Hypercube queuing model for emergency facility location problem considering travel and on-scene service times

The hypercube queuing model is a descriptive model for emergency systems in which servers are mobile and serve customers at their locations. In emergency systems, the service time of each server includes the travel time from the server station to the customer's location, the on-scene time and the travel time from the customer's location to the server station. The on-scene service time depends on factors such as server expertise and the severity of the customer’s situation while the travel times depend on factors such as vehicle type, the path, and the traffic volume. Therefore, it is necessary to consider and analyze these two times separately. In the hypercube queuing model presented in this study, the service time is divided into two sections, the travel time and the on-scene service time, both of which follow independent exponential distributions with known rates. A new system state is defined in which the status of servers is classified into idle, serving at the customer's location and traveling. By solving the equilibrium equations with the Gaussian- Elimination method (for small size examples) and simulation (for larger examples), limiting probabilities are obtained, and performance measures (such as the ratio of the on-scene time to the total server busy time) are evaluated. A case study of the road emergency stations of the Red Crescent, which are based in Hamadan province, Iran, is also used to check the model's real-world performance.