Stochastics and StatisticsOn the exact distribution of a delayed renewal process with exponential sum interarrival times
Introduction
Consider an electricity distribution network rife with circuit breakers whose purpose is to mitigate the effects of power surges. Circuit breakers, however, go silently out of commission unbeknownst to the network’s operators; breakdowns to be discovered only through the impact of subsequent power surges. Smoke detector malfunctions and fires, hospitals emergency generator breakdowns and blackouts, weapon systems misperformance and hostilities breakouts, are further examples of belated awareness of an event through imputation by another during a given time period. The statistical behavior of the number of unobservable circuit breaker breakdowns is important since each occurrence incurs a substantial increase in the collateral damage of the subsequent power surge, which is otherwise comparably negligible, and it is thus the undertaking of this paper.
We model the situation by viewing it as two independent but interrelated processes. The imputing process is the sequence of blackouts that shed light on the imputed process which is the string of unobservable circuit breaker failures. We assume that the imputing process is Poisson and that circuit breakers have independent exponential life spans with a parameter that likely differs from that of the blackouts. Had the imputed process been observable it would have been Poisson as well, but due to the excision of the time intervals between failure to detection it is not. It can be nonetheless regarded as a thinned out Poisson process. The tally of the number of occurrences of either process during a given time period is of much interest. It is trivially Poisson for the imputing process alone and the joint count was resolved by Cuffe and Friedman (1996). This article concerns itself with the distribution of the number of occurrences of the imputed process during a prescribed time period at whose onset the process is on; i.e., the circuit breaker is operative. But for the unobervability of the process this number would have been Poisson and thus may be perceived as a Poisson offshoot, or an imputed Poisson.
The Poisson random variable lends itself to generating many variants and ancillary random variables. Cohen, 1960a, Cohen, 1960b examined the truncated Poisson and the modified Poisson distributions, respectively. Rao and Rubin (1964) suggested the generalized Poisson distribution which was greatly expanded upon by Consul (1989). Staff (1964) inquired into the displaced Poisson distribution, and Cohen (1963) explored various Poisson mixtures. It is well known that a sum of independent Poisson random variables with identical parameters is also distributed Poisson but Adelson (1966) researched the behavior of such sums with unequal parameters and called it the stuttering Poisson. Feller (1966) widely discusses random sums of Poisson variables, i.e., the compound Poisson distribution, and in particular the special case where the number of terms itself is distributed Poisson. Schmidt (1977) deemed the distribution of the number of perceived conflicts generated by aircraft around airports to be such compound Poisson until Friedman (1988) showed that it is much more complicated and named it the Extended Poisson. A multivariate extension of the Poisson distribution was propounded by Krishnamoorthy (1951).
The particular objective of this paper merits regarding the number of unobservable circuit breaker failures as a delayed renewal process.2 But for the initial one all interarrival times between consecutive failures are sums of two independent exponential random variables. The first term of the sum is the delay until detection and the second one is the life span of the item. The exponentiality of the first variable is preserved due to its memoryless property. The initial arrival time, though, is simple exponential since the circuit breaker is assumed operable at the beginning of the observation period.
Section snippets
The model and the approach
Let N(τ) be the number of unobservable failures of an item whose life spans Yj consist of i.i.d. exp(β) random variables during the time period (0, τ] assuming that the item is operating at t = 0. A failure becomes known through a subsequent occurrence of an independent Poisson process whose interarrival times Xi are exp(λ) and otherwise has no effect on the operativeness of the item. We assume the item is fixed instantaneously upon discovery of a failure. It is beneficial to view N(τ) as a
The case β = λ
We first dispense with the special case β = λ. For it Fn(τ), Fn+1(τ) is the Erlang distribution with 2n−1, 2n + 1 degrees of freedom, respectively. From the particular form of the Erlang, see Feller (1966, p. 10), it immediately follows thatEq. (3) also follows from the inversion of (2) with β = λ, and is trivially verifiable as a probability function.
pτ(n) of (3) is not a special case of the general β≠λ case to be now taken up. Its Laplace transform
The general β≠λ case
The Laplace transform of (2) does not lend itself to direct inversion for finding pτ(n). We therefore take the recursive route. Recall that trivially pτ(0) = e−βτ. Applying (1) for n = 1, realizing that F1(τ) is the c.d.f. of Y1 ∼ exp(β) and F2(τ) is the c.d.f of Y1 + X2 + Y2, where Y1,Y2 ∼ exp(β), X2 ∼ exp(λ), all independent, it follows after plowing through the algebra, thatEq. (4) follows also by inverting (2) for n = 1.
pτ(n) for some more small values of n can be
Extensions
Suppose that one is interested in the detection rather than the failure count of the item in question. Then the process becomes regular renewal with interarrival times Yn + Xn, n ⩾ 1, where the Yn,Xn are i.i.d. random variables exponentially distributed with parameters β, λ, respectively. Denoting by N′(τ) the number of detections during (0, τ] and by , and following the same logic and steps as above it turns out that for the case β = λ,
Discussion
This paper develops the exact distribution function of a delayed renewal process whose interarrival times are sums of two independent exponential random variables. The answer greatly differs if these two variables have equal or unequal parameters. Obviously the latter case is much more challenging and interesting. The variable Yn ∼ exp(β) of this sum is unobservable which renders the estimation of β a viable statistical question.
The issue is whether the statistical theory of missing data can
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Currently at Wingate University, Wingate, North Carolina.