Max-plus algebra modeling for outpatient service queues in Yogyakarta private hospitals

Abstract

This study aims to model the outpatient service queue system in a private hospital in Yogyakarta using the Max-Plus algebra approach and to represent the model using a Petri net. The research was motivated by the complexity and inefficiency observed in outpatient queues, which often lead to long and unpredictable waiting times. Data were collected through direct observation of the outpatient process on the fourth floor of a private hospital. The sequence of services was first illustrated using a flowchart to map each stage experienced by patients. The time data for each process were then used to construct a Max-Plus algebra matrix, providing a mathematical model of the system. This model was further expressed in the form of a Petri net to illustrate the discrete and sequential nature of the service flow. The simulation was performed using Scilab software to analyze the system's dynamics. Results from the simulation revealed that the service system is non-periodic, indicated by the absence of an eigenvalue. This suggests that the total service time may increase as the number of patients grows. The findings provide insight into the structure of the outpatient queue system and offer a mathematical framework for future system analysis or optimization.