Statistical
Kingman's VUT Equation: G/G/1 queueing wait time from variability & utilization
Estimate real-world factory and service queue wait times Wq = ((Ca^2 + Cs^2)/2) * (u / (1 - u)) * ts when arrivals and processing times are NOT exponential.
Estimate mean waiting at one service station
Kingman’s approximation estimates steady-state mean queue waiting for a single-server GI/G/1 model. It assumes independent identically distributed interarrival times and service times, independent arrival and service sequences, finite variability and a stable load below one. It is an approximation, not an individual wait-time promise, and should not be applied unchanged to multiple parallel servers or priority queues.
Use Ca = standard deviation of interarrival times / mean interarrival time and Cs = standard deviation of service times / mean service time. With utilization u and mean service time ts, Wq is approximately ((Ca² + Cs²)/2) × u/(1-u) × ts. All times must use the same unit. Ca is based on gaps between arrivals, not on hourly arrival counts.
Zero variability produces zero in this idealized steady-state calculation; it does not eliminate transient backlogs, failures or other real-world delays. Utilization must be consistent with arrival rate times mean service time.
| VUT Lever | Cell | Workstation Input | Factor Contribution |
|---|---|---|---|
| Arrival & Service CV (Ca, Cs) | B2 = 1.0, B3 = 1.2 | Burst arrivals + variable setup | V = (1 + 1.44) / 2 = 1.22 |
| Server Utilization (u) | B4 = 0.80 (80%) | u / (1 - u) = 0.80 / 0.20 | U = 4.00x nonlinear amplification! |
| Mean Service Time (ts) | B5 = 5.0 min | 5 minutes active work | Wq = 1.22 * 4 * 5 = 24.40 min |
| Total Cycle Time (Wq + ts) | 24.40 + 5.00 | 29.40 min total lead time | 83% of time spent waiting! |
Kingman VUT expected wait time in queue Wq
The model is outside its stable-load domain at u >= 1. The formula returns a message rather than extrapolating through a zero or negative denominator. CVs must be nonnegative and mean service time positive.
=IF(OR(COUNT(B2:B5)<>4,B2<0,B3<0,B4<0,B5<=0),"Invalid inputs",IF(B4>=1,"Outside stable-load domain",ROUND(((B2 ^ 2 + B3 ^ 2) / 2) * (B4 / (1 - B4)) * B5, 2)))Total flow / lead time in system W = Wq + ts
Adding mean service time estimates mean time at this one station, not end-to-end time through an entire process. Both outputs use the same validity checks.
=IF(OR(COUNT(B2:B5)<>4,B2<0,B3<0,B4<0,B5<=0),"Invalid inputs",IF(B4>=1,"Outside stable-load domain",ROUND(((B2 ^ 2 + B3 ^ 2) / 2) * (B4 / (1 - B4)) * B5 + B5, 2)))Try the example
Use the original values or change them to test your understanding.