In the past farmers had to do everything by hand. In particular they had to milk their cows twice a day all by themselves without help from machines. But as time passed by, more and more techniques and machines have been developed to make life easier.
Nowadays there are robots that can milk cows without human intervention. The cow walks into the machine, the teats of the cow are found by sensors in the machine and the milking begins automatically. There are at least two good reasons for using robots. First, it saves a lot of labor and second, it makes it much easier to go from milking twice a day to three or more times a day. When cows are milked three times a day their production is increased with 10 to 15 percent. You can see a milking robot in the following picture.
An important facility in a barn with robots is the concentrate feeder. Cows are very fond of concentrate food. The concentrate feeder is used to lure the cows into the milking robots. The only way they can get to the concentrate feeder is by passing through the milking robots. The cows are allowed to receive only a limited amount of concentrate food per day. So the concentrate feeder has equipment to be able to distinguish the cows and to decide to give or not to give concentrate food to the cow. It is even possible to adapt the amount to the specific needs of each cow. The three other (conventional) facilities in the barn are:
The space in between the facilities is used for walking, idling or grouping. This takes roughly 20 percent of their time, so 4 to 5 hours a day. In that time they cover only between 3 and 5 kilometers, so a better word for the facility might be standing. Anyway, the walking area should be large enough to accommodate roughly at least 20 percent of the herd. You can see the layout of a robotic barn in the picture below (the dimensions are in millimetres):
Milking robots are very expensive. A farmer likes to buy as few milking robots as possible. However, too few milking robots leads to cows waiting in front of the robots. When the cows are waiting, aggressive behavior may occur. So the discussion about the design focusses very much on the waiting times for the milking robots.
From the above description it is clear that the cows in the barn use 6 facilities. The capacity of these facilities is limited, so queueing may occur. The service durations and movement between the facilities is random. This suggests a queueing network model with 6 stations:
The station Walking is an artificial station for the movement of the cows between the various facilities. It can be modelled as an infinite server station. The other stations are single or multi servers. The service time per station can be obtained from measurements and specifications. The transition process can also be determined from measurements. And of course, the customers are the cows.
The above model of the robotic barn is a nice application of queueing networks. Given the visit frequencies, service times, capacities and herd size, important performance measures like waiting times, queue lengths and utilizations for the facilities can be very efficiently evaluated. The evaluation of the performance is based on an approximate mean value algorithm. In the paper Dairy barns and queueing networks, this algorithm is described in more detail. So the queueing network model is very useful to support the design of robotic barns. The model has been implemented in the program Cow 1.0. This program can be used by students or will-be students to get an idea of the possibilities of mathematics. Students can create and modify their robotic barn and try to obtain an `optimal design.' The program uses graphical representations for showing waiting times and queue lengths for the various facilities.
Cow 1.0 has been developed by Marko Boon and Michel Vollebregt for the Department of Mathematics and Computing Science at the Eindhoven University of Technology. Cow 1.0 may be used freely, but unauthorized copying is not allowed.