Pollaczek–Khinchine Service Variability
Expose the second service moment behind mean queueing delay and recognize instability.
Free theory articleExpose the second service moment behind mean queueing delay and recognize instability.
Concept & interpretation
The Pollaczek–Khinchine (P-K) formula relates the mean waiting time, queue length, and sojourn time of an M/G/1 queue (Poisson arrivals, General service distribution, 1 server) to the first two moments of the service time distribution. It shows that the mean waiting time depends not only on the mean service time but also significantly on its variance (or second moment).
Experiment
You can interactively explore the relationship between service variability and queue performance using the Pollaczek–Khinchine Service Variability module.
- Set the Service law to "Exponential", the Mean service to "1", and the Arrival rate to "0.5". Observe that the mean queue wait is 1.0.
- Change the Service law to "Two point zero two" while keeping the other settings identical. Notice that the mean queue wait is still 1.0, because both laws have the same mean and second moment E[S²] = 2m².
- Change the Service law to "Bursty zero four". The mean queue wait jumps to 2.0, demonstrating how higher variance in service times (even with the same mean) drastically increases queueing delays.
- Increase the Arrival rate to "1". The status changes to "No stationary mean" and metrics become unavailable since ρ = 1, indicating instability.