I'm thinking the same as Emma (Archie bean) that the reason might be an in season bitch, and at 8 months that could be happening to Millie.
11 year old William was castrated at 15 months because of an undescended testicle. He's never marked and doesn't cock his leg often, but he always aware if any bitches that he knows (like my friend's two Border Terriers) are coming into season. He's not interested in doing anything more than a delicate sniff but he's certainly aware. So perhaps that's what's happening with Murphy?
Strangely enough, it's an aspect of life that seems to have bypassed 6 year old entire Louis completely

. He cocks his leg endlessly on walks but has never shown any interest whatsoever in the ladies.