Technical note | Asymptotic Distribution of Rewards Accumulated by Alternating Renewal Processes

Abstract

This technical note considers processes that alternate randomly between 'working' and 'broken' over an interval of time. Suppose that the process is rewarded whenever it is 'working', at a rate that can vary during the time interval but is known completely. We prove that if the time interval is long then the accumulated reward is approximately normally distributed and the approximation becomes perfect as the interval becomes infinitely long. Moreover we calculate the means and variances of those normal distributions and formally, consider an alternating renewal process on the states 'working' vs 'broken'. 

Executive Summary

This technical note documents some research into processes that alternate randomly between 'working' and 'broken' over an interval of time. It supposes that the process is rewarded whenever it is 'working', at a rate that can vary during the time interval but is known completely. We study the reward that is accumulated over that time interval. For example, consider a solar panel that can earn money if it is exposed to the sun, at a rate of 5 dollars per hour before noon and 10 dollars per hour after noon. What is the amount of money that it will earn over a given 24 hour period?

The key finding is that if the time interval is long then the accumulated reward is approximately normally distributed, and the approximation becomes perfect as the interval becomes infinitely long. The research also calculates the means and variances of those normal distributions. The values are obtained from the rates at which the process is rewarded when working (the dollars per hour in the example given above), and statistics about the times to failure and times to repair (the durations to go from working to broken and from broken to working).

This technical note is the expanded version of an article that was prepared for the journal Statistics & Probability Letters [Hew 2017]. It provides the details of the proofs that were abridged for the journal article. The research was motivated by studies of a number of military operations. When collapsed to their essentials, the operations could be modelled in terms of a sensor that alternates randomly between working and broken, and is looking for a target that reluctantly gives away glimpses at random times. Consider in particular the probability of seeing the k-th glimpse. Intuitively, at any time, the glimpse provides some probability of being detected if the sensor is working at that time. The probability of seeing the glimpse is the accumulation of those probabilities over the time interval. Hence by using the results in this article, we know that the probability of seeing the k-th glimpse is approximately normally distributed, and we can use that knowledge to make predictions about operational performance. Full details will be reported separately.