Jackson Open Queueing Network: Total WIP & End-to-End Sojourn Time

Account for internal rework and routing loops across multi-stage queues!

In wafer fabs, emergency departments, and cloud microservices, jobs rarely pass straight through once—rework loops and branching mean each station i sees an effective total arrival rate lambda_i (including routed internal traffic) that can exceed external network arrivals gamma_ext! James R. Jackson proved a miraculous product-form theorem: as long as every station is stable (lambda_i < mu_i), each single-server exponential node behaves in steady state like an independent M/M/1 queue with local WIP Li = rho_i / (1 - rho_i) = lambda_i / (mu_i - lambda_i)!

Summing local WIP across all K stations gives exact total jobs in the network L = SUM(Li), and applying Little's Law at the network boundary divides total WIP L by external throughput gamma_ext (NOT SUM(lambda_i)!) to yield average end-to-end customer sojourn time W = L / gamma_ext! Across 3 nodes [(12, 20), (9, 15), (8, 16) jobs/hr] serving gamma_ext = 10 external orders/hr, Li = 1.5 + 1.5 + 1.0 = 4.0 jobs and end-to-end lead time is W = 4.0 / 10 = 0.40 hr (24 minutes)!

Station Queue + In-Service WIP Li = lambda_i / (mu_i - lambda_i) (With Stability Guard)

Checks ergodicity lambda_i < mu_i at each node; if lambda_i >= mu_i, that bottleneck queue explodes to infinity!

=IF(OR(B2 < 0, C2 <= B2), "Unstable Queue (lambda >= mu)", ROUND(B2 / (C2 - B2), 3))

Network End-to-End Sojourn Time W = SUM(D2:D4) / External_Arrival_Rate

Divides total multi-node WIP SUM(D2:D4) by external throughput gamma_ext in B5 via Little's Law.

=IF(B5 <= 0, "Invalid External Rate", ROUND(SUM(D2:D4) / B5, 3))

Further reading

• Microsoft Support — SUM function