This is going to be a description gleaned from a lot of sources and a lot of personal lab work. Here goes...
Fixing is a diffusion/kinetic/diffusion process. It starts by requiring the diffusion inward of fixing chemistry and usually takes between 10 and 20 seconds at 68 degrees F depending on thickness and hardness of the coating among other factors.
At this point, when the wave front of chemistry reaches the bottom of the coating, the top has begun to fix. This is a kinetic reaction involving the formation of silver complexes with hypo and ammonium ion (if present). With sodium hypo, about 5 complexes can form, but with ammonium ion present a huge number of combinations are possible.
As soon as silver halide begins to dissolve, it begins the third phase which is outward diffusion of the complexes. This is a different speed than that at the start if swell changes path length or porosity of the gelatin. Also, the hypo complexes are larger than hypo itself and therefore may move more slowly. Alkaline fixes or fixes that cause swell will enhance this speed. Total fixation is the sum of inward diffusion (~20 seconds), reaction rate, and outward diffusion (> 20 seconds). We take this to be 2x the time it takes to clear or 2x the time it takes to diffuse in, react and then move out, so it is logical. In + react + out = fix time. React and out are influenced by the nature of the complex and the fix formula (pH and etc). No fix can be faster than the inward diffusion rate, so beware of fixes that claim to work in 15" or things like that. It might clear the film, but you have to wait for things to begin outward diffusion to properly clear. It is inevitable though that some silver hypo complexes will remain in the coating. That is one reason why we wash.
Wash, as described by many, is a non-linear exponential function based on diffusion only. It is described as the change in concentration / unit time = a constant * (the amount of complex you have - the amount leaving the coating).
In this case, the constant is related to the type of complex (sodium or ammonium), the thickness of the coating, and the pH of the fix used. However, this is ideal assuming an infinite volume of water or infinite agitation.
If you have a small container of wash water and/or don't wash, the rate of washing slows down. The equation then beocmes this.....
the change in concentration / unit time = a constant * [(the amount of complex you have - the amount leaving the coating) - the amount building up on the surface of the film or in the wash water]
Now, what this means is that washing is dependant on time in the wash water, the amount of water used, the amount of fixer building up in the water and the amount of exchange rate of water at the surface of the thing being washed.
This may go against many practices and in fact may refute some posts here but is stated by Mason from Ilford and Mees and James, both from Kodak. And, this information refutes the Ilford wash method by one of Ilford's respected researchers.
The bottom line though is the objective. Get silver to dissolve and then get the complexation products out of the film or paper. There are two test for these.
One uses sodium sulfide to test for silver halide and complexes in the coating. A drop of the test solution (Sodium Sulfide) will turn brown or black if silver ion is present in the coating in any form. The other uses silver nitrate in acetic acid and turns varying shades of yellow brown as a function of the amount of hypo retained in the coating. Using both of these will tell you what is best for you. If your coating is washed enough to pass these two tests, then it will meed ISO standards for image stability. Overwashing can cause problems as described by Ctein. Underwashing will surely fail one or another of these tests.
If you use fresh fix, there is no problem, but if you reuse fix, then you should use the fixer exhaustion test which is Potassium Iodide solution added to a small quantity dropwise to see if a yellow precipitate or cloud forms. If it does, the fixer is exhausted or has gone bad.
This entire description is generic and mentions the caveats to keep in mind when one uses acid or alkaline fixes or sodium or ammonium fixes. It generalizes water variataions and a lot of other items. What it is meant to be is a guideline used with tests to verify the effectiveness of a wash.
Lets go on and add some variables. Metol and Hydroquinone are used in many developers as is ascorbic acid. All three are soluable in base due to formation of salts in alkali. Metol is soluable in acid as well because it is both acid and base (to an extent). There is no easy test for these, and so passing the above tests will not tell you how well you have done in removing these three chemcials. All of them can do things to the silver image. In addition, having a small amount of sulfur compounds present in a coating has been seen to be useful (Ctein, Agfa Sistan, Fuji stabilzer and etc), and so overwashing has been shown to be bad.
There is also discussion on washing aids. Well, these are controversial. They do help remove hypo and silver complexes. OTOH, they can cause their own problems if you do not wash them out properly and so textbooks on this subject often caution the photographer to insure that the wash aids themselves get a good wash after their use.
Hope this helps.
I have posted other viewpoints of this in several threads as well as extended descriptions of the diffusion process and hypo build up on the surface of the film. I might add that Bill Troop's mention of HQ retention and mine are probably the first mention of this anywhere. I doubt if anyone knew of it.
I have also posted a scan of the Kodak residual hypo test kit. I mention it here for those that missed it.
PE
Fixing is a diffusion/kinetic/diffusion process. It starts by requiring the diffusion inward of fixing chemistry and usually takes between 10 and 20 seconds at 68 degrees F depending on thickness and hardness of the coating among other factors.
At this point, when the wave front of chemistry reaches the bottom of the coating, the top has begun to fix. This is a kinetic reaction involving the formation of silver complexes with hypo and ammonium ion (if present). With sodium hypo, about 5 complexes can form, but with ammonium ion present a huge number of combinations are possible.
As soon as silver halide begins to dissolve, it begins the third phase which is outward diffusion of the complexes. This is a different speed than that at the start if swell changes path length or porosity of the gelatin. Also, the hypo complexes are larger than hypo itself and therefore may move more slowly. Alkaline fixes or fixes that cause swell will enhance this speed. Total fixation is the sum of inward diffusion (~20 seconds), reaction rate, and outward diffusion (> 20 seconds). We take this to be 2x the time it takes to clear or 2x the time it takes to diffuse in, react and then move out, so it is logical. In + react + out = fix time. React and out are influenced by the nature of the complex and the fix formula (pH and etc). No fix can be faster than the inward diffusion rate, so beware of fixes that claim to work in 15" or things like that. It might clear the film, but you have to wait for things to begin outward diffusion to properly clear. It is inevitable though that some silver hypo complexes will remain in the coating. That is one reason why we wash.
Wash, as described by many, is a non-linear exponential function based on diffusion only. It is described as the change in concentration / unit time = a constant * (the amount of complex you have - the amount leaving the coating).
In this case, the constant is related to the type of complex (sodium or ammonium), the thickness of the coating, and the pH of the fix used. However, this is ideal assuming an infinite volume of water or infinite agitation.
If you have a small container of wash water and/or don't wash, the rate of washing slows down. The equation then beocmes this.....
the change in concentration / unit time = a constant * [(the amount of complex you have - the amount leaving the coating) - the amount building up on the surface of the film or in the wash water]
Now, what this means is that washing is dependant on time in the wash water, the amount of water used, the amount of fixer building up in the water and the amount of exchange rate of water at the surface of the thing being washed.
This may go against many practices and in fact may refute some posts here but is stated by Mason from Ilford and Mees and James, both from Kodak. And, this information refutes the Ilford wash method by one of Ilford's respected researchers.
The bottom line though is the objective. Get silver to dissolve and then get the complexation products out of the film or paper. There are two test for these.
One uses sodium sulfide to test for silver halide and complexes in the coating. A drop of the test solution (Sodium Sulfide) will turn brown or black if silver ion is present in the coating in any form. The other uses silver nitrate in acetic acid and turns varying shades of yellow brown as a function of the amount of hypo retained in the coating. Using both of these will tell you what is best for you. If your coating is washed enough to pass these two tests, then it will meed ISO standards for image stability. Overwashing can cause problems as described by Ctein. Underwashing will surely fail one or another of these tests.
If you use fresh fix, there is no problem, but if you reuse fix, then you should use the fixer exhaustion test which is Potassium Iodide solution added to a small quantity dropwise to see if a yellow precipitate or cloud forms. If it does, the fixer is exhausted or has gone bad.
This entire description is generic and mentions the caveats to keep in mind when one uses acid or alkaline fixes or sodium or ammonium fixes. It generalizes water variataions and a lot of other items. What it is meant to be is a guideline used with tests to verify the effectiveness of a wash.
Lets go on and add some variables. Metol and Hydroquinone are used in many developers as is ascorbic acid. All three are soluable in base due to formation of salts in alkali. Metol is soluable in acid as well because it is both acid and base (to an extent). There is no easy test for these, and so passing the above tests will not tell you how well you have done in removing these three chemcials. All of them can do things to the silver image. In addition, having a small amount of sulfur compounds present in a coating has been seen to be useful (Ctein, Agfa Sistan, Fuji stabilzer and etc), and so overwashing has been shown to be bad.
There is also discussion on washing aids. Well, these are controversial. They do help remove hypo and silver complexes. OTOH, they can cause their own problems if you do not wash them out properly and so textbooks on this subject often caution the photographer to insure that the wash aids themselves get a good wash after their use.
Hope this helps.
I have posted other viewpoints of this in several threads as well as extended descriptions of the diffusion process and hypo build up on the surface of the film. I might add that Bill Troop's mention of HQ retention and mine are probably the first mention of this anywhere. I doubt if anyone knew of it.
I have also posted a scan of the Kodak residual hypo test kit. I mention it here for those that missed it.
PE
.
I had no appreciation for how complex an issue this is. Clearly, the only way to know is to do some testing. Just like we have to do our own EI testing, we also have to do our own wash testing. 
