In Hierarchical Web Caching Systems: Modeling, Design and Experimental Results by Hao Che, Ye Tung and Zhijun Wang, the authors used
$$ g(t) = \frac{\exp(- (t - \tau)/(T - \tau))}{T - \tau} $$
to approximate
$$ f(t) = \sum\limits_{n = 1}^\infty ( - 1)^{n + 1}\frac{(t - n\tau )^{n - 1}}{T^n(n - 1)!}u(t - n\tau ), $$ where $u$ is the Heaviside function.
I did some numerical tests. Results show when $T/\tau$ is large enough, the value of these two functions is almost the same.
Is there any mathematical explanation?