Operations Research & Queueing
Jackson Queueing Network Rework Traffic Equations and Lead Time
Solve Jackson’s (1957) rework traffic balance lam_eff = lam_ext / (1 - p_rework), station utilizations, total WIP, and Little’s Law customer cycle time without throughput errors.
Avoid the two classic rework-loop traps: arrival dilution and Little’s Law throughput
Consider an open Jackson (1957) two-station line where external orders arrive at lam_0 = 18 jobs/hr (B2) into Fabrication (mu_1 = 30 jobs/hr, B3) followed by Inspection (mu_2 = 25 jobs/hr, B4), and fraction p_rew = 0.10 (10%, B5) fail inspection and loop back to Fabrication. Trap #1 is plugging lam_0 = 18 into rho = lam / mu: because failed jobs revisit both stations, true effective arrival rate is lam_eff = lam_0 / (1 - p_rew) = 18 / 0.90 = 20 jobs/hr!
At lam_eff = 20, Station 1 utilization is rho_1 = 20 / 30 = 0.6667 (WIP L_1 = rho_1 / (1 - rho_1) = 2.00 jobs) and Station 2 utilization is rho_2 = 20 / 25 = 0.8000 (WIP L_2 = 4.00 jobs), giving total network WIP L_net = 6.00 jobs. Trap #2 hits Little’s Law: total customer lead time divides L_net = 6.00 by EXTERNAL exit throughput lam_0 = 18 (NOT internal 20!), yielding W_net = 6 / 18 = 0.3333 hr = 20.00 minutes!
| Jackson Network Metric | Exact Formula | Sample Output |
|---|---|---|
| Effective Internal Arrival lam_eff | lam_0 / (1 - p_rew) = 18 / 0.90 | 20.0000 jobs/hr |
| Station WIPs L_1 and L_2 | rho_1 / (1 - rho_1) + rho_2 / (1 - rho_2) | 2.00 + 4.00 = 6.00 jobs |
| End-to-End Lead Time (Minutes) | 60 * (L_1 + L_2) / lam_0 | 20.00 minutes (vs 18.00 trap!) |
| Saturated Rework Threshold | p_rew = 1 - lam_0 / MIN(mu_1, mu_2) | 0.2800 (28% rework blows up mu_2) |
Total Jackson Network WIP (L_1 + L_2) Formula
Check station stability (lam_eff < mu_1 AND lam_eff < mu_2) before summing M/M/1 product-form queue lengths.
=LET(l_ext, B2, mu_a, B3, mu_b, B4, p_rw, B5, IF(OR(l_ext <= 0, p_rw < 0, p_rw >= 1), "Invalid Routing Input", LET(l_eff, l_ext / (1 - p_rw), IF(OR(l_eff >= mu_a, l_eff >= mu_b), "Bottleneck Saturated (rho >= 1)", LET(rh_1, l_eff / mu_a, rh_2, l_eff / mu_b, ROUND(rh_1 / (1 - rh_1) + rh_2 / (1 - rh_2), 4))))))End-to-End Customer Lead Time (Minutes) Formula & Rework Surge Test
Divide total WIP by external arrival rate B2 (multiplied by 60 for minutes). Test doubling rework p_rew in B5 from 0.10 to 0.20: lam_eff rises to 22.5 jobs/hr, Station 2 utilization hits 90% (L_2 = 9.00 jobs), total WIP jumps from 6.00 to 12.00 jobs, and customer lead time doubles from 20.00 to 40.00 minutes!
=IF(ISNUMBER(E4), ROUND(60 * E4 / B2, 2), E4)Try the example
Use the original values or change them to test your understanding.