Queueing Theory, Computer Systems Latency & Operations Research
Takagi M/G/1 Server-Vacation Queue Waiting-Time Formula
Calculate Server Utilization rho = lambda*E[S], Pollaczek-Khinchine Non-Vacation M/G/1 Waiting Time W_q0, Mean Residual Vacation Life E[V]*(1 + CV_V^2)/2, and Takagi / Fuhrmann-Cooper Decomposed Queue Delay.
Model JVM garbage-collection pauses and CNC maintenance windows with Takagi’s Vacation Decomposition!
Whenever a server finishes draining its queue and leaves for background work—such as JVM stop-the-world compaction, SSD flash wear-leveling, cyclic polling, or CNC spindle tool-change vacations—newly arriving requests must wait even at modest utilization! The Fuhrmann-Cooper and Takagi Stochastic Decomposition Theorem proves that under exhaustive multiple vacations, expected M/G/1 queue waiting time splits cleanly into TWO additive terms: Standard Pollaczek-Khinchine Waiting Time W_q(non-vacation) = (lambda * E[S]^2 * (1 + CV_S^2)) / (2 * (1 - rho)) PLUS the Inspection-Paradox Mean Residual Vacation Life R_V = (E[V] * (1 + CV_V^2)) / 2! Notice the variability trap: highly erratic vacation durations (high CV_V) inflate waiting time even when average vacation length E[V] stays constant!
Verify on a Microservice Worker Thread—Poisson_Arrival_Rate lambda = 4.0 req/ms (B2), Mean_Service_Time E[S] = 0.15 ms (B3 -> Utilization rho = 0.6000 in E2), Service_CV = 1.0 (exponential service, B4 -> Base Pollaczek-Khinchine W_q0 = (4 * 0.0225 * 2) / (2 * 0.4) = 0.2250 ms in E3), Mean_GC_Vacation E[V] = 0.80 ms (B5), and Vacation_CV = 0.50 (B6): Mean Residual Vacation Delay is (0.80 * (1 + 0.25)) / 2 = 0.5000 ms (E4), yielding Total Decomposed Queue Wait W_q = 0.2250 + 0.5000 = 0.7250 ms (E5)!
| M/G/1 Vacation Queue Metric (lam=4/ms, E[S]=0.15ms, CVs=1, E[V]=0.8ms, CVv=0.5) | Formula | Verified Output |
|---|---|---|
| Traffic Intensity Utilization rho | Arrival_lambda * Mean_Service_ES | 0.6000 (60.0% Busy) |
| Base M/G/1 Pollaczek Wait (ms) | (lam * ES^2 * (1 + CVs^2)) / (2*(1 - rho)) | 0.2250 ms |
| Mean Residual Vacation R_V (ms) | (Mean_Vac_EV * (1 + CVv^2)) / 2 | 0.5000 ms (0.4000 @ CVv=0) |
| Total Decomposed Wait W_q (ms) | Base_Pollaczek_Wait + Residual_Vac_RV | 0.7250 ms (0.6250 @ CVv=0) |
Takagi / Fuhrmann-Cooper M/G/1 Vacation Waiting-Time Formula
Validate lambda > 0, E[S] > 0, strictly stable traffic rho = lambda * E[S] < 1, non-negative CV_S and CV_V, and E[V] > 0!
=LET(lm_r, B2, sv_m, B3, cv_s, B4, vc_m, B5, cv_v, B6, rh_u, lm_r * sv_m, IF(OR(lm_r <= 0, sv_m <= 0, rh_u >= 1, cv_s < 0, vc_m <= 0, cv_v < 0), "Unstable or Invalid Vacation Queue", LET(wq_b, (lm_r * (sv_m ^ 2) * (1 + cv_s ^ 2)) / (2 * (1 - rh_u)), vr_m, (vc_m * (1 + cv_v ^ 2)) / 2, ROUND(wq_b + vr_m, 4))))Deterministic Fixed-Timer Vacation Check (Set Vacation CV = 0)
Replace variable GC pauses with a deterministic 0.80 ms timer (B6 = 0): Residual Vacation Delay E4 drops from 0.5000 ms to exactly E[V]/2 = 0.4000 ms, trimming Total Wait E5 to 0.6250 ms!
=IF(ISTEXT(E2), E2, ROUND((B5 * (1 + B6 ^ 2)) / 2, 4))Try the example
Use the original values or change them to test your understanding.