It's more than that... when you are living on a tight budget you really want a line graph that tells you what your checking account is going to look like in a week, two weeks, three weeks. Your problem is not necessarily income but cash flow, and you do all of this cash-flow reasoning informally, “I'd better postpone any doctor’s visit until the 15th because usually things are less tense in the middle of the month and I’d like to keep my copays out of it,” but it would be really handy to say “look, if you need to you can spend up to $200 and things will get tight in 9 days time but you will not go broke, assuming you have put everything in properly.”

This sort of diagram is only easily possible if you can specify “here are the deposits that I think are going to happen, and here are the expenses that I think are going to happen.”

And for that, those envelopes are crucial, especially the ability to graph the amounts in the various envelopes and the amount that’s not in any envelope.

That is a matter of fixed recurring bills. I know when I have to pay rent, insurance, loan payments, to a degree utilities ect. Once you have plotted those in the system, you know how much to set aside for that every month and those are funds you cannot touch without a very very good reason. And should you do so anyway because you're terribly ill, you make a plan for how to make up the deficit before the payment happens.

For my personal economy it come out such that the "account" for all of those recurring payments are always significantly in the positive - in other words, I can "borrow" from that "account" without it interfering with my ability to pay the bills. E.g. you have two annual bills of $600, one has a payment date of Jan 1st and the other have Jul 1st. I need to set aside $100 monthly to pay them, but at any payment date there will be $900 in the account, so it's always at least $300 in the positive. It's just a peculiarity that can happen.

So that leaves all the variable expenses, food, clothing, going out, ect, things you can control the magnitude of. Those are well suited for the envelope system. I would personally treat necessaries such as food and transportation as a special envelope in the sense that it's only variable down to a minimum but no further. Clothing on the other hand can be turned down for $0 for quite a while, and "going out" certainly can be $0 for as long as it need to.

How does all that fit in a cash flow analysis then? Well for bills and fixed income it's easy - just make the graph. But for variable expenses it's a bit more hand wavy, because I cannot not buy food for my children, but I can certain not go to that sushi restaurant. And that is where the envelope system is helping me. If I have an unforeseen expense that is greater than what I have in my emergency fund, and there is not enough to borrow from the fixed expense account without compromising payments. Well I have to transfer some funds from the envelops that have a surplus that i can control, such as the "going out" envelope.

Yes, exactly – when I was using GnuCash I tried to create my own 'cash flow' tool using data exported from GnuCash because this was exactly what seemed to be missing in GnuCash. I miss double-entry bookkeeping but YNAB has turned out to be better for me overall.