Transferring large quantities of data from S3 bucket with direct connect

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We need to regularly transfer large quantities of data from/to an AWS S3 bucket to/from our datacenter. We thought of using a direct connect (pricing for AWS Direct Connect) and I have a question: normally to transfer e.g. 450 GB from S3 to our datacenter we get billed at 0.09$ per GB in the Europe (Frankfurt) region for S3 (pricing for AWS S3).

If I use a dedicated Direct Connect with 10Gigabit capacity, which is billed at 2.25$ per hour, will I also pay the AWS S3 0.09$ per GB? Or will I just pay the AWS direct connect hourly fee (whatever percentage of those 10G bandwidth I use)?

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Yes. You will pay egress data transfer costs (egress = from S3 to Direct Connect location). The documentation you linked to also says this explicitly under "AWS Direct Connect data transfer".

Data Transfer OUT pricing is dependent on the source AWS Region and AWS Direct Connect location. [...] If you are using Direct Connect gateway, you will pay applicable egress data charges based on the source AWS Region and AWS Direct Connect location.

As rightly mentioned by @Maurice in the comments, with S3 -> Direct Connect you will pay only a fraction of the data transfer costs compared to the S3 -> Internet transfer costs mentioned under the "Data transfer" in section "Data Transfer OUT From Amazon S3 To Internet" of the S3 Pricing site.

There, you will be charged depending on the total monthly amount of data transferred out from S3 to the Internet. So, if you have e.g. < 10 TB per month (which your case of 450 GB falls into), you will pay $0.09 per GB, which is more than $0.02 per GB with Direct Connect.

Given that with Direct Connect you also have hourly costs that don't depend on the actual amount of data you transfer, there is obviously some break-even point in pricing.

So, if you actually did the math and solve the equation:

2.25 * 730 + 0.02 * x = 0.09 * x

for x (amount of GBs), with an assumed maximum transfer of 10TB/month (your 450 GB case) out of S3 (costs for the non-Direct-Connect case) then the break-even point is at ~23,464GB (',' is the thousands-separator here) per month.

So, if you are planning to transfer only up to 10TB (your 450 GB case) per month, Direct Connect will be more costly than S3 alone, but will of course give you higher throughput, if you didn't already achieve 10 GB through the internet to S3.