G/G/1 queue
In queueing theory, a discipline within the mathematical theory of probability, the G/G/1 queue represents the queue length in a system with a single server where interarrival times have a general (meaning arbitrary) distribution and service times have a (different) general distribution.[1] The evolution of the queue can be described by the Lindley equation.[2]
The system is described in Kendall's notation where the G denotes a general distribution for both interarrival times and service times and the 1 that the model has a single server.[3][4] Different interarrival and service times are considered to be independent, and sometimes the model is denoted GI/GI/1 to emphasise this.
Waiting time
Kingman's formula gives an approximation for the mean waiting time in a G/G/1 queue.[5] Lindley's integral equation is a relationship satisfied by the stationary waiting time distribution which can be solved using the Wiener–Hopf method.[6]
Multiple servers
Few results are know for the general G/G/k model as it generalises the M/G/k queue for which few metrics are known.
In a G/G/2 queue with heavy-tailed job sizes, the tail of the delay time distribution us known to behave like the tail of an exponential distribution squared under low loads and like the tail of an exponential distribution for high loads.[7][8][9]
References
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| Single queueing nodes | |
|---|---|
| Arrival processes | |
| Queueing networks | |
| Service policies | |
| Key concepts | |
| Limit theorems | |
| Extensions | |
| Information systems | |
