Here is a hypothetical example of a G/G/S queueing model for supermarket check-out lines.  [Note: if you are not familiar with Kendall Notation for queueing models, please read our introductory queueing optimization page before continuing.]  This is a G/G/S system because interarrival times and service times both follow a general (nonexponential) random probability distribution.

As the table below indicates, the supermarket has 10 human cashier-served lines and four self check-out stations where the customer scans items from the shopping cart, bags the items and pays at a machine that accepts cash, credit cards or debit cards as payment.  Customers enter the store at an average rate of 320 per hour; 70% choose a check-out line served by a human cashier, while 30% choose self-service.  Average service time checking out is about 2.6 minutes for cashier-attended lines and about one minute for self check-out lines.  (Self check-out lines usually take less time because customers who choose these lines typically have a much smaller number of items as well as less bulky items.)

As currently configured, cashier-served lines average a total check-out time of 8.94 minutes, of which about 6.34 minutes are spent waiting in line behind other customers.  The self-serve checkout lines average just over a minute to get through, with almost no waiting in line.  Thus, there is an 89% probability of having to wait in line for a human cashier, but only a 15% probability of having to wait for the self check-out stations, for an overall averge of 67%.

Management would like to decrease the wait times for human-serviced check-out lines.  The self check-out stations have been installed only recently, and many customers probably don't use them because they aren't familiar with them.  A queueing model is developed to determine how much impact increased usage of self check-out would have on overall system crowdedness and wait times.  As the graph below shows, if the percent of customers using self check-out increased from 30% to 35%, then the overall number of customers in the system and the overall average time in the system would be reduced by more than half:

So a training system is set up whereby store personnel assist customers in learning how to use the stations efficiently.  After about a week, 35% of customers are using the self check-out stations.  Here is the impact on various measures: 

Among other things, we can see that average time spent waiting in line for a human cashier has dropped dramatically, from 6.34 minutes to 1.43 minutes.  Continued training manages to boost the percent using self check-out to 40%, with the following results:

Once again we have managed to cut waiting times in the cashier-serviced lines by more than half; and there is little negative impact on the self check-out lines.  The store manager believes, based on the line graph above, that there is little to be gained from trying to increase the percent of customers using self check-out beyond about 40%; and the effort might not be very successful anyway, because there are a limited number of customers having grocery carts with few items and/or few bulky items.

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