In the paper, we study the service process $S(t)$ of an independent and identically distributed (\textit{i.i.d.}) Nakagami-$m$ fading channel, which is defined as the amount of service provided, i.e., the integral of the instantaneous channel capacity over time $t$. By using the Characteristic Function (CF) approach and the infinitely divisible law, it is proved that, other than certain generally recognized curve form {or a stochastic process}, the channel service process $S(t)$ is a deterministic linear function of time $t$, namely, $S(t)=c_m^\ast\cdot t$ where $c_m^\ast$ is a constant determined by the fading parameter $m$. Furthermore, we extend it to general \textit{i.i.d.} fading channels and present an explicit form of the constant service rate $c_p^\ast$. The obtained work provides such a new insight on the system design of joint source/channel coding that there exists a coding scheme such that a receiver can decode with zero error probability and zero high layer queuing delay, if the transmitter maintains a constant data rate no more than $c_p^\ast$. Finally, we verify our analysis through Monte Carlo simulations.