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：当到达过程为泊松过程时，排队网络在单个队列中作业数 $N_i$ 的产品形式结构方面已得到充分理解；而关于整个网络的等待时间 $W$ 则知之甚少。对于非泊松到达，无论 $N_i$ 还是 $W$ 的相关结果均十分有限。本文考虑串联网络 $$GI/G/1\rightarrow \cdot/G/1\rightarrow\dots\rightarrow\cdot/G/1$$，其中到达过程具有一般性，服务时间具有轻尾特性。主要结论是，通过构建形如 $$(a_{I}x^{I}+\dots+a_1x+a_0)e^{-\theta x}$$ 的上界，尾部概率 $\P(W>x)$ 具有多项式指数（Poly-Exp）结构。多项式的次数 $I$ 取决于瓶颈队列的数量、其在串联网络中的位置以及服务时间的“轻尾程度”。该界限在非渐近框架（即针对有限 $x$）中成立，被证明是紧的，且相比基于大偏差理论的替代结果在数量级上实现了多个量级的改进。整体方法具有高度鲁棒性，可立即推广至非更新到达等其他场景。