Anyways, to really understand what happens when R->0 I think we need to make a new model that takes into account what we just learned from your chart above (that miners won't necessarily be hashing all the time).
That seems to be an interesting point illustrating how the best interest of users and miners incentives could diverge.
For users, an empty block is always better than no block because it adds work to the chain and increases the security of the previous transactions.
Miners being financially motivated to shutdown for a period of time may not be an issue though.
Let's say that coinbase rewards are zero and miners live on fees only, and given a difficulty level it does not make sense to spend electricity until X number of fees/transactions are published.
In such a situation the difficulty will adjust/decrease until 10 minute blocks are restored. This might mean that after a block is found miners turn off for 5 minutes and only turn on after 5 minutes of fees are sent, but the difficulty will have adjusted so that miners are likely to find the next block 5 minutes after turning on. Yes this would also mean that if all miners keep running we would have 5 minute blocks, but they wouldn't be. And if they did then difficulty would adjust back up.
Since these issues develop slowly, I believe we would see that difficulty will continue to adjust to maintain 10 min blocks regardless of the financial incentives of the time.
Yeah, I was thinking about this too. Interesting to imagine a global network of bitcoin miners switching their hashing farms off and on as transactions build up.
Although we could expect the 10 minute block period to stay the same, it seems that the
variance in the time between blocks should go down.
As you say, for the first few minutes after a block is published, very few miners would hash until a certain threshold of transaction fees could be reaped, so very few blocks would be found soon after the previous one. At the other end of the curve, the further beyond 10 minutes that a block is not found, we could expect miners to throw every last hash at the block, willing to burn lots of power to get the chance to collect the richer reward from blocks with more transactions than average. So the chances that a block takes much longer than 10 minutes would also decrease as miners frantically burn energy in the hopes of earning richer blocks full of fees.
