^^+ U(oo,s) = (5.19)
B
Further, U(j,s) is an entire function complying with
lim U(j,s)e?'^ = 0 for I n < arg ? < |lt. (5.20)
Subsequently a trial solution is substituted in (5.19) leading
to the conclusion that a solution of (5.19), satisfying (5.20)
is given by (6.3.14).
Summarising, we can say that the method presented in this
thesis as compared with the three discussed above is the more
probabilistic of the four, except possibly for Takacs'
combina-torial approach, to which it is probably closest in spirit. In
as far as usefulness is concerned, it should be borne in mind,
that, although our method provides us with the distributions
of many Markov times, for Markov processes in discrete time it
can only be applied if the state space of the process is
crete. We have not yet been able to extent the method to
dis-crete time parameter processes with a continuous state space.
pro-cesses as the Markov process imbedded in the G/G/1 queue. This
restriction does not apply to the Green's function and Analytic
Method described.
1.5. Some notational and anal.ytical detail.
In the sequel we will be lead to consider the equation
in p,
p = sP(pq), (6.1)
where P(.) is the generating function characterizing the input
for the discrete dam (cf. (1.7)). Prom Rouche's theorem (6.1)
is easily seen to have a unique root w = w(q,s) in the domain
|pq| ^ 1 provided that iqsj ^ 1.
Similarly, for dams in continuous time the relevant
equa-tion is
f - s = ^(tf-H?), (6,2)
where ^(,) is defined in (2.13). V,'e quote from Takacs (1967)
plane Re p ;^ - Re o- if Re(s-i-cr) > 0 and note that Cti(a,s) = => 6>(0,a-+s) - a.
In order to reduce the complexity of notation we introduce
the following short hand conventions.
ig(u)du = ^ ^ /g(u)du, (6.3)
C
where C is a simple closed contour encircling.the.origin,
tra-versed in counterclockwise direction and such that |uq| .^ 1 on
C. The dummy variable p will always be used if C should not
en-close the point w, while the dummy variable t refers to a
con-tour C encircling the point t = 1 but again not encircling w.
Analogously, integrals without limits and Greek dummy
variables are defined by
/g(e)de = lim ^ / g(e)de, (5.4)
"^ x-iy
for specified x = Re ö. The dummy variable f will always be utilized when x > Re co; the dummy variable ^ refers to a
simi-lar integral along a line parallel to the imaginary axis such
zero, an appropriate indentation of the contour is assumed so as to let it pass to the left of the origin and to the right of a and always such that the contour is in the semi plane Re^j > - Re ff. These contours are displayed in figures 3 and 4.
Imp Rep Figure 3 A p contour -Reff Imr| Req -Recx Imri
t^
Req
Reu < 0 Rew > O Figure 4 An Ti contourThe absolute values of the complex variables p, q, r and s
occuring in Chapter II are always less than unity and nonzero
unless specifically stated otherwise or specifically implied by
the definition of the contours just defined; often analytic
continuation to zero or unit absolute values of the variables
is possible. Similarly, the real parts of p, a ,X and s in Chapters III and VI are always positive unless specifically
Por the discrete dam processes of Chapter II a star
indi-cates that the quantity refers to the finite dam; if no
super-script is attached the quantity refers to the infinite dam. For
the continuous dams of Chapters III to VI the absence of a
su-perscript indicates that a quantity refers to the unrestricted
dam. The superscripts minus, plus and star refer to the
infinite-ly deep, the infiniteinfinite-ly high and the finite dam respectiveinfinite-ly.
In the sequel all intervals are left open right closed;
thus (.,.) should be read as (.,.].
Finally, we refer to (1.4) for the definition of the
The dam processes described in the first section of the
previous chapter will now be investigated. These processes,
being Markov Chains, have imbedded in them Renewal Processes.
This fact may be exploited either to derive the distribution of
various Markov times from the transient behaviour or
alterna-tively the latter from the former. Such derivations are based
on Feller's Theory of Recurrent Events. Both approaches will
actually be used in the course of the calculation of the
tran-sient behaviour of the finite dam from same of the infinite
dam.
The present chapter is included mainly in order to
illustra-te the underlying ideas in relation to simple models. The
ap-proach will be used extensively in the next chapters.
2.1. The transient behaviour and first entrance times for the
infinite dam.
Consider the Markov chainj z^ , n = 0, 1, 2, ...j defined
^o
z , = z - H X - l i f z > 0 , (1.1) —n+1 —n —n -n '^
= X if z = 0.
—n —n
This is the content process of the infinite dam defined in
(1.1.1), where we have taken M = 1, K = <», Further, we will
in-clude in the subsequent derivations the processes X(n) and
d(n), defined for n = 0 , 1, 2, ... by and X(n-fl) = X(n) + X (1.2) — — —n d(n+l) = d(n) if z > 0, (1.3) d(n) + 1 if z = 0, — —n
whereX(O) = d(0) = 0. These have been discussed in section !•!
and are the total input and total dry time for the infinite
dam in the time interval (0,n).
The recurrence relations (1.1), (1.2) and (1.3) yield for
the trivariate probability generating function of the three
processes considered that
, z^ , X(n-fl) d(n+l)| ) , f z X(n) d(n) I 1 E[p""'q r |z^=zJ=iE[p"q r ( V ° ) U o = ^ J '
X
-I- rE|q r (z^=0)| z^=zJE ((pq) " j , (1.4)
where (A) stands for the indicator function of the event A.
On definition of Z (p,q,r,s) = X_ E'jp q r \z =z\s ^ n=o
we have from (1.4) and Z (p,q,r,0) = p that
p^*-'--s(l-pr)P(pq)Z^(0,q,r,s)
h^P''i'^'^^ = p-sP(pq) • (1-5)
The quantity Z (0,q,r,s) is easily determined from the fact
that z +n = X +d(n) + z, so that Z (p,q,r,s) is analytic for
pq c 1, lqs| < 1, Jl <i 1 and |pr| < 1, |sr| < 1, |^| ^ 1. \
The unique root of p = sP(pq) in this domain w = w(q,s) thus
must be a zero of the numerator of (1.5) and hence (of.
(1.6.1))
Z^(0,q,r,s) = Y ^ . (1.6)
The transient behaviour (1.5) with (1.5) of the infinite
dam may now be used to find the probability generating
func-tions of the first entrance times of same. A first entrance
time f is the time it takes the content process to evolve -xy
precisely, the first entrance time f for the infinite dam is
defined for nonnegative integer x and y by
f^y = minjn : z^ = y, n > 0 ) z^ = x|. (1.7)
jz^, n = 0, 1, 2, ...I i!
Since the process jz , n = 0, 1, 2, ...V is a Markov Chain,
the event z^ = y constitutes a recurrent
event(Peller(1950)).Con-sidering the content process only in as far as the entrance of
a certain state is concerned, we obtain a Delayed Renewal
Pro-cess for which we have from Feller's Theory of Recurrent Events
that ^
The left hand side of this relation is easily found from (1.5),
since it is the coefficient of p'^ in Z (p,0,0,s).- p^,
A convenient means of selecting this coefficient is through
division of Z (p,0,0,s) - p by p^ and subsequent integration,
so that we may write for the left hand side of (1.8)
/ z (p,0,0,s) - p''
where the contour integral is to be taken as described in
sec-tion 1.6 • Inputs and dry times during disjoint time
inter-vals are independent, particularly so for same during disjoint
first entrance times. Therefore expanding (1.8) we obtain for
X < y that E I S ^ V ^ ^ ^ ^ - ^ ^ ^ ^ I = 1 - I I W L / (l-pr)P(pq) jpl~\ (1^10)
' (l-rw;(sP(pq)-p)p''^^ I
I f x(4y) d(4^)| _ (i-^)w-ytiP(^^F^i
r ' 5 S I (l-pr)P(pq)
'
^ (sP(pq)-p)py^-^ and ( f X(f ) d(f )1:ls-y^q--y^ r--y^ t = w^-^
(1.11) E)s '"q ^" r •»" i = w-'^. (1.12)From these probability generating functions, we may now
derive analogous functions for taboo first entrance times.
A taboo first entrance time f is a first entrance time -u;xy
from a state x to y with the extra requirement that the taboo
state u is not entered in the mean time; that is
On noting the fact that a first entrance form a state
X to y occurs either without entering the state u in the mean
time or after the state u has been entered, it is found that
E J S M
=
EIS^'^^^
+ Els-y-'^Els^y^. (1.14)
Interchange of u and y in (1.14) yields a similar
expres-sion. The joint probability generating function of the taboo
first entrance time f , the input X(f ) and the dry time -u;xy' '^ u;xy
d(f ) in the intervening time then follows from these ex--^—u;xy
pressions, using (1.10), (1.11) and (1.12). Note that not all
possible combinations of u, x and y result in true taboo first
entrance times; for instance f is simply the first entrance -u;xy
time f if u < y <: X, due to the character of the content
pro--xy •' 3SS [ z ^ , n = 0, 1, 2, ...!,
2.2. First surpassage times for the infinite dam.
For the description of the relation between the infinite
and finite dam we further require the notion of surpassage. A
first surpassage time £ is the time required by the content xy
process ^ to go from the state x, via some state not exceeding
y if X > y, for the first time to a state exceeding y. Thus,
such a Markov time may be defined for any nonnegative integer
X and y by
£^y = minfn : z^.-^ * y< z^.n^oUo = xj . (2.1)
Note that we only need consider x i. y, since in the
comple-mentary case the first surpassage time is the sum of the two
independent times f and p . For the calculation of p ,x ^ v.
-xy -^yy -"^xy* ^ ^'
we note firstly that the first surpassage time £ does not
exceed n either if the content process, starting in x, is at
time n in a state exceeding y, while the state y + 1 has not
been entered in the mean time, or if the content process,
star-ting in X, has entered the state y + 1 in the Interval (0,n),
so that ( 2 . 2 ) n - 1 r ( ^ y ^< \ »z > y | z = x [ - ^ P | Z >y z = y-nlV P t f = ml + p | f -,<n\ = >- S ^ P I Z > y | z = y-i-lfplf = mi + P f ^^nf
) £^0 r''""' ° J (-x.y+i J [-x,y+i- J
P<z >y z = X (j-n ' —oIn order to obtain the joint distribution of £ , X(p ) and
d(D ) we note secondly that d^(D ) = d(f . ) , since there
is a one to one correspondence between relevant overlapping
first entrance and first surpassage times, while during the
(pessibly infinite) time interval from elapse of the first
sur-passage time D to elapse of its overlapping first entrance
^ocy
time f ., the d^(.) process is constant. Hence, when
expan-ding (2.2), we will consider the input process throughout the
interval (0,n), but restrict our attention to the d(.) process
to (0,f .. ) or (0, D ) , whichever is more convenient. It
follows that
V-„r X(n-m)]„[, , X(m) d(m)f „[, , X(n) d(n)|
\
,„ .,,
2_E q-'' 'JE (p =m)q-' 'i-' 'j =E|(z^>y)q-' 'r-' U Q ^ X J + (2.3)
r^ï-f/ V ^ X(n-m) d(n-m)| •,l„J/, N X(m) d(m)l - 2_E[(ln-m^y)l ^ l^o=y*y((^x,y+l="'^'l- ^ J ^
On taking generating functions with respect to n it is seen that
1 f £xy i(£xv^ -^-^v^l /"z (P,q,r,s)
- ^ Z ^ , , ( l . q , r , s ) - ; . ^ ^ i ^
. - ^ (;-tr)P(tq) ^,J„x_^y^iEf3^.y.i/^^.y^i^/^^.y.i^j|,
l-^/(l-t)(sP(tq)-t)ty^l ( 1^ '^ -"^^
where the last two integrals are to be taken along a circular
contour in the t-plane such that tq < 1 and that the contour
encircles the point t = 1 but not the point w, the latter
con-dition may require a local indentation of the otherwise
cir-cular contour. Thus we obtain from (1.11) for x < y that
( ^ y ^ <
[s^^q
! i
i / ( i - t
^x-y
) ( s P ( t q )
xy^
J-t)^
l-
(l-tr)P(tq)
dt -
9 ^^'^'^
dp Ml-t)(3P(tq)-t)t^^^"
^* ^sP(pq)-p^P / (l-pr)P(pq) ^^
/(sP(pq)-p)py'-2
and
T
L
(l-tr)P(tq)^,_
£ , X(£^^) d ( £ ^ ) l-sP(q) 7(l-t)(sP(tq)t"^^_
\ ' '^ "^ J ^ SP(0) / (l-pr)P(pq) ,„ •
X-H2^P(sP(pq)-p)p-We reinforce a previous remark by noting that the trivariate
(2.5)
generating function of P ^ , X ( p ^ ) and ^ ( o ^ ) is the product
of (1.12) and (2,5),
Finally, we define the taboo first surpassage time
simi-lar to the first surpassage and taboo first entrance time; the
taboo state must not be entered in the intervening time. The
taboo first surpassage time D _ for the infinite dam is
de-fined by
^ ; x y
"
'"^"[''=^n-l ^ y ^ -Sn' -^ ^ "• ° ^ " ^ ''1^0 = ^ j ' ^^'"^^
where u, x and y may have any nonnegative integer values, only
nine distinct cases are of interest however. The generating
functions for these variables follow from an argument not
un-like the one indicated in relation with the taboo first
en-trance times as follows. A first surpassage from x to y may
eventuate in two exclusive and exhaustive v/ays. Either the
surpassage occurs without the state u being entered in the me.;n
time or after state u has been entered at least once, that
As before, trivariate generating functions may be obtained
through inclusion of input and dry time in (2.8) during
rele-vant time intervals.
2.3. First entrance times and transient behaviour for the
finite dam.
'J^he behaviour of a finite dam of integer capacity K may
now be deduced from the results of the previous sections. The
recurrence relations which govern the content process
j ^ , n = 0, 1, 2, ...7 were given in (1.1.1), but we will not
actually use them. Instead, consider the behaviour of the
finite dam at the upper boundary and note that this is the
only aspect in which it differs from the infinite dam.
When-ever an input occurs which were to cause the infinite dam to
assume a content exceeding K, the subsequent behaviour for the
finite dam follows by reducing this amount to K. From this
ob-servation we derive the first entrance times for the finite
dam defined by
A first entrance of the state y by the content process z
starting from an initial value z, = x may occur after a number of overflows. Actually, this number of overflows may serve as the discreminator for an exhaustive system of mutually exclu-sive events leading to such a first entrance, so that
JlA] - z[r^'^'^^] .
E[s^y'-K]E[s-^^l'^|£E[s^y'^^Jj"
Substitution of the relevant generating functions, indicated in the previous sections, yields for x •< y that
\ f* X(f* ) d*(f* ) | x y ^ y xy j -E(_S q r ,x-K
/ t^-^ / — ^ ^ - ^ ^ ) ^ ^ ^ q ) — d t / p ^ - y
/(l-t)(sP(tq)-t)^^ _7(l-t)(sP(tq)-t)t'^^^ /sP(pq / ( l - t ) ( s P ( t q ) - t ) ^ * ƒ (l-t)(sP(tq)-t) r f* X ( f * ) d''(f* ) \f
(l-pr)P(pq) sP(pq)-p)p y+1 dp (l-tr)P(tq) (1-1;)(nP(tfi)-t)t K-Hl dt / ty-^ / (l-Pr)P(pq) ƒ (1-t) (sP(tq)-t)'** J (sP(pq)-p)py*^ -dp (3.2) (3.3) (3,4)and
f" X(f* ) d*(f* )
s-y^q- -y^ r " "y^
h^
y-K
(l-t)(sP(tq)-t) dt .x-K
-dt
(l-t)(sP(tq)-t)
By a reversal of the argument we now find the transient
behaviour for the finite dam from these generating functions
of the first entrance times. The finite dam content process ^
is in the state y if it is so for the first time or second
time or third time etc. It follows that
• ^ n„f, 3€ - X(n) d*(n)l » y s E (z = y)q-' 'r- ')z^ = xl =
n ^ L
r f* x(f* )
Els-^yq- - ^y r d*(f* )] . . ( f» x(f* ) (ïXf* )]g-yyq- -yy y- -yy '
For the tetravariate generating function of content z^ , input
X(n) and dry time d (n) we find
^ ^ r z* X(n) d*(n)| 5,q,r,s) = ^ S ' ^ E | ^ P '^q r |z^ = 'l-(2) K+1 u-p u-sP(uq) z+1 (l-ur)P(uq) j'(l-t)(sP(tq)-t)'^^ (l-tr)P(tq) ,K+1 •dt (3.5) (3.6) (3.7) ;du, (l-t)(sP(tq)-t)t
where the integral with dummy variable u should be taken along
origin but neither encircling the points u = w nor u = p. Note
that, if the dam is initially full, there is a simple relation
between the transient behaviour of the process relating to the
finite and infinite dam as follows
n f z* X(n) d"(n)| 1
>\jp "q r |z^ = KJ
n=o ^ „ f z X(n) d(n) , ^2 _ S " E [ P
"q r (V^^l^o = ^j
£_s\U^^n
- £:s"Efq^(")r^('^)(z^>K)|z^ = K } '
n=o
(3.8)In the present chapter we consider an unrestricted Random
Walk in continuous time which is skip free in the negative
direction. It can be thought of as a dam with additive input,
where there is no restriction on the content. For this model
we derive first entrance time distributions, using the relation
from the first chapter and Renewal Theory. Subsequently first
skip time distributions are calculated. The transforms of
these, apart from being of interest in their own right, are
used in the following chapters to lead eventually to the
tran-sient behaviour of the finite dam.
3.1. Introduction.
The sample functions of the content process | v(t), t ^ 0 f
of the unrestricted dam are the familiar succession of line
segments with slope minus one. They are right continuous and
the upward jumps are due to inflows of water which always
occur instantaneously.
As we will be considering the renewal processes imbedded
^ as the time it takes the v(t) process to evolve from an
initial state x to a state y for the first time. In terms of
the sample functions of v(t), the state y is entered at t if o
the line t = t^ intersects a sloping line segment of the sample
function at v(t) = y. The state y is usually approached from
above; if a jump in the v(t) process is such that
v(t-) < y < v(t-f), the state y is said to be skipped at t. The distribution of ^ may be defective.
xy V(t)
k
^h
si
\ .
*W*x.yy « y .\
t « X = 1 y\
» y y«
\
X,
l \
Ttuu =TtuyN
Ttyxs
N
» y yK
1 •»u.yy\
\
t1«u = "^xy = ty.«u ='^y TWy =Tlu, Ky
Figure 5
First entrance and skip times
defined similar to ^ with the requirement added that the ^xy
state u is not entered in the mean time (but may be skipped at
some stage). Taboo first entrance times generally have defective
distributions,
We further require for the description of the relation
be-tween the unrestricted and restricted dam processes the first
skip time X . This is the time required by the v(t) process to "Tcy
go from the state x to the first jump such that y is skipped.
Taboo first skip times ir are defined similar to taboo first -u;xy
entrance times; the taboo state must not be entered but may be
skipped at some stage in the mean time. The distributions of
first skip times are usually proper if x 4 y, those of taboo
first skip times usually not.
3.2. The transient behaviour of the unrestricted dam.
let v(t) be the content of the dam at t and let the initial
content be ^(0) = v, -00 <v < 00. The content at any time t equals
the initial content v plus the input during the intervening time
interval (0,t) minus t, the release rate being unity. Thus we
v(t) = V + X(t) - t. (2.1) Define oo '"'"'"''"" l(0) = vidt, (2.2) oo „ ^ ^.^ f -st„f -^v(t)-crX(t)l , V^(9,a.t) = je E^e ^-' ' -' ' | v(
for Re (s-j')>0. Re (a--i-i')>0. It follows (cf. (1.2.13)) that
Re (s-i'+€(o-+y)) > O in the domain of definition of (2.2) so that
in this domain V^(?.<r.t) = e-y^j e(^-^)*E{e-(^*y)^(*)]dt (2.3) O CO
^.j>vJ^-(s-y + ^(^+f))t ^^
e o e-^^ s - y + ^ i"+f)'Note that this expression is continuous in the initial
condi-tion V = V so that, with the other properties imposed, the
process is strong Markovian (cf. the end of section 1.4).
On behalf of the strong Markov property we have from the
imbedded in the content process and the distribution function of the latter (cf. (1.4)) for Re ? = 0 that
<o CO
|e-*'y |e-^*d^EJ2^y)(t)e-'^^(*)lv(0) = xj
dy = V (5.,^,t), (2.4)where ^ y ( t ) is the number of downward level y crossings in the time interval (0,t) of the content process v ( t ) . That i s ,
v^'^''(t) is the number of times the state y is entered in (0,t). It follows from the inversion formula of the Fourier Transform (cf. Widder (1945) p. 241) for any real x and y that
^ ( y ) ( ^ , s ) = j e-^\E|v^(y^t)e-'^2(t)|,(o) = ^ j = (2.5)
,'!?'(y-x)
Xi-f+'C^)^'^^ " f(x = y ) , Re(s+«-) > 0, Re'i?'= - Recr.
It follows for X < y that
( y ) . , „ . f e « y - )
A x (-• = > ° y s - V . ^ ( . . » ) ^ ^ ' (2.6)
.
^^h^ .\ - I '^'^'^
i _ 1
-
, V(g+^) ,, .,x
Px ^"^'^^ ' i s -'t- -I- \{<^+^ " 2 ~ 1 - t'(5 + c^) " -^ = 1 -T'(a-^>^)' ^ ' ,(x), ^(o-+^l 2 1 - ^'(»+^) 1 - T'r)'^'^'^^ =/s-;-.T(^^.t»^'^ = i - V ( x . . ) ' (^•«^
where the last two integrals have been evaluated using the fact
that %{<}) = 0(?) for |?| ^ oo , Re ? > 0, (cf. Takacs (1967) p. 42) and our knowledge of the location of the zeros of the
denominator (of. section 1.6).
The present elementary derivation from which, it will be
realised, the first entrance times follow (cf. next section),
clearly indicates the power of the theorem derived in section
1.3. Indeed, one only needs to compare the present apparatus
with those required by Kemperman (1963), Hasofer (1954, 1955),
Phatarfod (1969) and Cohen (1969) to obtain only some of the
results of the next four sections.
3,3. Calculation of the first entrance times ± •
The first entrance time * for the unrestricted dam is
defined by
±^ - infft : v(t) = y|v(0) • xj,
where x and y may have any real value. Writing for brevity's
1 -s4v -'^^(^v^i
'f'^yU.s) =
E[e ^ y ^ y j , (3.2)
we have from Renewal Theory for any real x and y that
(v) ï'xv^"''^^
Tx (--^^ °l-7(^.s)- (^-3)
^yy
Thus from (2.7) it follows that
^^^(•r,s) = 1 - [ l
+/^^'^''(<y,s)j
= f(<3--Hu>) = f'(u:(0,o-+s)). (3.4)
Note that the latter transform is only a function
of the sum (r+ s. This is due to the fact that, at moments
the content returns to the initial value, the time and
accumu-lated input equalize. We employ (3.3) again, but now with
(2.6) and (2.8) to yield for x < y that
t^/^.^)
= (1 - ^(a.u3))l^
.^.^(a^i^^'^=
(3.5)
= e
^ „ ( ^ . s ) - e'^(^"y^ (3.5) yx
3,4. Calculation of the taboo first entrance times * 2LL.
The taboo first entrance time <f> for the unrestricted ^ ; x y
dam is defined by
^ . ^ y = inf[t : v(t) = y, v(ir) + u, 0 < f < t|v(0) = x j , (4.1)
where u, x and y may assume any real values. Taking into
account that the process v(t) is skip free in the negative
direction, some values are ruled out either because the
re-sulting taboo time would be an ordinary first entrance time or
because the route implied is impossible. We are left with the
following eight taboo first entrance times of interest, where
u < x < y:
^ ; x y -y;xu
•^;uy % ; u x
^ ; x x •^x;xu
The variables as listed are found pairwise. We appeal to
the usual arguments (cf. Chung (1950)) when dealing with taboo
states. A first entrance from x to y may be effectuated in the
following two exhaustive and mutually exclusive ways. Either
by passing from x to y avoiding the state u in the mean time,
or by passing from x to u avoiding y and subsequent passage
from u to y. The v(t)process, being a strong Markov process,
has independent non-overlapping (taboo) first entrance times
É. . » .É, ^y virtue of the strong Markov property. We conclude for any real values of u, x and y that
-s
e
'"^y) - E(e"^^'^yJ + E[e"^^'H E[e"^^yJ. (4.2)
Writing again
4> (,r,s) =E(e"'^'^y "''-^^'^y^(,
^u;xy ' ' t . J '
(4.3)
we have by the afore mentioned equations (4.2) and (3.4), (3.5)
and (3.6) for u -c x < y that
t^„('^.s) - <= (cr,s)* (cr,s)
