Stochastic Processes · 免费预览
大数定律
更新过程保留了泊松过程的独立同分布到达间隔结构，但允许采用任意的寿命分布；长期更新速率为平均寿命的倒数；与各周期相联系的奖励则按期望奖励与期望周期长度之比累积。
1. Beyond the exponential clock
Beyond the exponential clock
2. The counting law and its sandwich
The counting law and its sandwich
3. Rewards riding the cycles
Rewards riding the cycles
4. Alternating between two states
Alternating between two states
5. What makes a renewal process
Which assumption defines a renewal process, as opposed to a Poisson process?
6. The rate set by the mean alone
A renewal process has mean interarrival time mu equal to 2 days. What is its long-run renewal rate?
7. Reading the reward ratio
The renewal reward theorem gives a limiting rate equal to which ratio?
8. The economics of replacing a car
In Mr. Brown's problem with uniform lifetimes on zero to ten, why is the optimal replacement age about 8.8 years rather than at the ten-year maximum?
9. The on-off time share
A machine works for random periods with mean 3 days and is down for random repairs with mean 1 day, cycling independently. What fraction of time does it work in the long run?
10. The janitor's three answers
With bulb lifetime mean 100 hours and Poisson inspections at rate one fiftieth per hour, what is the limiting fraction of visits on which the janitor finds the bulb dead?
11. Fifteen cycles a month
A machine's complete work-plus-repair cycle is random with mean 2 days, cycles independent. Compute the expected number of completed cycles in a 30-day month and write the number.
12. The best trade-in age
In Mr. Brown's problem with car lifetimes uniform on the interval from 0 to 10 years, purchase cost 10 thousand dollars and breakdown cost 3 thousand, compute the optimal replacement age in years and write the answer as a decimal.
13. A third of the time dark
A bulb lasts a mean of 100 hours, and a janitor inspects at Poisson rate one fiftieth per hour, replacing dead bulbs instantly. Compute the long-run fraction of time the bulb is working and write the answer as a fraction or decimal.