Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How bad is time variability for users in mobility services?

Zhaoqi Zang, David Z. W. Wang, Xiangdong Xu, Shaojun Liu

Abstract

Time variability is a pervasive feature of mobility services and a major source of welfare loss. Although literature has quantified the cost of time variability (COTV), it remains theoretically unclear how bad time variability can be in the worst case. Without such a benchmark, quantified variability costs lack a principled reference for assessing whether they are economically meaningful. Meanwhile, this benchmark is critical for strategic prioritization in transport appraisal, service design, and pricing -- particularly in early-stage decision making where detailed valuation is often infeasible. To fill this gap, this paper develops an expected utility (EU) framework to quantify the cost of time (COT) and COTV, establishing theoretical upper bounds on the ratio $COTV/COT$. For users with quadratic utility, we show $COTV/COT \le 1/2 CV^2$, where $CV$ is the coefficient of variation of service time. For Poisson processes, a common assumption, this bound simplifies to $COTV/COT \le 1/2$, implying the total cost of a stochastic service is at most 1.5 times that of an otherwise identical deterministic service. In more general settings, the ratio depends on three interpretable factors: $CV$ and users' second- and third-order risk preferences, captured by relative risk aversion (RRA) and relative prudence (RP). We identify benchmark values of RRA and RP that characterize preferences over mean-, variance-, and skewness-related reductions. Our analysis extends to non-EU frameworks, including dual theory and rank dependent utility, showing that key structural insights remain robust. By quantifying the cost induced by time variability and the $COTV/COT$ ratio, this study provides a data-light benchmark for early-stage decision making and a principled upper bound on users' willingness to pay for reliability improvements, informing the pricing and design of reliability-oriented services.

How bad is time variability for users in mobility services?

Abstract

Time variability is a pervasive feature of mobility services and a major source of welfare loss. Although literature has quantified the cost of time variability (COTV), it remains theoretically unclear how bad time variability can be in the worst case. Without such a benchmark, quantified variability costs lack a principled reference for assessing whether they are economically meaningful. Meanwhile, this benchmark is critical for strategic prioritization in transport appraisal, service design, and pricing -- particularly in early-stage decision making where detailed valuation is often infeasible. To fill this gap, this paper develops an expected utility (EU) framework to quantify the cost of time (COT) and COTV, establishing theoretical upper bounds on the ratio . For users with quadratic utility, we show , where is the coefficient of variation of service time. For Poisson processes, a common assumption, this bound simplifies to , implying the total cost of a stochastic service is at most 1.5 times that of an otherwise identical deterministic service. In more general settings, the ratio depends on three interpretable factors: and users' second- and third-order risk preferences, captured by relative risk aversion (RRA) and relative prudence (RP). We identify benchmark values of RRA and RP that characterize preferences over mean-, variance-, and skewness-related reductions. Our analysis extends to non-EU frameworks, including dual theory and rank dependent utility, showing that key structural insights remain robust. By quantifying the cost induced by time variability and the ratio, this study provides a data-light benchmark for early-stage decision making and a principled upper bound on users' willingness to pay for reliability improvements, informing the pricing and design of reliability-oriented services.
Paper Structure (31 sections, 14 theorems, 69 equations, 6 figures, 1 table)

This paper contains 31 sections, 14 theorems, 69 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review
  3. Time variability in mobility service systems
  4. Modeling the cost of time variability
  5. A theoretical framework for modeling users' responses to time variability
  6. Utility function of time and risk attitudes
  7. Modeling users' responses to time variability
  8. Bounding the inefficiency of time variability relative to no variability
  9. The costs of time and time variability
  10. The cost of time
  11. The cost of time variability
  12. The ratios
  13. The ratio of the COTV to the COT
  14. The ratio of the VOTV to the VOT
  15. Theoretical results
  16. ...and 16 more sections

Key Result

Proposition 1

Consider a service whose service time $t$ is a random variable with mean $\mu$ and variance $\sigma^2$. For a user with utility function of $u(t)$, the variability premium is given by

Figures (6)

  • Figure 1: Service instances without time variability $(T_0)$ and with time variability $(t, T_0,\mathbb{E}[\tilde{\varepsilon}^2])$.
  • Figure 2: Indifferent between two service instances and variability premium
  • Figure 3: Two instances of a mobility service with a certain loss and zero-mean risk.
  • Figure 4: Illustration of service instances with EU and DT variability where $\xi_1$ is a zero-mean variability and $\psi$ is probability.
  • Figure 5: Service instances before/after dual zero-mean $n$ state variability in which $\xi _1\le \cdots \le \xi _i=0\le \cdots \le \xi _n$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Definition 1: Risk attitudes
  • Definition 2: Variability premium
  • Proposition 1
  • proof
  • Remark 1: Differences and relationship among $VOT(T)$, $VOT$, and $COT$
  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 18 more