On-Off Systems with Strategic Customers

TL;DR

The paper analyzes a fluid, multi-queue, single-server system with on-off service cycles under strategic customer joining. It develops two regimes: exogenous on-off durations and endogenous durations under exhaustive service; for the former, it derives closed-form equilibria and a compact LP to maximize throughput, while for the latter it proves that optimal policies involve at most one nonzero post-clearance duration and presents a $2\bar n$-step algorithm to compute it. A linear complementarity framework governs the endogenous regime, yielding a scalable method to compute equilibria in $n$ queues. Across both regimes, customers exhibit cyclical herd behavior rather than threshold strategies, and the model connects exogenous and endogenous results through structural insights, including conditions under which exogenous patterns can be implemented endogenously. The findings have practical implications for traffic control, make-to-order systems, and dynamic polling, offering actionable optimization tools with rigorous equilibrium reasoning.

Abstract

Motivated by applications such as urban traffic control and make-to-order systems, we study a fluid model of a single-server, on-off system that can accommodate multiple queues. The server visits each queue in order: when a queue is served, it is "on", and when the server is serving another queue or transitioning between queues, it is "off". Customers arrive over time, observe the state of the system, and decide whether to join. We consider two regimes for the formation of the on and off durations. In the exogenous setting, each queue's on and off durations are predetermined. We explicitly characterize the equilibrium outcome in closed form and give a compact linear program to compute the optimal on-off durations that maximizes total reward collected from serving customers. In the endogenous setting, the durations depend on customers' joining decisions under an exhaustive service policy where the server never leaves a non-empty queue. We show that an optimal policy in this case extends service beyond the first clearance for at most one queue. Using this property, we introduce a closed-form procedure that computes an optimal policy in no more than 2n steps for a system with n queues.

Figures (11)

• Figure 1: Illustration of the Waiting Time Dynamics under Exogenous on-off Durations with Strictly Positive Not-Joining Duration $\bar{J}_i$. Note. The amount of the upward jump at $\clubsuit$ is $\bar{L}_i$. The post-clearance duration $T_i$ can be zero. In the other case where the not-joining duration is zero ($\bar{J}_i=0$), we have $W_i(\diamondsuit)<\theta_i$ while the pattern remains unchanged.
• Figure 2: Queueing Dynamics of Exhaustive Equilibrium Outcomes under on-off Durations $(L_i,\bar{L}_i)$.
• Figure 3: Queueing Dynamics of Non-Exhaustive Equilibrium Outcomes under on-off Durations $(L_i,\bar{L}_i)$.
• Figure 4: Queueing Dynamics of Queue $i$ under the Exhaustive Service Policy $T$. Note. The switchover time $\tau_i$ can be larger than $\bar{J}_i$. Besides, the not-joining duration $\bar{J}_i$ can be zero, in which case $\diamondsuit$ coincides with $\clubsuit$.
• Figure 5: Throughput and Equilibrium of the Exhaustive Service Policy $T=(T_j,T_{-j}=0)$, where $j \in \mathop{\rm arg\,max}_{i \in \bar{\mathcal{I}}(0)} \lambda_i$. Note. The figures are based on the parameters $n = 4$, $\lambda = [1.0, 1.2, 0.4, 0.2]$, $\mu = [5.0, 3.0, 2.0, 4.0]$, $1^\top \tau = 1.5$, and $\theta_0 = [3.5, 0.5, 2.5, 2.0]$. It turns out that in all three cases, the queue $j \in \mathop{\rm arg\,max}_{i \in \bar{\mathcal{I}}(0)} \lambda_i$ is queue $2$. The values of the non-plotted variables $\alpha_i$ remain fixed at 1 throughout. The equilibrium joining sets under the pure exhaustive service policy are $\mathcal{I}(0) = \emptyset$ in (a), $\mathcal{I}(0) = \{1, 3\}$ in (b), and $\mathcal{I}(0) = \{1, 3, 4\}$ in (c).

Theorems & Definitions (56)

• Definition 1: Equilibrium
• Lemma 1: Waiting Time
• Remark 1: Unique Equilibrium Joining Strategy
• Lemma 2
• Theorem 1: Exhaustive Equilibrium Outcomes
• Theorem 2: Non-Exhaustive Equilibrium Outcomes
• Corollary 1
• Proposition 1
• Theorem 3
• Lemma 3: Equilibrium Structure