Two-stage Online Reusable Resource Allocation: Reservation, Overbooking and Confirmation Call

TL;DR

This work studies a two-stage online reusable resource allocation problem with advance reservations, walk-in demand, overbooking, and a confirmation call that refines no-show predictions. It introduces Decoupled Adaptive Safety Stocks (DASS) to hedge overbooking risk using only single-day information, enabling decoupling of inter-day occupancy dynamics. Under a busy-season condition, DASS achieves constant regret by reducing a multi-day problem to independent per-day subproblems and shows that Stage II regret decays exponentially with the time between confirmation and day end; late confirmations remain near-optimal. The framework is extended to heterogeneous, multi-class customers with nested capacity protection, with rigorous regret bounds and an empirical validation on synthetic data and Algarve resort hotel data demonstrating practical revenue gains. Overall, the approach provides actionable guidance on reservation thresholds, confirmation timing, and walk-in utilization to achieve near-optimal revenue in dynamic, uncertain service settings.

Abstract

We study a two-stage online reusable resource allocation problem over T days involving advance reservations and walk-ins. Each day begins with a reservation stage (Stage I), where reservation requests arrive sequentially. When service starts (Stage II), both reserved and walk-in customers arrive to check in and occupy resources for several days. Reserved customers can cancel without penalty before or during a confirmation call initiated by the decision maker (DM) before day's end. The DM must immediately accept or reject each booking or check-in request, potentially overbooking by accepting more reservations than capacity. An overbooking loss occurs if a reserved customer's check-in is rejected in Stage II; a reward is obtained for each occupied resource unit daily. Our goal is to develop an online policy that controls bookings and check-ins to maximize total revenue over the T-day horizon. We show that due to cancellation uncertainties and complex correlations between occupancy durations, any online policy incurs a regret of Ω(T) compared to the offline optimal policy when the \textit{busy season} assumption does not hold. To address this, we introduce decoupled adaptive safety stocks, which use only single-day information to hedge against overbooking risks and reduce resource idling. Under the busy season condition, our policy decouples the overall offline optimal into single-day offline optimal policies. Consequently, the regret between our policy and the offline optimal decays exponentially with the time between the confirmation call and day's end, suggesting the DM can delay confirmation calls while maintaining near-optimal performance. We validate our algorithm through sythetic experiments and empirical data from an Algarve resort hotel.

Figures (14)

• Figure 1: Timeline of two-stage control for Day $k$. Reservations arrive during Stage I (days $k-k_0$ to $k$). On Day $k$, both reserved and walk-in customers arrive during Stage II. The confirmation call at time $k+\nu$ reveals which reservations will actually show up, enabling the manager to finalize capacity allocation.
• Figure 2: Customer journey flow. Type I customers (red) make advance reservations during Stage I and may cancel; those who confirm at $k{+}\nu$ proceed to check-in. Type II customers (blue) are not yet present during Stage I; they arrive directly as walk-ins during Stage II. Both types undergo capacity checks at check-in.
• Figure 3: The overbooking tradeoff. A conservative policy (left) accepts fewer reservations, leaving rooms idle and forgoing revenue $r^{(k)}$ per empty room. An aggressive policy (right) overbooks, filling all rooms but incurring penalty $\ell^{(k)}$ when confirmed arrivals exceed capacity.
• Figure 4: Retention probability as a function of booking time. The probability $p^{(k)}(t)$ of keeping a reservation increases as booking time $t$ approaches Day $k$. Early bookings have low retention probability; late bookings rarely cancel. After Day $k$, $p^{(k)}(t) = 1$: at the confirmation call ($k{+}\nu$), remaining customers show up with certainty.
• Figure 5: Capacity evolution with multi-day stays. Each day's occupancy combines continuing stays (light gray) and new check-ins (dark gray). Approximately $\delta C$ rooms turn over each day---this turnover rate determines both available capacity for new arrivals and the scale of walk-in demand needed for the busy season assumption.

Theorems & Definitions (20)

• Example 1: Notation in context
• Example 2: Busy season threshold calculation
• Example 3: Lower bound illustration
• Proposition 1: Loss lower bound
• Example 4: DASS-I in action
• Theorem 1: Regret upper bound for Stage I
• Lemma 1
• Example 5: Decoupling in action
• Example 6: DASS-II in action
• Example 7: DASS-II after confirmation