Exact formula is not possible to derive, however, non homogeneity in a media creates situation of multiple resonance response due to creation of multiple LC network and also it creates secondary source radiation. Length/width are more on a gut feel/trial bases. As a rule of thumb you may take of the order of 1/20th part of wavelength
There is a book talking about slots. I'll try to find it and send you the title and the authors.
In the other hand, you may find something in the book of Balanis, regarding slots cut in a rectangular waveguide. You can go to IEEE and search for slot antennas, I am sure you'll find plenty of équations. Some people use trial error method too.
slot impacts on the resonance is based on radiating element which u choose . i.e rectuangular ,square, circular or triangle patch . and another parameter is feed method. which method u feeding the antenna whether the probe feed or microstrp feed etc. so wch type of radiating elements and feed method u used..?
The purpose of cutting the slot is to enhance the bandwidth of particular design of microstrip antenna. Also the slots width & length depends on radiating element as well as feeding techniques.