Three capacity rules
Section of Birth–death chains and queues, part of Stochastic processes
Three capacity rules
| Type | Section |
| Taught in | Birth–death chains and queues |
| Part | Stochastic processes |
The three basic queues share the arrival rate \lambda and differ in the departure rate:
| Model | Departure rate in state n |
|---|---|
| M/M/1 | \mu when n\ge 1, and 0 when n=0 |
| M/M/c | \min(n,c)\,\mu |
| M/M/∞ | n\mu |
In M/M/1, once the server is busy, further customers do not increase the service rate. In M/M/c the rate increases until all c servers are busy and then stops increasing. In M/M/∞ it keeps increasing. The stability conditions are the statement that the birth–death series of weights converges, read off this tail.
Take \lambda=4 and \mu=3 per hour. One server is overloaded because 4>3. Two servers have maximum departure rate 6, so M/M/2 is stable. The infinite-server system is stable with Poisson mean 4/3. The three answers describe three different amounts of service capacity.
Used in
See also
Little’s law · Birth–death balance · M/M/1 queue · M/M/∞ queue · Multi-server and finite-capacity queues · Machine-repair model · Blocking, throughput, and Little’s law · Local balance