---
type: claim
domain: internet-finance
description: "Using max or average rate instead of time-varying rate leads to chronic under or overstaffing"
confidence: proven
source: "Liu et al. (NC State), 'Modeling and Simulation of Nonstationary Non-Poisson Arrival Processes' (2019)"
created: 2026-03-11
supports:
- non-stationary-service-systems-require-dynamic-worker-allocation-because-fixed-staffing-wastes-capacity-during-low-demand-and-creates-bottlenecks-during-peaks
- time-varying-arrival-rates-require-dynamic-staffing-not-constant-max-workers
reweave_edges:
- non-stationary-service-systems-require-dynamic-worker-allocation-because-fixed-staffing-wastes-capacity-during-low-demand-and-creates-bottlenecks-during-peaks|supports|2026-04-18
- time-varying-arrival-rates-require-dynamic-staffing-not-constant-max-workers|supports|2026-04-20
sourced_from:
- inbox/archive/internet-finance/2019-00-00-liu-modeling-nonstationary-non-poisson-arrival-processes.md
---

# Constant rate approximation of time-varying arrivals causes systematic staffing errors

Replacing a time-varying arrival rate λ(t) with a constant approximation—whether the maximum rate, average rate, or any other single value—leads to systematic capacity planning failures. Systems sized for maximum rate are chronically overstaffed during low-demand periods, wasting resources. Systems sized for average rate are chronically understaffed during high-demand periods, creating congestion.

This is not a minor efficiency loss but a structural mismatch: the constant-rate approximation discards the temporal structure of demand, making it impossible to match capacity to load.

## Evidence

Liu et al. explicitly state that "replacing a time-varying arrival rate with a constant (max or average) leads to systems being badly understaffed or overstaffed." This is a direct consequence of nonstationary arrival processes where demand varies predictably over time.

The paper demonstrates that "congestion measures are increasing functions of arrival process variability," meaning that even if average load is manageable, temporal concentration of arrivals creates congestion that constant-rate models cannot predict.

## Implications for Pipeline Architecture

For capital formation pipelines with session-based arrival patterns, this means:

1. Sizing capacity for peak (research session active) rate wastes resources during quiet periods
2. Sizing capacity for average rate creates backlogs during research sessions
3. Optimal capacity must be time-varying or must use queueing/buffering to smooth demand

The MMPP framework provides tools to size capacity for the mixture of states rather than for a single average state, enabling more efficient resource allocation.

---

Relevant Notes:
- domains/internet-finance/_map

Topics:
- core/mechanisms/_map