Application of Methods in Sequential Analysis to Dam Theory

Abstract

Wald's Fundamental Identity in sequential analysis [15] has been widely used for various applications apart from sequential sampling. Bartlett [1] used it for the insurance risk problem and also for the random walk and gambler's ruin problem. It is the purpose of this paper to show how the Fundamental Identity can be used to derive certain results in Dam Theory. The paper is purely of an expository nature and considers only simple examples; more difficult problems will be dealt with in a future paper. We consider here a continuous time dam model due to Gani and Prabhu ([5], [6], [7]), based on Moran's ([10], [11]) discrete time model. Briefly it is as follows: (a) The dam has finite capacity $K$. (b) Let $X(T)$ represent the input (including the amount overflowed, if any) during a time interval of length $T$; we assume that $X(T)$ is an additive process with stationary increments. It is known that for such processes the m.g.f. is given by $E\{e^{\theta X(T)}\} = e^{-T\xi(\theta)}$ where $\xi(\theta)$ is a function of a specified type (Levy [9]). (c) The release is continuous and occurs at a unit rate except when the dam is empty. It then follows that the net input (including the amount overflowed, if any) in the dam during the time interval $(0, T)$ is $Y(T) = X(T) - T$ whose m.g.f. is $M_T(\theta) = e^{-\theta T-T\xi(\theta)}$. For $T = 1$ this gives $M_1(\theta) = e^{-\theta-\xi(\theta)}$. Let us denote the dam content at time $t$ by $Z(t)$, with the initial content $Z(0) = u (0 < u < K)$; we have, then $Z(t) = u + X(t) - t$. It follows from the above assumptions that the stochastic process $Z(t)$ is a temporally homogeneous Markov process. Following Moran's first paper, attention was concentrated mainly on investigating the stationary properties of the system. Later, Kendall [8], Gani [4], Prabhu [13] and Weesakul [16] considered the problem of emptiness of the dam; it is with this second problem that the present paper is concerned. Considering the dam process described above as a random walk with barriers at $Z = 0$ and $Z = K$, the process starting at $Z(0) = u$, we obtain the probability that the dam becomes empty (i.e. the process terminates at 0) before it overflows (i.e. the process terminates at $K$) and the probability of the reverse situation; further, we derive the probability distribution of the time at which the dam becomes empty. We do this by making use of the following extension of Wald's Fundamental Identity to continuous time parameter by Dvoretzky, Kiefer and Wolfowitz [3]. If $\{Y(t); T \geqq 0\}, Y(0) = 0,$ is a process with stationary independent increments and if $M_1(\theta) = E\lbrack e^{\theta Y(1)\rbrack}$ exists for all $\theta$, then \begin{equation*}\tag{1.1}E\lbrack e^{\theta Y)t + 1)} M_1(\theta)^{-t}\rbrack = 1,\end{equation*} where $t$ is such that $Y(t) \geqq b$ or $Y(t) \leqq a$, while $a < Y(\tau) < b \text{and all} \tau < t$. (Here $a$ and $b$ are constant, $a < 0, b > 0$.) We also require a lemma due to Bartlett [1]. Lemma. Let $Y$ be a random variable such that: (a) $E(Y)$ exists and is not equal to zero; (b) $M(\theta) = E\lbrack^{Y\theta}\rbrack$ exists for $\theta$ in a finite interval (c) $M(\theta) \rightarrow \infty$ as $\theta \rightarrow \pm \infty$. Then there exists one and only one real $\theta_0 \neq 0$ such that $M(\theta_0) = 1$. Since the Identity as stated above is applicable only when the process starts from zero, we put $b = K - u$, and $a = -u$. Applying the above lemma to the input process of the dam model, we find that there exists a nonzero real solution $\theta_0$ of the equation \begin{equation*}\tag{1.2}\theta + \xi(\theta) = 0;\end{equation*} we then have, putting $\theta = \theta_0$ in (1.1) \begin{equation*}\tag{1.3}E\lbrack^{\theta_0Y(t + 1)}\rbrack = 1.\end{equation*} If the boundaries $a$ and $b$ are exactly reached at stage "$t$" we put $Y(t) = a$ with probability $P_a$ and $Y(t) = b$ with probability $P_b = 1 - P_a$; then (1.3) gives $P_ae^{a\theta_0} + (1 - P_a)e^{b\theta_0} = 1,$ so that \begin{equation*}\tag{1.4}P_a = (1 - e^{b\theta_0})/(e^{a\theta_0} - e^{b\theta_0}).\end{equation*} Using (1.3) the characteristic function (c.f.) of the time "$t$" at which the dam becomes empty for the first time before touching the barrier $K$ can also be determined. It is known that the eqauation $\theta + \xi(\theta) = i\phi$ has two roots $\theta_1(\phi)$ and $\theta_2(\phi)$ such that $\theta_1(\phi) \rightarrow 0$ and $\theta_2(\phi) \rightarrow \theta_0$ as $\phi \rightarrow 0$. Hence, in the case where the boundaries "$a$" and "$b$" are exactly reached, we obtain from (1.1) \begin{align*} P_ae^{a\theta_1(\phi)} C_a(\phi) &+ P_be^{b\theta_1(\phi)} C_b(\phi) = 1,\\ \tag{1.5} \\ P_ae^{a\theta_2(\phi)} C_a(\phi) &+ P_be^{b\theta_2(\phi)} C_b(\phi) = 1,\end{align*} where $C_a(\phi)$ is c.f. of $t$ conditional on $Y(t) = a$, and $C_b(\phi)$ is defined similarly, so that $E\lbrack^{i\phi t}\rbrack = P_aC_a(\phi) + P_bC_b(\phi)$. Equations (1.4) and (1.5) give $P_aC_a(\phi)$, the c.f. of "$t$" the time at which the dam becomes empty for the first time without overflowing, (cf. Bartlett [1] pp. 18-19). Before we use the Fundamental Identity in any problem let us consider the advantages of this method. The main advantage is its simplicity and easy applicability. Moreover, by a slight modification we can derive the more important probability distribution of the times of emptiness, no matter how often the dam has overflowed in the meantime. In random walk terminology this means that we have an absorbing barrier at 0 and a reflecting barrier at $K$. Another advantage of this method is that it can be used even when the inputs are not independent, as assumed in the above model, but Markovian. In this case the extension of Wald's Fundamental Identity to Markov Processes (Bellman [2], Tweedle[14], Phatarfod [11]) has to be applied; this problem will be considered in a future paper.