Operations & Logistics
M/M/1/K Finite-Buffer Queue Blocking Probability, Drop Rate & Mean Occupancy
Model finite-capacity single-server queues (API connection pools, router packet buffers, call trunks) with exact M/M/1/K blocking probability P_K, admitted throughput lambda_eff, mean occupancy L, and rho = 1 singularity handling.
Why finite buffer K stabilizes overloaded queues (rho >= 1) while shedding P_K arrivals
Infinite-buffer M/M/1 formulas explode to infinity at offered utilization rho = lambda / mu >= 1 and pretend zero requests are ever rejected below rho < 1. Real network switches, worker-thread pools, and drive-thru lanes have finite system capacity K (1 active server + K - 1 waiting slots): arrivals finding all K slots full are shed immediately with tail blocking probability P_K = (1 - rho) * rho^K / (1 - rho^(K + 1)) for rho <> 1, and P_K = 1 / (K + 1) by L’Hopital’s limit at exact unit load rho = 1.
Because blocked requests never enter the buffer, effective admitted throughput is lambda_eff = lambda * (1 - P_K), mean system occupancy is L = rho / (1 - rho) - (K + 1) * rho^(K + 1) / (1 - rho^(K + 1)) (or K / 2 at rho = 1), and Little’s Law for mean sojourn time of admitted jobs divides L by lambda_eff—NOT raw offered lambda!
| M/M/1/K Metric (K = 4 System Slots) | Sub-Capacity Load (lambda=8, mu=10, rho=0.8) | Unit-Load Singularity (lambda=10, mu=10, rho=1.0) |
|---|---|---|
| Blocking / Rejection Probability P_K | 0.1218 (12.18% shed) | 0.2000 (1 / (4 + 1) = 20.00% shed) |
| Effective Admitted Rate lambda_eff | 7.0252 req/s | 8.0000 req/s |
| Mean System Occupancy L | 1.5631 jobs | 2.0000 jobs (K / 2) |
Singularity-Safe M/M/1/K Blocking Probability P_K Formula
Handle ABS(rho - 1) < 1E-9 with exact L’Hopital limit 1 / (K + 1) and evaluate (1 - rho) * rho^K / (1 - rho^(K + 1)) for all other positive traffic intensities.
=LET(rho_val, B2 / B3, cap_k, B4, IF(ABS(rho_val - 1) < 0.000000001, ROUND(1 / (cap_k + 1), 4), ROUND(((1 - rho_val) * (rho_val ^ cap_k)) / (1 - (rho_val ^ (cap_k + 1))), 4)))Exact M/M/1/K Mean System Occupancy L Formula
Return K / 2 when rho = 1, and rho / (1 - rho) - (K + 1) * rho^(K + 1) / (1 - rho^(K + 1)) otherwise.
=LET(rho_val, B2 / B3, cap_k, B4, IF(ABS(rho_val - 1) < 0.000000001, ROUND(cap_k / 2, 4), ROUND((rho_val / (1 - rho_val)) - (((cap_k + 1) * (rho_val ^ (cap_k + 1))) / (1 - (rho_val ^ (cap_k + 1)))), 4)))Try the example
Use the original values or change them to test your understanding.