I'm not sure about the street grimoire variations, with which you can change the karma costs. Do they grow the time to get the initiation? Maybe not, but without them the time to get the initiation is given in rules: it is Arcana + intuition [Astral] (initiate grade, 1month) Extended test.
Basic mage have typically intuition 5, and arcana 3. This means that the average times to initiate are 30 days per initiations to get the first 2 initiations.
The third grade is slow. But if you want to do it faster, it is possible to use increase intuition spell and buy 1 arcana skill point more. This means that your pool is 13, on average you can get third and fourth grade again with 1 month.
On average you get with 13 and 12 dices 6 successes => 7th grade will take more than 2 month with probability higher than 50%.