M/M/∞ queue
Section of Birth–death chains and queues, part of Stochastic processes
M/M/∞ queue
| Type | Section |
| Taught in | Birth–death chains and queues |
| Part | Stochastic processes |
| Questions using it | 1 |
With infinitely many servers, every customer starts service at once. In state n there are n independent services in progress, so the departure rate is \mu_n=n\mu. Local balance gives
\pi_n=\pi_0\prod_{k=0}^{n-1}\frac{\lambda}{(k+1)\mu}=\pi_0\frac{(\lambda/\mu)^n}{n!}.
The series is the exponential series, so \pi_0=e^{-\lambda/\mu} and the stationary number in system is \mathrm{Poisson}(\lambda/\mu). There is no finite-server overload threshold, because the total service rate grows without bound as n grows. The mean is \lambda/\mu for every \lambda>0 and \mu>0.
Used in
- Taught in Birth–death chains and queues
- Questions that use it (1) — An M/M/∞ system
- All 3 questions on Birth–death chains and queues
See also
Little’s law · Birth–death balance · M/M/1 queue · Multi-server and finite-capacity queues · Machine-repair model · Three capacity rules · Blocking, throughput, and Little’s law · Local balance