Two-Product Make-to-Stock System: Strategic Joining and Optimal Inventory Levels

TL;DR

This work analyzes a two-product make-to-stock system with a single shared production server where two customer classes decide to join without observing inventory. It derives stationary distributions and closed-form expressions for waiting times $\mathbb{E}[W_i]$, on-hand inventories $\mathbb{E}[I_i]$, and backlogs $\mathbb{E}[B_i]$, andCharacterizes Nash equilibria in joining strategies, proving existence and, under positive inventory, uniqueness. It formulates a Stackelberg producer problem to choose base-stock targets $S_1,S_2$ anticipating customer responses and compares it with a welfare-maximizing centralized planner who selects $(\lambda_1,\lambda_2,S_1,S_2)$; it also derives natural inventory thresholds and a toll scheme to implement welfare-optimal outcomes from decentralized decisions. Numerical experiments illustrate how waiting-cost asymmetry between types ($\kappa$) and utilization ($\rho$) shape profits and social welfare, revealing regions where delay-sensitive customers are excluded under decentralization and emphasizing potential welfare gains from proper pricing/tolicy interventions. Overall, the paper provides a comprehensive framework for analyzing strategic joining in multi-product queueing-inventory systems and offers actionable insights for aligning decentralized decisions with social welfare.

Abstract

This paper analyzes a two-product make-to-stock queueing system where a single production facility serves two customer classes with independent Poisson arrivals. Customers make strategic join-or-balk decisions without observing current inventory levels. The analysis establishes the existence and uniqueness of Nash equilibria in customer joining strategies for various inventory scenarios. Optimal base-stock levels are characterized from both profit-maximizing and welfare-maximizing perspectives, with closed-form expressions for key performance measures.

Figures (36)

• Figure 1: Two-Product Make-to-Stock System: Customer Flow and Production Process
• Figure 2: Queue Position for Backlogged Arrival
• Figure 3: Producer's profit $\Pi^{\textsc{dec}}$ under decentralization over the $(\kappa,\rho)$ plane. Thick black contours mark 1.98, 2.34, 2.69 (25%, 50%, 75% of range); thin gray contours provide detail.
• Figure 4: Social welfare $\mathrm{SW}^{\textsc{cen}}$ under centralization over the $(\kappa,\rho)$ plane. Thick black contours mark 3.16, 3.54, 3.92 (25%, 50%, 75% of range); thin gray contours provide detail.
• Figure 5: Ratio of decentralized to centralized welfare, $\mathrm{SW}^{\textsc{dec}}/\mathrm{SW}^{\textsc{cen}}$ over the $(\kappa,\rho)$ plane. Contours mark $0.70$ (solid), $0.80$ (dotted), and $0.90$ (dashed); thin gray contours provide detail.

Theorems & Definitions (23)

• Proposition 3.1
• Lemma 4.1: Continuity and Monotonicity of $U_i$
• Lemma 4.2: Unique Fixed Points in One Variable
• Proposition 4.1: Nash Equilibrium Classification
• Proposition 4.2: Existence of Nash Equilibrium
• proof : Proof sketch
• Proposition 4.3
• proof : Proof sketch
• Proposition 4.4
• proof : Proof sketch