N will always be more than 1. It is impossible to accurately judge exactly how much more than 1, but here are some factors that will contribute to increasing the time between a Post request and an HTTP response:
I'm going to do this from the perspective of a single Post and a single HTTP response compared to a single Echo Request and a single Echo Reply. The time increase between 1 ICMP Exchange and HTTP Exchange verses 3 ICMP/HTTP Exchanges would be linear. Each exchange would be affected by each of factor belows.
1. There are more packets sent/received in the HTTP Post/Response
A ping 'sequence' is this:
01 Client --> Server ICMP Request
02 Client <-- Server ICMP Response.
Where as a HTTP Post/Response sequence would look like this:
01 Client --> Server HTTP Post
02 Client <-- Server TCP ACK
03 Client <-- Server HTTP Response
The Server doesn't need wait for the ACK (packet #2) to get to the client before sending the HTTP Response (packet #3). Both of those packets are sent directly one after the other. But either way, it is still a second packet that must be sent, so an increase to your value
2. Additional Crypto processing causes additional overhead
Even if the TCP Ack (packet #2 above) didn't need to be sent, the simple fact that the packets are secured with HTTPS is going to require both parties to validate the packet and decrypt the content before it gets passed to the HTTP interpreter.
Which is something the ICMP packet would not require. Hence, additional time added to
N for the HTTP Request/Response.
3. HTTP packets might call for additional inspection by transient devices.
If this communication happens through a Firewall, it is likely the Firewall (or some sort of WAF/IPS) is performing additional processing/inspection/validation on the HTTP packets. In particular, since this a Post request.
This is less likely if it is HTTPS. But many transient firewalls can now decrypt HTTPS, inspect the underlying HTTP, and then re-encrypt the HTTPS before sending it on its way. Again, less likely, but not something we can dutifully rule out entirely.
Such inspection would not be necessary for simple ICMP packets. And not just because the packets themselves would more than likely be smaller. There is just less malicious intent that can exist with the far simpler ICMP protocol, as such more 'attention' is put in to deeper inspection of the HTTP/HTTPS.
4. An HTTP Post requires more processing by the server
Responding to an ICMP Response is (relatively) easy. At best, there is a little bit of data in the ICMP echo Payload to copy to the ICMP Response. Depending the content of the HTTP Post, there is more to consider.
Is there data entry validation? Does the content that is posted need to be processed and somehow included in the HTTP Response? Does it have to be processed 'as is', or does it require some analyzing before a useful HTTP Response can be produced?
There are literally endless possibilities for what sort of analyzing and/or data is presented in the Post. And this additional processing would also account for an increase to your
I'm sure there are others, but those are the ones I could come up with thus far.
Edit: added #4