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When using simplewallet and bitmonerod transaction fees are calculated based on the mixin and outputs being used in a transaction.

During times where network usage is high, is there an ability to choose a larger transaction fee than the fee automatically calculated for me?

If the answer is yes, does choosing a higher fee provide me with a highly level on transaction confirmation priority from Monero miners?

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You can select a higher fee if you want. This choice is intentionally limited to a 1x, 2x, and 3x multiplier to avoid leaking too much information via the fee (ie, if you're the only one using a 2.7x fee, your transactions will stick out).

NOTE: from two weeks after the v3 fork, the fees multipliers will change from 1/2/3 to 1/20/166. The simplewallet setting blow is renamed from fee-multiplier to priority, so 1/2/3 are still used in simplewallet, but do not express actual multipliers anymore. The base fee will also decrease by 5 times at the same time.

This behavior is a global setting in simplewallet, and can be changed with: set fee-multiplier 1 to reset to default, same with 2 and 3.

Miners will select transactions based on decreasing fee per byte, so transactions with a higher fee multiplier will get priority over transactions with lower fee multipliers.

As J Dog says, a glut of transactions in the pool cannot be expected to last long term, as the block size will increase to meet demand.

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Fees are rounded to 0.01 XMR per KB. Because of the way outputs are selected, calculating the size of a transaction in advance (and choosing your own fee) would be very difficult and time consuming.

The transaction fee priority issue is not really important to Monero because its dynamic blocksizes will grow to meet demand. The need to pay a higher fee than someone else in order to gain priority with miners is more important when there is a fixed block size (such as in Bitcoin).

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