## 4 End-to-End Probabilistic Bounds in Multi-Hop Networks  
An essential aspect of a network performance analysis is scaling of end-to-end delay and PDV bounds across a series of \(n\) servers. We considered the scenario shown in Figure 3, where performance bounds are derived for the through flows that traverse all \(n\) servers in series. Cross traffic joins and leaves the server at each stage.  
For \((\sigma(\theta),\rho(\theta))\) through traffic subjected to a series of constant-rate servers, the convolution of the MGFs of individual stages was derived in [19], and the results can be extended for LR servers.  
For the single-node \((n=1)\) case:  
\[\begin{split} M_{A\O\!S}(\theta,-d)&\leq\sum_{ \tau=d}^{\infty}e^{\theta(\sigma(\theta)+\rho(\theta)(\tau-d))}e^{-\theta(C- \rho_{c}(\theta))(\tau-T^{\prime}-T)}\\ &\leq\sum_{\tau=d}^{\infty}e^{\theta(\sigma(\theta)+\rho(\theta) (\tau-d))}e^{-\theta(C-\rho_{c}(\theta))(\tau-T^{\prime})-\sigma_{c}(\theta)} \\ &=e^{\theta(\sigma(\theta)+\sigma_{c}(\theta)-\rho(\theta)d+(C- \rho_{c}(\theta))T^{\prime}}\,\sum_{\tau=d}^{\infty}[e^{-\theta(C-\rho(\theta )-\rho_{c}(\theta))}]^{\tau}\\ &=e^{\theta(\sigma(\theta)+\sigma_{c}(\theta)-\rho(\theta)d+(C- \rho_{c}(\theta))T^{\prime}}\,\frac{e^{-\theta(C-\rho(\theta)-\rho_{c}(\theta) )d}}{1-e^{-\theta(C-\rho(\theta)-\rho_{c}(\theta))}}\end{split} \tag{17}\]  
where \(\theta>0\) and \(C>\rho(\theta)-\rho_{c}(\theta)\) for stability. This means that the average arrival rate of traffic should be less than the service rate of the server. Thus, the stability condition maintains the server utilization below a certain level.  
For the multi-node \((n)\) LR server case:  
The equivalent latency for \(n\) servers \(T^{\prime}_{n}\) can be defined as  
\[T^{\prime}_{n}=\sum_{i=1}^{n}T^{\prime}_{i}. \tag{18}\]  
Assuming the same latency \(T^{\prime}\) for all servers,  
\[T^{\prime}_{n}=nT^{\prime}. \tag{19}\]  
The equivalent rate for \(n\) servers is given by  
\[\rho_{c^{\prime}_{n}}(\theta)=min(\rho_{c_{1}(\theta)},\rho_{c_{2}(\theta)}, \cdots,\rho_{c_{n}(\theta)}). \tag{20}\]  
The second-order moment can be obtained as  
\[E[\hat{W}^{2}(t)]=\sum_{d=0}^{\infty}d^{2}[M_{AOS}(\theta,-\{d+1\})-M_{AOS}(\theta,-\{d\})]. \tag{35}\]  
The moment computation can be used to further determine the bounds for PDV.  
**Theorem 1** (End-to-end delay and PDV).: _Consider \(n\) work-conserving, latency rate servers in series, each with capacity \(C\), latency \(T^{\prime}\), and independent \((\sigma_{c}(\theta),\rho_{c}(\theta))\) cross traffic under the general scheduling model. Consider the \((\sigma(\theta),\rho(\theta))\) aggregate of through flows that traverses all \(n\) servers in series. If \(d\) and \(v\) are the delay and PDV bounds with violation probability \(\epsilon\in(0,1]\) and for any \(\theta\in(0,\infty)\) such that \(C>\rho(\theta)+\rho_{c}(\theta)\) for stability, then_  
\[d\geq\frac{\theta(\sigma(\theta)+n\sigma_{c}(\theta)+nCT^{ \prime}-n\rho_{c_{n}^{\prime}}(\theta)T^{\prime})-\ln(1-e^{-\theta(C-\rho( \theta)-\rho_{c_{n}^{\prime}}(\theta))})-\ln(\epsilon)}{C\theta-\rho_{c_{n}^{ \prime}}(\theta)}\] \[v\leq\frac{e^{\theta(C-\rho(\theta)-\sigma(\theta)+n\sigma_{c}( \theta)+(C-\rho_{c_{n}^{\prime}}(\theta))T^{\prime})}(e^{C\theta}+e^{\rho_{c _{n}^{\prime}}(\theta)\theta})}{(1-e^{-\theta(C-\rho(\theta)-\rho_{c_{n}^{ \prime}}(\theta))})(e^{C\theta}+e^{\rho_{c_{n}^{\prime}}(\theta)\theta})^{2}}\]  
The proof for this theorem is covered in Appendix A.