Birth–death balance
Section of Birth–death chains and queues, part of Stochastic processes
Birth–death balance
| Type | Section |
| Taught in | Birth–death chains and queues |
| Part | Stochastic processes |
| Questions using it | 1 |
For states n=0,1,2,\ldots, let \lambda_n be the birth rate in state n and let \mu_n be the death rate in state n\ge 1, with \mu_0=0. The generator has q_{n,n+1}=\lambda_n, q_{n,n-1}=\mu_n, and q_{nn}=-(\lambda_n+\mu_n). Local balance equates the flow across the cut between n and n+1:
\pi_n\lambda_n=\pi_{n+1}\mu_{n+1}, \qquad \pi_n=\pi_0\prod_{k=0}^{n-1}\frac{\lambda_k}{\mu_{k+1}}.
These weights are a probability distribution when they can be normalised:
\pi_0=\Biggl(1+\sum_{n=1}^{\infty}\prod_{k=0}^{n-1}\frac{\lambda_k}{\mu_{k+1}}\Biggr)^{-1},
provided the sum is finite. A finite sum is a stability condition. It is the statement that the chain returns often enough to have a stationary probability, not a bookkeeping step after the fact.
Used in
- Taught in Birth–death chains and queues
- Also used in Classes, recurrence, and reversibility
- Questions that use it (1) — Birth–death balance on three states
- All 3 questions on Birth–death chains and queues
See also
Local balance · Little’s law · M/M/1 queue · M/M/∞ queue · Multi-server and finite-capacity queues · Machine-repair model · Three capacity rules · Blocking, throughput, and Little’s law