Experimenting under Stochastic Congestion

Abstract

We study randomized experiments in a service system when stochastic congestion can arise from temporarily limited supply or excess demand. Such congestion gives rise to cross-unit interference between the waiting customers, and analytic strategies that do not account for this interference may be biased. In current practice, one of the most widely used ways to address stochastic congestion is to use switchback experiments that alternatively turn a target intervention on and off for the whole system. We find, however, that under a queueing model for stochastic congestion, the standard way of analyzing switchbacks is inefficient, and that estimators that leverage the queueing model can be materially more accurate. Additionally, we show how the queueing model enables estimation of total policy gradients from unit-level randomized experiments, thus giving practitioners an alternative experimental approach they can use without needing to pre-commit to a fixed switchback length before data collection.

Figures (13)

• Figure 1: A queueing system with an outside option. Agents are heterogeneous in their preferences, and consider both queue length and the price $p$ in choosing whether to join the queue. However, once agents join the queue, they are all processed at the same rate $\mu$.
• Figure 2: Illustration of the queue length process as a continuous-time Markov chain.
• Figure 3: Illustration of the two types of switchback experiments. Every dot indicates a new arrival. Blue dots/curve correspond to a price of $p- \zeta$, and red dots/curve correspond to a price of $p+\zeta$. In regenerative switchback experiments, the price is only switched when the queue is empty. In interval switchback experiments, the price is only switched at some fixed time (when $t = 25z$, for $z \in \mathbb{Z}$ in this example).
• Figure 4: In an $M/M/1$ queue, $\sigma_{\operatorname{WDE}}^2(p)$ and $\sigma_{\bar{\lambda}}^2(p)$ are the same.
• Figure 5: In a zero-deflated or zero-inflated $M/M/1$ queue, $\sigma_{\operatorname{WDE}}^2(p)$ is strictly smaller than $\sigma_{\bar{\lambda}}^2(p)$ except when $\lambda_0 = 0.5$ (corresponding to an $M/M/1$ queue). When $\lambda_0$ is large, $\sigma_{\pi_0}^2(p)$ is smaller than $\sigma_{\bar{\lambda}}^2(p)$.

Theorems & Definitions (54)

• Example 1: Waiting cost
• Example 2: Group conformity
• Remark 1: State-dependent pricing
• Theorem 1
• Theorem 2: Model-free estimator
• Theorem 3: Idle-time-based estimator
• Theorem 4: Weighted direct effect estimator
• Remark 2: Choice of $\zeta_T$
• Remark 3: Interval length
• Theorem 5: Variance comparison