Average Renewal Loss Rates

Abstract

Many problems in the theory of decision processes give rise to an unending sequence of cycles whose lengths are given by a sequence of independent and identically distributed positive random variables $X_1, X_2, \cdots$, constituting a renewal process. Typically $X_i$ will represent the number of items passing an inspection point in a production process or the length of time elapsing in some continuous time process until a decision point is reached. In many such problems another sequence of random variables, say $Y_1, Y_2, \cdots$, arises where $Y_i$ is the profit or loss associated with the $i$th cycle. Examples of problems having this structure may be found, for instance, in [1], [3], [6], [9] and [10]. In most cases $X_i$ and $Y_i$ will not be independent for the same index $i$ and in this note we will permit $Y_i$ to depend on $X_{i-q}, \cdots, X_{i-1}, X_i$ for any fixed finite $q$. In all such problems the appropriate index of merit for the decision procedure under consideration is the "average" profit (or loss) per unit time (or per item) for a large number of cycles. The notion of "average" profit rate can be mathematically defined in four distinct and apparently equally plausible ways, and it is the purpose of this note to show that the various definitions are not necessarily equivalent and to determine the conditions under which they are equivalent. First, the profit rate up to time $t$ may be defined in terms of $N(t)$, the number of cycles completed by time $t$, as $(1/t) \sum^{N(t)}_{i=1} Y_i$. Then the average profit rate may be defined as either the limit of the expected value of the profit rate, or as the almost sure limit of the profit rate, as $t \rightarrow \infty$. The use of the expected value definition is consistent with the principles of utility theory and the notions of risk that underlie the formulation of problems in decision theory. The almost sure limit definition has, of course, considerable intuitive appeal. It is shown in this note that, as an immediate consequence of well-known results from renewal theory, both of the above definitions lead to the average profit rate $\eta/\mu$ where $EX = \mu$ and $EY = \eta$, assuming that $\eta$ is finite. Alternately, the profit rate may be defined for $n$ cycles as $\sum^n_{i=1} Y_i/ \sum^n_{i=1} X_i$ and again the average profit rate may be defined as either the limit of the expected values, or the almost sure limit of this ratio, as $n \rightarrow \infty$. By the strong law of large numbers the almost sure limit is always $\eta/\mu$ but in Section 2 an example is given for which the limit of the expected values is not $\eta/\mu$. In Section 3 necessary and sufficient conditions are obtained for the limit of the expected values of the ratios to be $\eta/\mu$ if $\eta$ is finite.