Pianoannie is right onthe spot.

Here are a few more thoughts on this subject:
In my humble opinion, you are approaching this problem the wrong way. The question is not one of fairness. Fairness does not really enter the equation (partly because who are we to judge what is and what is not fair, and partly because of: fairness to whom?).
This is how I go about setting a price for my services:
I ask myself two simple questions:
1. How many hours do I want to work per week?
2. How much money do I need to live a comfortable and dignified existence (or if you want to word it in another way: How much money do I need in order not to be resentful with my lot in life?)
Answer these two questions honestly, and your hourly price becomes a question of simple maths.
Here is an (hypothetical) example:
1. You want to work 20 hours per week.
2. You will live happily on US$ 5000 per month.
Then you need to charge US$ 60 per hour. (I rounded it).
Now let us suppose that the realistic market price in your area for piano lessons is US$ 20 per hour. If you asked $60, you will be out of the market and have no students. If so, you have three choices:
1. Reconsider how many hours you are willing to work per week. Would you prepared to work 60 hours per week instead of 20 ?
2. Reconsider how much money per month would make you happy. Would you be happy making $1600/ month?
3. Reconsider your professional choice: would you be happy to go into a profession with a better payment rate than piano lessons?
That is pretty much it. Fairness or unfairness does not enter the equation at all. By the way, the same reasoning applies to any paid activity.
Now for the matter of holidays, travelling and so on. This is what I do:
1. Payment is per month, in advance, all months of the year.
2. My policy explains very clearly that all months must be paid. I do not do lessons, I do courses. I expect commitment. If parents want to pay per lesson, that is fine, but then there are 3 consequences:
a. I cannot guarantee a time slot, so lessons must be arranged and a slot must be reserved one month in advance (payment for the lesson must be made at the time of reservation).
b. Rates for single lessons are 5 times the rate for a course lesson. This means that the student will be paying for one lesson more than what he pays per month.
c. A student coming for isolated lessons does not have any of the benefits of regular students (there are a lot of benefits). The student comes, has the lesson and goes home, and that is it.
3. Every month is paid because the total yearly cost is spread through 12 months. So it does not matter if the month has 4 or 5 weeks.
4. The student has the right to 53 lessons in a year (the year has 52 weeks). If the student misses one or several lessons for whatever reasons, the monthly payment still stands. However I will make up any missed lessons. Likewise, if I cannot give a lesson for whatever reason, the payment still is due, and I will make up the lesson.
5. I take one month holiday during the year. I make this holiday coincide with the student’s holidays. Payment is still due during that month. During the year every student has a compulsory one week intensive course with five daily one hour lessons. This intensive week makes up for the four lessons missed during holidays. This intensive course can be taken at any convenient week during the year. Since there is one extra lesson, this explains why there are 53 one hour lessons during the year, rather than 52.
6. There are no cancellations. If a lesson is cancelled it must be made up.
7. The whole point is simple: payment is due every month no matter what happens.
This is roughly how things are at the moment. I reserve myself the right to change my policy (and have done so in the past). At the moment I am very satisfied how things are working out. Perhaps the most important result of this policy was that it selected the students. There are always people prepared to take you for a ride if you will let them. You are always better of without that sort of student. They are the ones who squirm when they see this sort of policy. The nice, honest, appreciative students, never bat an eyelid. They see straightaway that there is nothing unreasonable about such a policy. These are the students you want.
I hope this helps.
Best wishes,
Bernhard.