So, this all assumes that all packets are the same length and all links are the same bandwidth.
I think a mistake you are making is that your formula looks at the end-to-end delay for a single packet, then multiplies it by the number of packets. In this case, the first packet would be sent, end-to-end, then the second packet would be send end-to-end etc.
In a real network, one the first link has finished processing the first packet, the second packet can be sent onto the first link, while the first packet is being sent across the second link etc.
So really, you need to look at the time taken to send all packets across the first link, one-by-one, then look at the time taken for the last packet to traverse the remaining links.
The formula for this would be:
(P + (N-1)) * (L/R)