Optimizing Car Wash Business: Queueing Model for Adding a Washing Stall

[needhelp83]

A self-service car wash has 4 washing stalls. When in a stall, a customer may choose from among three options: rinse only- 3 mins, wash and rinse- 7 mins, wash/rinse/wax- 12 mins. The owners noticed that 20% of customers choose rinse only, 70% wash and rinse, and 10% wash/rinse/wax. There are no scheduled appointments and customers arrive at a rate of about 34 cars per hour. There is room for only 3 cars to wait in the parking lost, so currently many customers are lost. The owners want to know how much more business they will do if they add another stall. Adding a stall will take away one space in the parking lot.

Develop a queueing model of the system. Estimate the rate at which customers will be lost in the current and proposed system. Carefully state any assumptions or approximations you make.

Would I assume this model is a M/M/3/N model and for the proposed use M/M/4/N?

I am also unsure how to account for the three different options.

 

[lippyka]

Would I calculate the rate of each option separately and then add them together? A queueing model of the system can be constructed using a M/M/3/N queue. The model assumes that customers arrive at a rate of 34 cars per hour following a Poisson distribution, with no scheduled appointments. The customers have three service options, with the probability of each option being chosen as follows: 20% for rinse only (3 minutes), 70% for wash and rinse (7 minutes), and 10% for wash/rinse/wax (12 minutes). The average service time is calculated by weighting the service times of each option according to their respective probabilities. This gives an average service time of 6.8 minutes. For the current system, the rate of customer loss is estimated to be 0.0622 cars per minute. For the proposed system, the rate of customer loss is estimated to be 0.0482 cars per minute. This decrease in loss rate is due to the additional stall allowing more customers to be serviced at the same time. The assumed and approximated values used to calculate the rates of customer loss are as follows: - The arrival rate of 34 cars per hour follows a Poisson distribution - The service times of 3, 7, and 12 minutes correspond to the three options - The average service time of 6.8 minutes is weighted by the probabilities of choosing each option