Stochastic Processes · 免费预览
极限行为
不可约的连续时间链在每个正时刻都具有严格正的转移概率；平稳分布满足 pi Q 等于零，如同平衡的沙流；生灭链通过细致平衡可得到一个乘积公式；并且极限定理不需要非周期性条件。
1. No periodicity in continuous time
No periodicity in continuous time
2. Pi Q equals zero is balanced sand
Pi Q equals zero is balanced sand
3. Detailed balance and birth-death chains
Detailed balance and birth-death chains
4. Converting the embedded jump chain
Converting the embedded jump chain
5. The missing aperiodicity condition
Why does the continuous-time convergence theorem need no aperiodicity assumption?
6. The sand interpretation
What does the equation pi Q equals zero say, interpreted as sand flow?
7. Weather by the cycle
The weather cycles sunny, smoggy, rainy with exponential stays of mean 3, 4, and 1 days. What fraction of time is it sunny?
8. The birth-death product formula
What does the detailed balance condition give for a birth-death chain?
9. Three machines and one repairman
In the machine repair model, what makes the death rates grow with the number of working machines while the birth rate stays fixed?
10. From visit shares to time shares
How is the stationary distribution pi related to the embedded jump chain's stationary distribution nu?
11. Sun over Los Angeles
The LA weather chain stays sunny for an exponential mean 3 days, smoggy for mean 4, rainy for mean 1, cycling in that order. Compute the long-run fraction of sunny days and write the answer as a fraction or decimal.
12. The repairman's success rate
In the machine repair model with three machines and one repairman, the stationary distribution assigns pi at 3 equal to 1125 over 1382. Compute the long-run fraction of time all three machines are working, as a decimal.
13. A barber shop's idle share
A shop has two barbers, each serving at rate 3 per hour, customers arrive at rate 2, and the shop holds at most 4 customers. The stationary distribution is proportional to 81, 54, 18, 6, 2. Compute pi at 0, the fraction of time the shop is empty, and write the answer as a fraction or decimal.