A Appendix
1.1 A.1 Diamond Network
By using the conjecture, we have obtained so far that
$$\begin{aligned}&\text {P}\! \left( d(t)>T\right) \\&\quad \le \sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] e^{-\theta c_{1}(t+T-\tau _{1})}\text {E}\! \left[ e^{\theta \left( D_{2}^{(2)}+D_{3}^{(3)}\right) (\tau _{1},t+T)}\right] \\&\quad \le \sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] e^{-\theta c_{1}(t+T-\tau _{1})}\\&\qquad \times \text {E}\! \left[ e^{\theta \left( \left( A_{2}\oslash \left[ S_{4}-A_{3}\right] ^+\right) \oslash S_{2}\right) (\tau _{1},t+T)}\right] \text {E}\! \left[ e^{\theta \left( \left( A_{3}\oslash S_{4}\right) \oslash S_{3}\right) (\tau _{1},t+T)}\right] . \end{aligned}$$
This leads to
$$\begin{aligned}&\text {P}\! \left( d(t)>T\right) \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] e^{-\theta c_{1}(t+T-\tau _{1})}\left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta \left( A_{2}\oslash \left[ S_{4}-A_{3}\right] ^+\right) (\tau _{2},t+T)}\right] \text {E}\! \left[ e^{-\theta S_{2}(\tau _{2},\tau _{1})}\right] \right\} \\&\cdot \left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta \left( A_{3}\oslash S_{4}\right) (\tau _{2},t+T)}\right] \text {E}\! \left[ e^{-\theta S_{3}(\tau _{2},\tau _{1})}\right] \right\} \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] e^{-\theta c_{1}(t+T-\tau _{1})}\\&\cdot \left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\left\{ \sum _{\tau _{3}=0}^{\tau _{2}}\text {E}\! \left[ e^{\theta A_{2}(\tau _{3},t+T)}\right] \text {E}\! \left[ e^{\theta A_{3}(\tau _{3},\tau _{2})}\right] \text {E}\! \left[ e^{-\theta S_{4}(\tau _{3},\tau _{2})}\right] \right\} \text {E}\! \left[ e^{-\theta S_{2}(\tau _{2},\tau _{1})}\right] \right\} \\&\cdot \left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\left\{ \sum _{\tau _{3}=0}^{\tau _{2}}\text {E}\! \left[ e^{\theta A_{3}(\tau _{3,}t+T)}\right] \text {E}\! \left[ e^{-\theta S_{4}(\tau _{3,},\tau _{2})}\right] \right\} \text {E}\! \left[ e^{-\theta S_{3}(\tau _{2},\tau _{1})}\right] \right\} , \end{aligned}$$
after applying the Union bound for each usage of the deconvolution. Further assuming all \(A_i \) to be \((\sigma _A, \rho _A)\)-bounded yields a closed-form for the delay bound under the stability condition
$$\begin{aligned} \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )<&c_{1},\\ \rho _{A_{2}}(\theta )<&c_{2},\\ \rho _{A_{3}}(\theta )<&c_{3},\\ \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )<&c_{4}: \end{aligned}$$
$$\begin{aligned}&\text {P}\! \left( d(t) > T\right) \\ \overset{\left( \text {Definition}~5\right) }{\le }&\sum _{\tau _{1}=0}^{t}e^{\theta \left( \rho _{A_{1}}(\theta )(t-\tau _{1})+\sigma _{1}(\theta )\right) } e^{-\theta c_{1}(t+T-\tau _{1})}\\&\cdot \Bigg \{ \sum _{\tau _{2}=0}^{\tau _{1}}\Bigg \{ \sum _{\tau _{3}=0}^{\tau _{2}}e^{\theta \left( \rho _{A_{2}}(\theta )(t+T-\tau _{3})+\sigma _{A_{2}}(\theta )\right) }e^{\theta \left( \rho _{A_{3}}(\theta )(\tau _{2}-\tau _{3})+\sigma _{A_{3}}(\theta )\right) }e^{-\theta c_{4}(\tau _{2}-\tau _{3})}\Bigg \}\\&\quad e^{-\theta c_{2}(\tau _{1}-\tau _{2})}\Bigg \} \\&\cdot \left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\left\{ \sum _{\tau _{3}=0}^{\tau _{2}}e^{\theta \left( \rho _{A_{3}}(\theta )(t+T-\tau _{3})+\sigma _{A_{3}}(\theta )\right) }e^{-\theta c_{4}(\tau _{2}-\tau _{3})}\right\} e^{-\theta c_{3}(\tau _{1}-\tau _{2})}\right\} \\ \le&e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{1}\right) T+\sigma _{1}(\theta )+\sigma _{A_{2}}(\theta )+2\sigma _{A_{3}}(\theta )\right) }\\&\cdot \sum _{\tau _{1}=0}^{t}e^{\theta \left( \rho _{A_{1}}(\theta )-c_{1}\right) (t-\tau _{1})}\left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\frac{e^{\theta \rho _{A_{2}}(\theta )(t-\tau _{2})}}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{4}\right) }}e^{-\theta c_{2}(\tau _{1}-\tau _{2})}\right\} \\&\cdot \left\{ \sum _{\tau _{2}=0}^{\tau _{1}}\frac{e^{\theta \rho _{A_{3}}(\theta )(t-\tau _{2})}}{1-e^{\theta \left( \rho _{A_{3}}(\theta )-c_{4}\right) }}e^{-\theta c_{3}(\tau _{1}-\tau _{2})}\right\} \\ \le&e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{1}\right) T+\sigma _{1}(\theta )+\sigma _{A_{2}}(\theta )+2\sigma _{A_{3}}(\theta )\right) } \\&\cdot \sum _{\tau _{1}=0}^{t}\frac{e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{1}\right) (t-\tau _{1})}}{1-e^{\theta \left( \rho _{A_{2}}(\theta )-c_{2}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{3}}(\theta )-c_{3}\right) }}\\&\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{4}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{3}}(\theta )-c_{4}\right) }} \\ \le&\frac{e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{1}\right) T+\sigma _{1}(\theta )+\sigma _{A_{2}}(\theta )+2\sigma _{A_{3}}(\theta )\right) }}{1-e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{1}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )-c_{2}\right) }}\\&\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{3}}(\theta )-c_{3}\right) }}\\&\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{4}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{3}}(\theta )-c_{4}\right) }}, \end{aligned}$$
where we used the convergence of the geometric series.
1.2 A.2 The \(\mathbb {L}\)
We have that
$$\begin{aligned}&\text {P}\! \left( d(t)>T\right) \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\text {E}\! \left[ e^{\theta D_{3}^{(3)}(\tau _{1},t+T)}e^{\theta D_{2}^{(3)}(\tau _{1},\tau _{2})}\right] \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})} \text {E}\! \left[ e^{\theta D_{3}^{(3)}(\tau _{1},t+T)}e^{\theta D_{2}^{(3)}(\tau _{1},t+T)}\right] . \end{aligned}$$
With the conjecture, we compute
$$\begin{aligned}&\text {P}\! \left( d(t)>T\right) \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\text {E}\! \left[ e^{\theta D_{3}^{(3)}(\tau _{1},t+T)}\right] \text {E}\! \left[ e^{\theta D_{2}^{(3)}(\tau _{1},t+T)}\right] \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\text {E}\! \left[ e^{\theta \left( A_{2}\oslash S_{3}\right) (\tau _{1},t+T)}\right] \\&\quad \text {E}\! \left[ e^{\theta \left( A_{3}\oslash \left[ S_{3}-A_{2}\right] ^+\right) (\tau _{1},t+T)}\right] \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\left\{ \sum _{\tau _{3}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta A_{2}(\tau _{3},t+T)}\right] e^{-\theta c_{3}(\tau _{1}-\tau _{3})}\right\} \\&\cdot \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta A_{3}(\tau _{3},t+T)}\right] \text {E}\! \left[ e^{-\theta \left[ S_{3}-A_{2}\right] ^+(\tau _{3},\tau _{1})}\right] \right\} \\ \le&\sum _{\tau _{1}=0}^{t}\text {E}\! \left[ e^{\theta A_{1}(\tau _{1},t)}\right] \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\left\{ \sum _{\tau _{3}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta A_{2}(\tau _{3},t+T)}\right] e^{-\theta c_{3}(\tau _{1}-\tau _{3})}\right\} \\&\cdot \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}\text {E}\! \left[ e^{\theta A_{3}(\tau _{3},t+T)}\right] \text {E}\! \left[ e^{\theta A_{2}(\tau _{3},\tau _{1})}\right] e^{-\theta c_{3}(\tau _{1}-\tau _{3})}\right\} . \end{aligned}$$
If we again assume all \(A_i \) to be \((\sigma _A, \rho _A) \)-bounded, we obtain for
$$\begin{aligned} \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )<&\min \{c_1, c_2\},\\ \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )<&c_{3}, \end{aligned}$$
and \(c_1 \ne c_2\):
$$\begin{aligned}&\text {P}\! \left( d(t)>T\right) \\ \overset{\left( \text {Definition}~5\right) }{\le }&\sum _{\tau _{1}=0}^{t}e^{\theta \left( \rho _{A_{1}}(\theta )(t-\tau _{1})+\sigma _{A_{1}}(\theta )\right) }\sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\\&\cdot \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}e^{\theta \left( \rho _{A_{2}}(\theta )(t+T-\tau _{3})+\sigma _{A_{2}}(\theta )\right) }e^{-\theta c_{3}(\tau _{1}-\tau _{3})}\right\} \\&\cdot \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}e^{\theta \left( \rho _{A_{2}}(\theta )(\tau _{1}-\tau _{3})+\sigma _{A_{2}}(\theta )\right) }e^{\theta \left( \rho _{A_{3}}(\theta )(t+T-\tau _{3})+\sigma _{A_{3}}(\theta )\right) }e^{-\theta c_{3}(\tau _{1}-\tau _{3})}\right\} \\ \le&e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) \cdot T+\sigma _{A_{1}}(\theta )+2\sigma _{A_{2}}(\theta )+\sigma _{A_{3}}(\theta )\right) }\\&\cdot \sum _{\tau _{1}=0}^{t}e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) (t-\tau _{1})}\sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\\&\cdot \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}e^{\theta \left( \rho _{A_{2}}(\theta )-c_{3}\right) (\tau _{1}-\tau _{3})}\right\} \left\{ \sum _{\tau _{3}=0}^{\tau _{1}}e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{3}\right) (\tau _{1}-\tau _{3})}\right\} \\ \le&e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) \cdot T+\sigma _{A_{1}}(\theta )+2\sigma _{A_{2}}(\theta )+\sigma _{A_{3}}(\theta )\right) }\\&\cdot \sum _{\tau _{1}=0}^{t}\frac{e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) (t-\tau _{1})}}{1-e^{\theta \left( \rho _{A_{2}}(\theta )-c_{3}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{3}\right) }}\\&\cdot \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\\ \le&e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) \cdot T+\sigma _{A_{1}}(\theta )+2\sigma _{A_{2}}(\theta )+\sigma _{A_{3}}(\theta )\right) }\\&\cdot \sum _{\tau _{1}=0}^{t}\frac{e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )\right) (t-\tau _{1})}}{1-e^{\theta \left( \rho _{A_{2}}(\theta )-c_{3}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{3}\right) }}\\&\cdot \sum _{\tau _{2}=\tau _{1}}^{t+T}e^{-\theta c_{1}\cdot (\tau _{2}-\tau _{1})}e^{-\theta c_{2}\cdot (t+T-\tau _{2})}\\ \le&\frac{e^{\theta \left( \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-\min \{c_1, c_2\}\right) \cdot T+\sigma _{A_{1}}(\theta )+2\sigma _{A_{2}}(\theta )+\sigma _{A_{3}}(\theta )\right) }}{1-e^{\theta \left( \rho _{A_{1}}(\theta )+\rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-\min \{c_1, c_2\}\right) }}\\&\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )-c_{3}\right) }}\cdot \frac{1}{1-e^{\theta \left( \rho _{A_{2}}(\theta )+\rho _{A_{3}}(\theta )-c_{3}\right) }}\cdot \frac{1}{1-e^{-\theta |c_{1}-c_{2}|}}, \end{aligned}$$
where we used again the convergence of the geometric series.
