Mobile-edge computing casts the computation-intensive and delay-sensitive applications of mobile devices onto network edges. Task offloading incurs extra communication latency and energy cost, and extensive efforts have focused on offloading schemes. Many metrics of the system utility are defined to achieve satisfactory quality of experience. However, most existing works overlook the balance between throughput and fairness. This study investigates the problem of finding an optimal offloading scheme in which the objective of optimization aims to maximize the system utility for leveraging between throughput and fairness. Based on Karush-Kuhn-Tucker condition, the expectation of time complexity is analyzed to derive the optimal scheme. A gradient-based approach for utility-aware task offloading is given. Furthermore, we provide an increment-based greedy approximation algorithm with $\mathrm{1}\mathrm{+}{\displaystyle \frac{\mathrm{1}}{\mathrm{e}\mathrm{-}\mathrm{1}}}$ ratio. Experimental results show that the proposed algorithms can achieve effective performance in utility and accuracy.