When the arrival processes are Poisson, queueing networks are well-understood in terms of the product-form structure of the number of jobs $N_i$ at the individual queues; much less is known about the waiting time $W$ across the whole network. In turn, for non-Poisson arrivals, little is known about either $N_i$'s or $W$. This paper considers a tandem network $$GI/G/1\rightarrow \cdot/G/1\rightarrow\dots\rightarrow\cdot/G/1$$ with general arrivals and light-tailed service times. The main result is that the tail $\P(W>x)$ has a polynomial-exponential (Poly-Exp) structure by constructing upper bounds of the form $$(a_{I}x^{I}+\dots+a_1x+a_0)e^{-\theta x}~.$$ The degree $I$ of the polynomial depends on the number of bottleneck queues, their positions in the tandem, and also on the `light-tailedness' of the service times. The bounds hold in non-asymptotic regimes (i.e., for \textit{finite} $x$), are shown to be sharp, and improve upon alternative results based on large deviations by (many) orders of magnitude. The overall technique is also particularly robust as it immediately extends, for instance, to non-renewal arrivals.

翻译：当到达流程为泊松流时，排队网络可以很好地理解为各个队列中作业数$N_i$的乘积形式；但是对于整个网络的等待时间$W$了解甚少。而对于非泊松的到达流程，对于$N_i$和$W$的知识更少。本文研究了具有一般到达流程和服务时间轻尾特性的串联网络$$GI/G/1\rightarrow \cdot/G/1\rightarrow\dots\rightarrow\cdot/G/1$$ 的情况。最主要的结果是，尾部$\P(W>x)$具有多项式指数(Poly-Exp)结构，通过构造上界$$(a_{I}x^{I}+\dots+a_1x+a_0)e^{-\theta x}~.$$ 多项式的次数$I$取决于瓶颈队列的数量，其在串联中的位置，以及服务时间的“轻尾性”。这些上界在非渐进情况(即，对于“有限” $x$)成立，被证明是准确的，比基于大偏差的替代结果提高了(很多)数量级。此外，整体技术也特别强大，因为它立即扩展到非可重复到达情况。