What really sets the wait?

At a given load, two systems can wait wildly differently — because variability matters as much as utilization. Kingman's approximation factors the queue wait into utilization × variability × work, making it visible exactly how much the smoothness of arrivals and service times amplifies or dampens the delay.

The wait
Each job takesto handle.
Work arrives at% utilization.
Arrivals areand service is.
The equation
Queue wait Wq
16m
=
Utilization ρ/(1−ρ)The load amplifier: how much busier a server is than its available slack. At 80% load this is 4×; at 95% it's 19×. Reducing utilization is the most powerful lever — but not the only one.
4×80% busy
×
Variability (Ca²+Cs²)/2The variability amplifier: (Ca²+Cs²)/2. When arrivals and service are perfectly regular (CV=0) this is 0 and the queue vanishes. At CV=1,1 (Poisson/exponential) it's 1. At CV=2,2 it's 4 — quadrupling the wait.
1Ca=1, Cs=1
×
Work mean serviceMean service time τ: how long a single job takes to process. Kingman's formula gives the queue wait in the same units — a longer job always means a longer wait.
4m
16mqueue wait

At 80% busy with this variability, each job waits 16m before it even starts — 4× the work itself. Set both CVs to 0 (perfectly steady) and the wait would fall to 0s; at CV 1,1 it would be 16m.

16mqueue wait
20mtime in system
3.2jobs in line
Wait vs. utilization
99×0.1×1×10×
0%20%40%60%80% busy100%
current variability (V=1)M/M/1 (V=1)
Smoothing pays
VariabilityQueue wait
bursty (CV 2,2)
V = 4
1h 4m
random (CV 1,1)nearest
V = 1
16m
tightened (CV 0.5,0.5)
V = 0.3
4m
steady (CV 0,0)
V = 0
0s

At fixed load the wait scales directly with (Ca²+Cs²)/2 — halving the variability halves the queue wait, as surely as halving the load. Smaller batches, steadier arrival patterns, and consistent service times all reduce V. The queue cliff shows only the utilization term; this is the one it leaves out.

based on Kingman's formula · Wq ≈ ρ/(1−ρ) · (Ca²+Cs²)/2 · τ