Question

XYZ company operates an automatic car-wash facility in a city. The manager is concerned about the long lines of cars that build up while waiting for service. The service time for the system is machine-paced and thus constant. The manager has the opportunity to decrease the service time by increasing the speed of the conveyor that pulls the cars through the system. Of course, the quicker the pace, the lower is the quality of the car-wash. The manager wants to study the effects of setting the system for a 2-minute car-wash. The following data on customer arrivals have been gathered. Interarrival time (minutes) : 1 3 3 4 5 Number of occurrences: 136 34 102 51 17

Compare the traffic density for a service rate of 2 minutes per car.

Answer 1

To study the traffic density for a service rate of 2 minutes per car at an automatic car-wash facility, we compare the average arrival rate of customers to the service rate. If the service rate matches the average arrival rate, the traffic density could be balanced. However, variations in arrival rates can still cause queuing, necessitating more comprehensive analysis using queuing theory.

Step-by-step explanation:

To determine the traffic density for a service rate of 2 minutes per car at the XYZ company's automatic car-wash facility, we must first understand the concept of arrival rates and service rates in a queuing system. From the data provided, we have an average of 1 customer arriving every 2 minutes. This is calculated by considering the average number of customers arriving in a 60-minute period, which in this case would be 30 customers per hour. Therefore, if the service time is set for 2 minutes per car, the system can potentially match the average arrival rate, washing one car every time another car arrives.

However, we must account for variability in actual arrival times and the potential for periods where arrivals are more frequent than the service rate. We can find the traffic density by looking at the ratio of service rate to arrival rate. Since both rates are 2 minutes, under perfect conditions, the traffic density would be 1. Nonetheless, given the variability in arrivals, there might still be waiting lines.

The exponential distribution of waiting times between events will be essential for further analysis, as would concepts such as waiting time distribution. To fully evaluate the traffic density and potential queuing issues, more sophisticated queuing theory models would need to be employed.