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M/D/S Queueing Model


This hypothetical example shows how a partially automated car wash can improve service times and its bottom line by replacing post-wash cloth drying with automated blow drying.  Currently the single-queue car wash station has an automatic wash-rinse-wax cycle, after which employees manually cloth-dry each vehicle as it emerges from the station.

Management believes that they can significantly shorten service times, reduce waiting lines and improve their bottom line by adding an automatic blow-dryer to the end of the cleaning cycle.  Here is the current setup, before adding the automatic drying unit:

Car wash with hand drying

The current setup can be represented by a M/G/1 queueing model that has a Poisson distribution for customer arrival rates, a corresponding exponential distribution for interarrival times, and a general probability distribution for service times.  [Note: if you are not already familiar with Kendall Notation for queueing models, then before continuing you should read our introductory queueing optimization page.] 

The service rate has a general probability distribution because there is little variation in the automatic portion of the cleaning cycle: some customers may want more or less expensive cleaning options ("Bronze," "Silver" and "Gold" service packages); but the total time required for each cleaning package varies little.  The hand-drying portion of the cleaning cycle also follows a general probability distribution because there is not much variation in the time required to cloth-dry different vehicles.

In the table above, the numbers highlighted in yellow can change as a function of changes in the gray-highlighted values above them.  Here is the result of switching from cloth drying to automatic drying, shown next to the original model for comparison:

M-G-3 and M-D-3 car wash queueing models

We have shifted from a cloth-drying M/G/1 queueing model to an automated-drying M/D/1 model.  This new model still has exponentially distributed interarrival times, but it now has a deterministic service-time probability distribution because the fully automated cleaning cycle has a service-time standard deviation of zero.  This is because the blow-drying time required for a waxed car is less than for an unwaxed car, and the mechanical drying unit automatically adjusts drying time so that total cleaning time remains constant irrespective of which cleaning options the customer has selected.

From the above table we can see that average time spent waiting in line for a car wash has been slashed by more than 2/3, from 9.62 to 3.13 minutes; service times have been reduced; and total time in the system has been reduced my more than half, from 14.23 minutes to 6.88 minutes.  In addition, the shorter service time allows the system to service three more vehicles per hour.

Theoretically, this increased throughput and the shorter queues should attract more customers per hour when they see that the car wash is less crowded and waiting times have decreased significantly.  So we can build a third model that resets the time spent in queue back up to its original value of 9.62 as an upper limit, and then recalculate the customer arrival rate.  Here is the result:

M-G-3 and M-D-3 car wash queueing models with equal queue times

From the table above we can see that the customer arrival rate could increase from 10 to 13.39 vehicles per hour, leading to an 84% system utilization rate.  In reality, we may not always see a customer arrival rate of 13.39 cars per hour.  But we know that we now have the capability to handle higher peak volumes of traffic when they occur; and we have reduced the cost of labor by automating the drying cycle.  The automatic drying unit should pay for itself fairly quickly.

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