# Queuing Analysis of a Car Wash

Congestion at a car wash during peak hours can be a source of customer frustration as well as lost revenue.  We will perform a car wash queuing analysis of a proposed in-bay automatic car wash that is part of a convenience store.

An in-bay automatic car wash consists of a machine that passes back and forth over a parked vehicle in the wash bay.  This type of car wash can only wash one vehicle at a time, and the next person in line must wait for the previous wash to finish before entering the wash bay.

Queuing theory is the study of waiting lines.  The purpose of this car wash queuing analysis it to gain an understanding of the potential wait times before entering the wash.  When the line gets too long, customers will decide to not wash.  In queuing theory this is called balking.  Also, customers that are already in line may decide to leave.  In queuing theory this is called reneging.

The car wash queuing analysis could be part of a larger investment decision.  If the capacity of the in-bay automatic is too low, an analysis of whether it makes sense to make a larger investment in a tunnel car wash system, or a second in-bay automatic may be performed.  A tunnel car wash pulls vehicles through the wash on a conveyor and has more throughput than an in-bay automatic.  Both alternatives will require a larger investment, and more real estate.

## Car Wash Queuing Analysis Set-Up

We will use Monte Carlo simulation to generate a range of waiting time outcomes from the model shown below.  We are modeling the peak hours from 3:00 pm to 7:00 pm, and assuming the average number of washes in this period is 30.

The car wash will have three wash packages for sale, and each package has a different cycle time.  We will treat wash time as a random variable, mainly because the time of entering and exiting the wash bay will vary.  The triangular distribution will be used for wash time.

The mix of wash packages is modeled using a categorical distribution.  The basic wash has a 0.2 probability of being selected.  The standard wash has a 0.5 probability, and the premium wash has a 0.3 probability.

Customer arrivals will be treated as a Poisson process.  As a result, the time between arrivals is exponentially distributed.

The model recalculates at the time of each arrival.  The wash cycle time at the next arrival is for the vehicle already in the wash bay.  When cumulative wait time is zero, the arriving customer can enter the wash bay without waiting.

To simulate the four hour period, we keep track of elapsed time from 3:00 pm.  When elapsed time reaches 240 minutes (7:00 pm), cumulative wait time and elapsed time are set to zero so the simulation will reset to 3:00 pm.  This assumes that there is no wait at 3:00 pm since wait time is reset to zero.

## Simulation of the Model

The model was simulated for 50,000 iterations.  The output of the simulation is shown below.  Using the probability analysis tool, the probability of no wait time is 31.5%.  From the percentile data, half of customers will experience a wait time of 3.87 minutes or less.  90% of customers will have a wait time of 16.34 minutes or less.

## Sensitivity Analysis

A correlation report is run to study the effect input random variables have on wait time.  In the chart below, we see that time between arrivals is most correlated (negatively) with wait time.  As time between arrivals decrease, wait times tend to increase.

Wash cycle time and wash type have lower correlations to wait time.  As a result, arrival times are the dominant input to the model.

Let's look at the effect average car washes has on wait times.  First, we find the expected throughput of the car wash.

Based on the probability of each wash package and their mean cycle time, the expected value of wash time is 5.8 minutes.  In a four hour period, this results in an average throughput of 41.4 washes.  Depending on the actual mix of wash packages selected, the capacity will vary from this average.

With an idea of the throughput, we will run new simulations while changing the average car washes each time.  A summary is shown below.

As the number of car washes approach the average capacity of the car wash, wait times become increasingly unacceptable.  This illustrates that a theoretical capacity doesn't tell the whole story.  At 40 car washes, 25% of customers would be waiting more than 23.54 minutes.  Again, the expected throughput is 41.4 washes.

To explain why large wait times can occur despite being below capacity, consider that we are assuming the number of arrivals follows a Poisson distribution.  The time between arrivals, therefore follows an exponential distribution.

This means that the number of washes each day varies, and the time between washes also varies.

Since time between arrivals is a random variable, we could experience several consecutive arrival times that are less than the wash cycle time.  During these periods, wait time continues to build.  At other periods, time between arrivals is greater than wash cycle time, so there is a decrease in wait time or no wait time.

## Conclusion

We have performed a car wash queuing analysis using Monte Carlo simulation.  The results above show how important estimating the average number of cars washed in the four hour period is to our model.  It's also important to not rely on average throughput of the wash as a sole factor in selecting a system, since wait time can be significant for a large percentage of customers even when operating below capacity.

Simulations were performed using Simulation Master.

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