Staffing under Taylor's Law: A Unifying Framework for Bridging Square-root and Linear Safety Rules

TL;DR

This work addresses the mismatch between classical square-root staffing and observed over-dispersion in arrivals that follows Taylor's law. It introduces a parsimonious doubly stochastic Poisson process with intensity dynamics given by a generalized CIR process, capturing dispersion scaling via the parameter $\alpha$ and yielding a functional central limit theorem under heavy traffic. The authors derive alpha safety staffing rules, including a basic rule with a closed-form approximation and a refined rule calibrated by simulation, bridging square-root and linear safety as $\alpha$ varies. They validate the framework with stationary and non-stationary data, including a NYC 311 case study, and show improved staffing performance over alternative models, highlighting practical implications for large-scale service systems facing time-varying, over-dispersed arrivals.

Abstract

Staffing rules serve as an essential management tool in service industries to attain target service levels. Traditionally, the square-root safety rule, based on the Poisson arrival assumption, has been commonly used. However, empirical findings suggest that arrival processes often exhibit an ``over-dispersion'' phenomenon, in which the variance of the arrival exceeds the mean. In this paper, we develop a new doubly stochastic Poisson process model to capture a significant dispersion scaling law, known as Taylor's law, showing that the variance is a power function of the mean. We further examine how over-dispersion affects staffing, providing a closed-form staffing formula to ensure a desired service level. Interestingly, the additional staffing level beyond the nominal load is a power function of the nominal load, with the power exponent lying between $1/2$ (the square-root safety rule) and $1$ (the linear safety rule), depending on the degree of over-dispersion. Extensive numerical experiments with both simulated and real arrival data show that our proposed model and staffing rules significantly outperform various alternatives.

Figures (20)

• Figure 1: Daily Arrival Pattern of the NYC 311 Call Center.
• Figure 2: Taylor's Law and the Arrival Process of the NYC 311 Call Center.
• Figure 3: Simulated Intensity Process $X(t)$ and Arrival Count $A(t, t+\Delta]$.
• Figure 4: Nested Relationships Among $\mathcal{M}_1$ through $\mathcal{M}_5$.
• Figure 5: Performance of Staffing Rules in Infinite-Server Systems with Arrival Model $\mathcal{M}_5$ ($\lambda=150$).

Theorems & Definitions (25)

• Example 1: Square-root Safety Rule
• Example 2: Linear Safety Rule
• Definition 1: Doubly Stochastic Poisson Process
• Remark 1
• Proposition 1
• Theorem 1
• Remark 2
• Theorem 2
• Remark 3
• Theorem 3