Hospital referral and capacity strategies in the two-tier healthcare systems

•We explore the impact of referral payment scheme on implementing the healthcare referral program in the two-tier service systems.•We employ a game-theoretic queuing model to capture the interaction between the patients and two levels of the hospital.•Both equilibrium referral rate and equilibrium capacity first increase and then decrease according to the revisit rate.•A win-win situation can be attained when referral payments below the threshold.•Referral payments increasing always reduces the efficiency of the healthcare delivery system when referral payments above the threshold.