This paper firstly proposes a greedy online packet scheduling algorithm for the problem raised by Mao, Koksal and Shroff that allows arbitrary hard deadlines in multi-hop networks aiming at maximizing the total revenue. With the same assumption of $\rho _{M} / \rho _{m}={O}(1)$ where $\rho _{M}$ and $\rho _{\textit {}m}$ are the maximum and minimum revenue a packet may carry, our algorithm is ${O}$ ( ${P} _{M}$ )-competitive improving on MKS algorithm by a factor of ${O}$ (log ${P} _{M}$ ), where ${P} _{M}$ is the length of the longest path a packet may travel in the network. We prove that it is asymptotically optimal by presenting a lower bound of ${P} _{M}$ on the competitiveness for this problem. Secondly, this paper studies the extension of this problem that includes routing as a part of the solution. We prove that using the fastest path algorithm for the routing part, the greedy online algorithm also achieves asymptotically optimal competitiveness for the extended problem. Furthermore, we present a non-greedy online algorithm that not only is asymptotically optimal, but also can adaptively achieve a better competitiveness when the network has a larger ${C} _{\textit {min}}$ , where ${C} _{\textit {min}}$ is the minimum link capacity in the network. Finally, simulation results are reported, showing that not only do the greedy online algorithms achieve asymptotically optimal bounds, but also practically achieve better performance than the previously proposed algorithms.