An Algorithm to Calculate Transient Distributions of Cumulative Reward
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Reward model solution methods with impulse and rate rewards: an algorithm and numerical results. 27 the third example for a fee of $600. From the gure we note that there is less variability in the distribution of the cost for the second example for the range of fees plotted than for the rst example (recall that in both examples, there is also a cost per time unit when the system is down). By examining the gure it is clear that paying $100 per repair performed is better than performing repairs only when the system goes down and paying a fee of $300 when that occurs. However, the choice is not as clear when the fee is in the range of $130 to $150 per repair. For instance, consider the $140 curve in Figure 2. The probability that the total cost over one month is under, say, $4; 200 is higher than the corresponding probability for the rst example. As a consequence, if the expense budget has a hard limit, we may want to make repairs as components fail, since this policy implies a lower probability of exceeding the limited budget. On the other hand, the policy of the rst example has a higher probability of a cost under, say, $2; 700 (e.g., below the limiting expected cost). In summary, the choice will depend on the amount of risk the company wants to take. A higher possibility for savings implies a higher probability of exceeding the budget limit. 10 Summary We developed an algorithm to obtain the distribution of the reward accumulated during a given interval of time when both rate and impulse rewards are present. We then showed how to specialize this algorithm to the case of models with only rate based rewards and models with only impulse based rewards. Previous algorithms for these cases were shown to result. Only probabilistic arguments were applied throughout the development to obtain the main equations. The algorithm developed is simple to implement, robust, and has a low computational cost. Its computational complexity compares favorably with respect to previous methods. Calculating availability and performability measures of repairable computer systems using randomization. 26 In both examples we see that there is a considerable diierence in the distribution of the cost when a fee is paid each time the system goes down. For instance, in the rst example, the probability that the cost is less than …
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