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
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.
Back to the main Queueing Optimization page.