Stochastic Processes · 免费预览
排队网络
串联队列通过 Burke 定理将两个 M/M/1 级联起来；两个队列长度相互独立，并具有乘积形式分布；当流量方程给出的到达率低于服务率时，反馈网络是稳定的；而且乘积形式在反馈下依然成立。
1. Two queues in a row
Two queues in a row
2. The product form
The product form
3. Feedback and the traffic equations
Feedback and the traffic equations
4. The product form survives the loops
The product form survives the loops
5. The station that ignores its neighbor
In a tandem queue, why can the first station be analyzed alone?
6. Burke at the junction
What role does Burke's theorem play in analyzing the tandem queue?
7. Independence across the junction
Why are the two queue lengths independent in the tandem queue's equilibrium?
8. Worst case versus truth
In the feedback example, station 2's worst-case input rate is 4 while its service rate is 3.5. Why is the network still stable?
9. Solving the traffic equations
What do the traffic equations r sub 1 equals lambda 1 plus p2 r2 and r sub 2 equals lambda 2 plus p1 r1 assert?
10. The product form reads like independence
Once the effective rates are known, what is the stationary distribution of the feedback network?
11. Both chairs empty at once
A tandem queue has lambda 1 per hour, mu 1 equal 3, mu 2 equal 2. Compute the equilibrium probability that both stations are simultaneously empty, and write the answer as a fraction or decimal.
12. The true arrival rate at the hot station
In the feedback network with lambda 1 equal 1, lambda 2 equal 2, mu 1 equal 4, mu 2 equal 3.5, p1 equal one half, p2 equal one quarter, the traffic equations give r2 equal 20 over 7. Write this effective arrival rate as a decimal, rounding to two places.
13. The second queue's load
For the same tandem example with lambda 1, mu 2 equal 2, compute the average number of customers at the second station, using rho over one minus rho, and write the number.