Intensive care unit/step-down unit queuing game with length of stay decisions

In this paper, length of stay (LOS) competition between two servers in tandem without buffer between them is investigated using queuing games. This system typifies the relationship between the intensive care unit (ICU) and the step-down unit (SDU) of a hospital. We model and analyze the equilibrium LOS decision under four different games (one cooperative and three non-cooperative games) as follows: (i) both servers cooperate; (ii) the servers do not cooperate and make decisions simultaneously; (iii) the servers do not cooperate and the first server, the ICU, is the leader (ICU Stackelberg); (iv) the servers do not cooperate and the second server, the SDU, is the leader (SDU Stackelberg). The payoff of the ICU is expressed as the difference between the service benefit and the waiting in queue penalty, while that of the SDU is the difference between the service benefit and the overstay penalty. The results show that LOS decisions of each server depends critically on the payoff function’s form and the exogenous demand. Secondly, with a linear payoff function, the SDU is only beneficial to the system if the unit cost is greater than the unit reward at the ICU. Our results revealed also that payoffs depend on the substitutability in both ICU Stackelberg and SDU Stackelberg games. When most of the LOS is spent at the ICU unit. Our results suggest that the critical care pathway performs better under coordination and or leadership at the ICU level.