Load Balancer / Queueing Wait-Time Estimator (M/M/1) mechanism

Model scope

The stated M/M/1 idealized-model label is appropriate, but it does not resolve the metric-label error. As written, the formulas compute average number in the system and average time in the system, while the claimed outputs say average queue length and average wait time. Rename the outputs or replace the formulas with Lq=ρ²/(1−ρ) and Wq=ρ/(μ−λ).

• L=ρ/(1−ρ) is the average number in the system, not the average queue length. For average queue length excluding the customer in service, the standard formula is Lq=ρ²/(1−ρ).
• W=1/(μ−λ) is the average time in the system, including service time, not the average queue wait time before service. For queue wait time, the standard formula is Wq=λ/[μ(μ−λ)]=ρ/(μ−λ).
• The output labels are ambiguous: the supplied formulas are correct only if L is labeled average number in system and W is labeled average time in system or response time.
• At ρ≥1, no finite steady-state averages exist; L, Lq, W, and Wq must be reported as undefined rather than numerically evaluated.

Units and conventions

• λ and μ must use identical rate units, such as both jobs per second, jobs per minute, or jobs per hour.
• ρ, L, and Lq are dimensionless quantities; W and Wq have time units equal to the reciprocal of the rate unit.
• Inputs must satisfy λ≥0 and μ>0. A stable finite result additionally requires λ<μ, equivalently ρ<1.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "arrival_rate_lambda": 2,
    "service_rate_mu": 5,
    "rate_unit": "jobs/second"
  },
  "expected_outputs": {
    "utilization_rho": 0.4,
    "instrument_formula_L": 0.666666667,
    "correct_average_queue_length_Lq": 0.266666667,
    "instrument_formula_W_seconds": 0.333333333,
    "correct_average_queue_wait_Wq_seconds": 0.133333333,
    "stability_warning": false
  },
  "working": "ρ=λ/μ=2/5=0.4. Supplied L=ρ/(1−ρ)=0.4/0.6=0.666666667. Correct Lq=ρ²/(1−ρ)=0.16/0.6=0.266666667. Supplied W=1/(μ−λ)=1/(5−2)=1/3=0.333333333 seconds. Correct Wq=ρ/(μ−λ)=0.4/3=0.133333333 seconds."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

The specification becomes unsafe if an implementation blindly evaluates L and W at or above ρ=1: at equality it divides by zero, and above equality it can emit negative queue lengths and wait times unless it rejects the unstable case. Mixed rate units can also produce incorrect utilization and stability decisions if rates are not normalized.

Other records

Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

• VERIFIED_LEDGER.json
• evidence-tranche5/tranche5-results.json
• BOUNDARY_HAZARD_CATALOG.json