The Myth of Adding Servers: Why Mathematics Says It Won't Necessarily Clear the Queue

Have you ever sat in a hard plastic chair in a hospital, staring at a queue number display that seems frozen for hours? In such moments, our logic often makes a simple demand: “Why not just add more counters to speed things up?” But is it really that simple? Using a simulation in Maple software and data from the Indonesian Medical Association (IDI), I tested this assumption. The results were surprising. In the model I built, adding doctors did not necessarily reduce the queue if the overall rate of service was not increased.

Taking the waiting room problem into a mathematical simulation, I explored the variables affecting queues. The simulation utilised two main variables: service speed (total patients per hour) and operational time (when an additional counter opens). The combination of these variables was visualised in linear function graphs to find the intersection point representing the condition when the queue begins to be cleared. This approach is closely related to Queueing Theory, introduced by A.K. Erlang in 1909 to optimise telephone networks. The legendary theory explains that queues are controlled by two forces: the arrival rate of patients (λ) and the service rate (μ). If the service speed is equal to or slower than the arrival rate, the system becomes saturated and queues pile up.

The first scenario tested was based on IDI consultation data, where a general practitioner’s consultation time averages 11.5 minutes and a specialist’s averages 12 minutes. The simulation added a specialist counter at 09:30 to assist a general practitioner who started at 09:00. Although the plan sounded logical, the Maple simulation showed the two lines intersecting at a negative time point, meaning the additional counter could never catch up with the accumulated backlog. In a second scenario, the specialist was replaced by another general practitioner with identical speed. The result was two parallel lines that never intersected, indicating no acceleration in clearing the queue. Changing the patient arrival rate or the opening time of the second counter also failed to produce a significant effect.

Success finally came when the service speed of the additional doctor was increased slightly above the average consultation time. With the second counter still opening at 09:30, the graphs suddenly intersected at a stable point, representing the moment the service rate began to overcome the accumulated queue. The simulation demonstrated that a marginal improvement in service efficiency per patient had a far greater impact than simply adding more doctors without increasing service speed.