Machine-repair model
Section of Birth–death chains and queues, part of Stochastic processes
Machine-repair model
| Type | Section |
| Taught in | Birth–death chains and queues |
| Part | Stochastic processes |
Two identical machines each fail, while operating, at rate \lambda. One repairer works at rate \mu. Let n be the number of failed machines, so the states are 0,1,2. Only operating machines can fail, and only one repair can run at a time, so the birth rates are \lambda_0=2\lambda and \lambda_1=\lambda, and the death rates are \mu_1=\mu_2=\mu. Local balance gives the unnormalised weights
1,\qquad \frac{2\lambda}{\mu},\qquad 2\Bigl(\frac{\lambda}{\mu}\Bigr)^2.
Check the second weight: \pi_0\cdot 2\lambda=\pi_1\mu. Check the third: \pi_1\cdot\lambda=\pi_2\mu, so \pi_2=\pi_0\cdot 2(\lambda/\mu)^2. The state space is finite, so the weights can always be normalised. The rate of new failures drops as machines fail, because a failed machine is not exposed to failure. A single component’s lifetime distribution, and the constant hazard of an exponential life, are on Reliability. The chain here tracks how many of the two machines are down.
Used in
See also
Little’s law · Birth–death balance · M/M/1 queue · M/M/∞ queue · Multi-server and finite-capacity queues · Three capacity rules · Blocking, throughput, and Little’s law · Local balance