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It just came across my mind that if I am broadcasting a message in my network, is it my computer, router or another node that decides to replicate the message?

Also, how does that node how many times to replicate the message?

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The PC only sends a single frame with a broadcast MAC address and Broadcast IP address. Routers don't forward broadcasts, so there is no replication at L3. Switches do replicate the frame. They forward it out of all interfaces apart from the one it came in on. Spanning Tree ensures there are no loops in the L2 topology to ensure the packet does not loop indefinitely.

There is a feature at L3 called directed broadcast, but the packet still isn't replicated at L3. The destination IP is the broadcast address of a distant subnet. The packet gets routed to it's destination LAN then exits the router and gets broadcast on that network, with replication at L2 if necessary. This feature is usually disabled for security reasons.

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It doesn't work that way. Broadcast packets are sent to the broadcast address, which every host on the network should listen to. So there's only one packet, it does not get replicated.

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