For one-way authentication (where Bob is the server and Alice the client), I would rather do something like this:
Alice —> Open Connection —> Bob
Alice <— $R$ <— Bob
Alice —> $K_{A-B} (R)$ —> Bob
By using the once-in-a-lifetime value $R$, and then checking the returned value, $K_{A-B} (R)$, Bob can be sure that both Alice is who she says she is (since she knows the secret key value needed to encrypt $R$) and is "live" (since she has encrypted the nonce, $R$, that Bob just created).