% =========================
% Section 6: Numerical Solution & Identification
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\section{Numerical Solution and Parameter Identification}
\label{sec:numerics_id}

This section describes a reproducible computational implementation of the coupled
ODE--algebraic closure induced by the constant-power-load (CPL) constraint, and a
mechanism-driven parameter identification workflow. We employ an explicit fourth-order
Runge--Kutta integrator (RK4) with a nested algebraic evaluation of the discharge current
at each substage, adaptive step-halving for convergence control, and event detection for
the time-to-end (TTE). Parameter estimation is performed by targeted sub-experiments and
log-based regressions that preserve physical interpretability.

\subsection{RK4 with substage nested algebraic evaluation of $I$}
\label{subsec:rk4_nestedI}

The state vector is $\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top$ and the input vector is
$\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top$. For any $(\mathbf{x},\mathbf{u})$, we compute
the total power demand
\begin{equation}
P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L(t))+P_{\mathrm{cpu}}(C(t))+P_{\mathrm{net}}(N(t),\Psi(t),w(t)).
\label{eq:Ptot_def_sec6}
\end{equation}
The terminal voltage satisfies the ECM relation
\begin{equation}
V_{\mathrm{term}}(t)=V_{\mathrm{oc}}(z_{\mathrm{eff}}(t))-v_p(t)-I(t)\,R_0(T_b(t),S(t)),
\label{eq:Vterm_sec6}
\end{equation}
with the low-SOC protection $z_{\mathrm{eff}}(t)=\max\{z(t),z_{\min}\}$. Under the CPL assumption,
\begin{equation}
P_{\mathrm{tot}}(t)=V_{\mathrm{term}}(t)\,I(t)
=\big(V_{\mathrm{oc}}(z_{\mathrm{eff}})-v_p-I R_0\big)\,I,
\label{eq:CPL_sec6}
\end{equation}
which yields the discriminant
\begin{equation}
\Delta(t)=\big(V_{\mathrm{oc}}(z_{\mathrm{eff}})-v_p\big)^2-4R_0(T_b,S)\,P_{\mathrm{tot}}(t).
\label{eq:Delta_sec6}
\end{equation}
If $\Delta(t)\ge 0$, the physically consistent branch of the quadratic solution is
\begin{equation}
I_{\mathrm{CPL}}(t)=\frac{V_{\mathrm{oc}}(z_{\mathrm{eff}}(t))-v_p(t)-\sqrt{\Delta(t)}}{2R_0(T_b(t),S(t))}.
\label{eq:Icpl_sec6}
\end{equation}
To reflect device-side protection (PMIC current limiting / OS throttling), we apply a
temperature-dependent saturation
\begin{equation}
I(t)=\min\big(I_{\mathrm{CPL}}(t),\,I_{\max}(T_b(t))\big),
\qquad
I_{\max}(T_b)=I_{\max,0}\big[1-\rho_T\,(T_b-T_{\mathrm{ref}})\big]_+.
\label{eq:I_limit_sec6}
\end{equation}
When $\Delta(t)<0$, CPL delivery is infeasible. In such cases we record a ``collapse-risk''
event (see Section~\ref{subsec:event_detection}) and place the system in a strong
degradation regime by taking $I(t)=I_{\max}(T_b(t))$ (equivalently, one may cap
$P_{\mathrm{tot}}$), while the runtime termination (TTE) is still defined by voltage/SOC cutoffs.

Given the current mapping $I=I(\mathbf{x},\mathbf{u})$, the ODE right-hand side is evaluated using
the established dynamics:
\begin{align}
\dot z &= -\frac{I}{3600\,Q_{\mathrm{eff}}(T_b,S)}, \label{eq:zdot_sec6}\\
\dot v_p &= \frac{I}{C_1}-\frac{v_p}{R_1C_1}, \label{eq:vpdot_sec6}\\
\dot T_b &= \frac{1}{C_{\mathrm{th}}}\Big(I^2R_0(T_b,S)+\frac{v_p^2}{R_1}-hA\,(T_b-T_a)\Big), \label{eq:Tbdot_sec6}\\
\dot S &= -\lambda_{\mathrm{sei}}|I|^{m}\exp\!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right), \label{eq:Sdot_sec6}\\
\dot w &= \frac{\sigma(N)-w}{\tau(N)}, \quad \sigma(N)=\min(1,N), \quad
\tau(N)=\begin{cases}\tau_\uparrow,&\sigma(N)\ge w,\\ \tau_\downarrow,&\sigma(N)<w.\end{cases}
\label{eq:wdot_sec6}
\end{align}

We discretize in time with RK4:
\begin{equation}
\mathbf{x}_{n+1}=\mathbf{x}_n+\frac{\Delta t}{6}\left(\mathbf{k}_1+2\mathbf{k}_2+2\mathbf{k}_3+\mathbf{k}_4\right),
\label{eq:rk4_update_sec6}
\end{equation}
where each stage $\mathbf{k}_j=f(\mathbf{x},\mathbf{u},I(\mathbf{x},\mathbf{u}))$ is computed at
the corresponding intermediate state and time, and \emph{each} stage uses the nested algebraic
evaluation \eqref{eq:Delta_sec6}--\eqref{eq:I_limit_sec6}. Inputs $\mathbf{u}(t)$ at intermediate
times are obtained by interpolation (piecewise-constant or piecewise-linear) or by direct
evaluation if $\mathbf{u}(t)$ is generated procedurally.

\subsection{Projection, physical constraints, and robustness}
\label{subsec:projection_robustness}

To prevent numerical drift outside admissible ranges, we apply a mild projection after each
successful time step:
\begin{equation}
z\leftarrow \min(1,\max(0,z)),\quad
S\leftarrow \min(1,\max(0,S)),\quad
w\leftarrow \min(1,\max(0,w)).
\label{eq:projection_sec6}
\end{equation}
This projection is used only to suppress floating-point accumulation and does not change the
continuous model definition. Additionally, the low-SOC protection is enforced solely in the OCV
calculation via $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$ to avoid the $(1/z)$ singularity while preserving
the runtime termination criterion based on $z(t)$ and $V_{\mathrm{term}}(t)$.

Robustness safeguards include: (i) enforcing $Q_{\mathrm{eff}}(T_b,S)\ge 0$ via the $[\cdot]_+$ operator;
(ii) recording infeasibility when $\Delta<0$; and (iii) saturating current by
\eqref{eq:I_limit_sec6}, which prevents unrealistically large currents under low-voltage conditions.

\subsection{Step-size selection, stability, and convergence control}
\label{subsec:stepsize_convergence}

The fastest electrical time scale is the polarization time constant
\begin{equation}
\tau_p=R_1C_1.
\label{eq:taup_sec6}
\end{equation}
To resolve the polarization dynamics and avoid stage-level oscillations in an explicit method,
we impose the time-step bound
\begin{equation}
\Delta t \le 0.05\,\tau_p.
\label{eq:dt_bound_sec6}
\end{equation}
When tail dynamics are active, one may further restrict $\Delta t \le 0.05\,\tau_\uparrow$.

We enforce convergence through step-halving. Over a candidate step from $t_n$ to $t_{n+1}=t_n+\Delta t$,
we compute two solutions: one using a single RK4 step of size $\Delta t$ and another using two RK4
steps of size $\Delta t/2$. The step is accepted if
\begin{equation}
\left\|z_{\Delta t}-z_{\Delta t/2}\right\|_\infty < 10^{-4},
\label{eq:step_halving_soc_sec6}
\end{equation}
and, for runtime outputs, the inferred TTE changes by less than $1\%$ under step-halving in the
same scenario:
\begin{equation}
\frac{\big|\mathrm{TTE}_{\Delta t}-\mathrm{TTE}_{\Delta t/2}\big|}{\mathrm{TTE}_{\Delta t/2}} < 1\%.
\label{eq:step_halving_tte_sec6}
\end{equation}
If the criterion fails, we set $\Delta t\leftarrow \Delta t/2$ and recompute the step.

\subsection{Event detection and TTE interpolation}
\label{subsec:event_detection}

The runtime termination time is defined as
\begin{equation}
\mathrm{TTE}=\inf\left\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\right\}.
\label{eq:TTE_def_sec6}
\end{equation}
During integration, we monitor $V_{\mathrm{term}}(t)-V_{\mathrm{cut}}$ and $z(t)$ for sign changes. If
$V_{\mathrm{term}}(t_n)>V_{\mathrm{cut}}$ but $V_{\mathrm{term}}(t_{n+1})\le V_{\mathrm{cut}}$, we approximate
the crossing time by linear interpolation:
\begin{equation}
t_\star \approx t_n + \Delta t\,\frac{V_{\mathrm{term}}(t_n)-V_{\mathrm{cut}}}{V_{\mathrm{term}}(t_n)-V_{\mathrm{term}}(t_{n+1})}.
\label{eq:interp_voltage_event_sec6}
\end{equation}
Similarly, if $z(t_n)>0$ and $z(t_{n+1})\le 0$,
\begin{equation}
t_\star \approx t_n + \Delta t\,\frac{z(t_n)}{z(t_n)-z(t_{n+1})}.
\label{eq:interp_soc_event_sec6}
\end{equation}
If both events occur within the same step, we take the earlier of the two interpolated times as TTE.

In addition, we optionally record a CPL infeasibility (voltage-collapse risk) time
\begin{equation}
t_\Delta=\inf\{t>0:\ \Delta(t)\le 0\},
\label{eq:tDelta_def_sec6}
\end{equation}
which is useful for diagnosing ``sudden shutdown'' risk even when current limiting postpones the
actual cutoff event.

\subsection{Algorithm 3: Simulation procedure}
\label{subsec:algorithm3}

\begin{algorithm}[t]
\caption{RK4 simulation with nested CPL current evaluation and event handling}
\label{alg:rk4_cpl}
\begin{algorithmic}[1]
\REQUIRE Initial state $\mathbf{x}(0)=[z_0,0,T_a(0),S_0,0]^\top$, input trajectory $\mathbf{u}(t)$,
parameters $\Theta$, cutoff $V_{\mathrm{cut}}$, step bound $\Delta t_{\max}$.
\ENSURE Trajectories $\mathbf{x}(t)$, $V_{\mathrm{term}}(t)$, and $\mathrm{TTE}$ (and optionally $t_\Delta$).
\STATE Set $t\leftarrow 0$, $\mathbf{x}\leftarrow \mathbf{x}(0)$, choose $\Delta t\le \Delta t_{\max}$.
\STATE Initialize flags: $\texttt{risk\_recorded}\leftarrow \texttt{false}$.
\WHILE{$V_{\mathrm{term}}(t)>V_{\mathrm{cut}}$ \AND $z(t)>0$}
    \STATE Evaluate $\mathbf{u}(t)$ and (if needed) $\mathbf{u}(t+\Delta t/2)$, $\mathbf{u}(t+\Delta t)$.
    \STATE Compute $z_{\mathrm{eff}}=\max\{z,z_{\min}\}$, then $V_{\mathrm{oc}}(z_{\mathrm{eff}})$, $R_0(T_b,S)$,
    $Q_{\mathrm{eff}}(T_b,S)$, and $P_{\mathrm{tot}}$ via \eqref{eq:Ptot_def_sec6}.
    \STATE Compute $\Delta$ via \eqref{eq:Delta_sec6}.
    \IF{$\Delta<0$}
        \IF{\NOT $\texttt{risk\_recorded}$}
            \STATE Record $t_\Delta\leftarrow t$; $\texttt{risk\_recorded}\leftarrow \texttt{true}$.
        \ENDIF
        \STATE Set $I\leftarrow I_{\max}(T_b)$ \COMMENT{strong degradation / protection}
    \ELSE
        \STATE Compute $I_{\mathrm{CPL}}$ via \eqref{eq:Icpl_sec6} and apply saturation \eqref{eq:I_limit_sec6}.
    \ENDIF
    \STATE Perform one RK4 step \eqref{eq:rk4_update_sec6} with nested current evaluation at each substage.
    \STATE Apply projection \eqref{eq:projection_sec6}.
    \STATE Step-halving check: compare $\Delta t$ vs.\ two half-steps; if \eqref{eq:step_halving_soc_sec6} fails, set
    $\Delta t\leftarrow \Delta t/2$ and recompute this step.
    \STATE Update $V_{\mathrm{term}}$ via \eqref{eq:Vterm_sec6}; test event conditions.
    \IF{event detected within this step}
        \STATE Interpolate event time using \eqref{eq:interp_voltage_event_sec6} or \eqref{eq:interp_soc_event_sec6}.
        \STATE Set $\mathrm{TTE}\leftarrow t_\star$ and \textbf{break}.
    \ENDIF
    \STATE Update $t\leftarrow t+\Delta t$.
\ENDWHILE
\RETURN $\mathrm{TTE}$, trajectories, and optionally $t_\Delta$.
\end{algorithmic}
\end{algorithm}

\subsection{Overall strategy for parameter identification}
\label{subsec:id_strategy}

Parameters are grouped to enable targeted identification with minimal confounding:
\begin{itemize}
\item \textbf{Open-circuit voltage (OCV)} parameters $(E_0,K,A,B)$ are identified from an OCV--SOC curve.
\item \textbf{ECM electrical parameters} $(R_0,R_1,C_1)$ are identified from current pulse tests by separating the
instantaneous ohmic drop from the relaxation dynamics.
\item \textbf{Thermal parameters} $(C_{\mathrm{th}},hA)$ are identified from heating and cooling transients.
\item \textbf{Aging parameters} $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$ are identified from capacity fade data
under controlled current/temperature conditions.
\item \textbf{Device power-mapping parameters} (screen/CPU/network) are identified from controlled workload logs by
isolating each subsystem and fitting the prescribed mechanistic forms.
\item \textbf{Tail parameters} $(k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow)$ are identified from network-burst
experiments by fitting the post-burst decay shape.
\end{itemize}
This staged approach preserves physical interpretability and avoids black-box regressions.

\subsection{OCV fitting: $(E_0,K,A,B)$}
\label{subsec:ocv_fit}

Given OCV--SOC samples $\{(z_i,V_i)\}_{i=1}^M$ collected under quasi-equilibrium conditions, we estimate
$(E_0,K,A,B)$ by least squares using the protected SOC $z_{i,\mathrm{eff}}=\max\{z_i,z_{\min}\}$:
\begin{equation}
\min_{E_0,K,A,B}\ \sum_{i=1}^M\left[
V_i-\left(E_0-K\left(\frac{1}{z_{i,\mathrm{eff}}}-1\right)+A e^{-B(1-z_{i,\mathrm{eff}})}\right)
\right]^2.
\label{eq:ocv_ls_sec6}
\end{equation}
The resulting OCV model is then used in the time-domain simulations through \eqref{eq:Vterm_sec6}.

\subsection{Pulse-based identification: $(R_0,R_1,C_1)$}
\label{subsec:pulse_id}

\paragraph{Ohmic resistance $R_0$.}
At fixed SOC and temperature, apply a current step of magnitude $\Delta I$ and measure the instantaneous voltage
drop $\Delta V(0^+)$, yielding
\begin{equation}
R_0 \approx \frac{\Delta V(0^+)}{\Delta I}.
\label{eq:R0_pulse_sec6}
\end{equation}

\paragraph{Polarization branch $(R_1,C_1)$.}
After removing the ohmic drop, the remaining relaxation is approximately first-order with time constant
$\tau_p=R_1C_1$. Denote the relaxation component by
$V_{\mathrm{rel}}(t)=V_{\mathrm{term}}(t)-\big(V_{\mathrm{oc}}-\Delta I\,R_0\big)$. Then
\begin{equation}
V_{\mathrm{rel}}(t)\approx -\Delta I\,R_1\,e^{-t/\tau_p},
\label{eq:relax_exp_sec6}
\end{equation}
so that a linear fit of $\ln|V_{\mathrm{rel}}(t)|$ versus $t$ yields $\tau_p$ and $R_1$, and hence
\begin{equation}
C_1=\frac{\tau_p}{R_1}.
\label{eq:C1_from_tau_sec6}
\end{equation}

\subsection{Temperature/aging coupling: $(R_{\mathrm{ref}},E_a,\eta_R,Q_{\mathrm{nom}},\alpha_Q)$}
\label{subsec:temp_aging_coupling}

\paragraph{Arrhenius temperature dependence for $R_0$.}
Measure $R_0$ at multiple temperatures $T_b^{(j)}$ (e.g., by \eqref{eq:R0_pulse_sec6}) and fit
\begin{equation}
\ln R_0^{(j)}=\ln R_{\mathrm{ref}}+\frac{E_a}{R_g}\left(\frac{1}{T_b^{(j)}}-\frac{1}{T_{\mathrm{ref}}}\right),
\label{eq:arrhenius_fit_sec6}
\end{equation}
to obtain $R_{\mathrm{ref}}$ and $E_a$.

\paragraph{SOH correction for resistance.}
Using measurements across different SOH levels $S$, fit
\begin{equation}
\frac{R_0(T_b,S)}{R_0(T_b,1)}\approx 1+\eta_R(1-S)
\label{eq:etaR_fit_sec6}
\end{equation}
to obtain $\eta_R$.

\paragraph{Effective capacity parameters.}
From capacity tests across temperatures, estimate $Q_{\mathrm{nom}}$ and $\alpha_Q$ using
\begin{equation}
Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}\,S\left[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\right]_+.
\label{eq:Qeff_fit_sec6}
\end{equation}

\subsection{Power mapping identification: $(k_L,\gamma,k_C,\eta,k_N,\kappa,\ldots)$}
\label{subsec:power_mapping_id}

\paragraph{Screen mapping.}
Under controlled conditions with minimal CPU/network activity, vary brightness $L$ and measure total power.
After subtracting background and CPU baseline, fit
\begin{equation}
P_{\mathrm{scr}}(L)=P_{\mathrm{scr},0}+k_L L^\gamma,\qquad \gamma>1.
\label{eq:screen_fit_sec6}
\end{equation}

\paragraph{CPU mapping.}
With fixed brightness and network conditions, apply controlled workloads to vary CPU load $C$ and fit
\begin{equation}
P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu},0}+k_C C^\eta,\qquad \eta>1.
\label{eq:cpu_fit_sec6}
\end{equation}

\paragraph{Network mapping and signal-quality penalty.}
At fixed throughput proxy $N=N_0$, vary signal quality $\Psi$ and fit
\begin{equation}
P_{\mathrm{net}}(N_0,\Psi,w)\approx P_{\mathrm{net},0}+k_N\frac{N_0}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w.
\label{eq:net_fit_sec6}
\end{equation}
For steady experiments where $w$ is constant or negligible, define
$\Delta P_{\mathrm{net}}(\Psi)=P_{\mathrm{net}}-P_{\mathrm{net},0}$ and fit in log space:
\begin{equation}
\ln \Delta P_{\mathrm{net}}(\Psi)\approx \ln(k_N N_0)-\kappa\ln(\Psi+\varepsilon),
\label{eq:kappa_fit_sec6}
\end{equation}
yielding $\kappa$ (slope) and $k_N$ (intercept).

\subsection{Tail parameter identification: $(k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow)$}
\label{subsec:tail_id}

Conduct a network-burst experiment: drive $N(t)$ high for a short period and then reduce it rapidly. After the burst,
$N\approx 0$ and the tail state decays approximately exponentially with time constant $\tau_\downarrow$:
\begin{equation}
w(t)\approx w(t_0)\,e^{-(t-t_0)/\tau_\downarrow},\qquad
P_{\mathrm{tail}}(t)=k_{\mathrm{tail}}w(t).
\label{eq:tail_decay_sec6}
\end{equation}
A linear fit of $\ln P_{\mathrm{tail}}(t)$ versus $t$ yields $\tau_\downarrow$, and the amplitude identifies
$k_{\mathrm{tail}}$. The rise time $\tau_\uparrow$ is obtained by fitting the initial ramp-up segment during burst onset,
consistent with $\tau_\uparrow\ll\tau_\downarrow$.

\subsection{Thermal parameter identification: $(C_{\mathrm{th}},hA)$}
\label{subsec:thermal_id}

Using a heating--cooling experiment, identify the lumped thermal time constant. During the cooling phase where
$I\approx 0$ and $v_p\approx 0$, \eqref{eq:Tbdot_sec6} reduces to
\begin{equation}
\dot T_b \approx -\frac{hA}{C_{\mathrm{th}}}(T_b-T_a),
\label{eq:cooling_sec6}
\end{equation}
so that
\begin{equation}
T_b(t)-T_a \approx (T_b(t_0)-T_a)\,e^{-(hA/C_{\mathrm{th}})(t-t_0)}.
\label{eq:cooling_exp_sec6}
\end{equation}
Fitting the exponential decay yields $hA/C_{\mathrm{th}}$. Then, using the heating phase with known heat generation
$\dot Q \approx I^2R_0+v_p^2/R_1$, one can separate $C_{\mathrm{th}}$ and $hA$.

\subsection{Aging parameter identification: $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$}
\label{subsec:aging_id}

From controlled aging data providing $S(t)$ under known $(I,T_b)$ conditions, the SEI-driven degradation model
\eqref{eq:Sdot_sec6} can be identified by log-linear regression. Approximating $\dot S$ via finite differences,
\begin{equation}
\ln(-\dot S)\approx \ln \lambda_{\mathrm{sei}} + m\ln|I| - \frac{E_{\mathrm{sei}}}{R_g}\frac{1}{T_b}.
\label{eq:aging_loglin_sec6}
\end{equation}
A multi-condition fit across varying currents and temperatures yields $(\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}})$.
This procedure preserves the mechanistic form and avoids black-box regression.

\subsection{Parameter table: nominal values, ranges, and sources}
\label{subsec:param_table}

Table~\ref{tab:param_summary} summarizes the parameters used in simulations, including nominal values and uncertainty
ranges for sensitivity analysis. Nominal values are obtained via the identification procedures above or from
manufacturer specifications / literature when direct measurements are unavailable. Ranges should be selected to
reflect measurement uncertainty and device-to-device variability (e.g., $\pm 10\%$--$\pm 20\%$ for power-map gains,
and temperature-dependent parameters constrained by Arrhenius fits).

\begin{table}[t]
\centering
\caption{Parameter summary (to be finalized): nominal values, uncertainty ranges, and sources.}
\label{tab:param_summary}
\begin{tabular}{llll}
\hline
Category & Parameter & Nominal / Range & Source / Method \\
\hline
OCV & $E_0,K,A,B$ & (fill) / (fill) & OCV--SOC LS fit \eqref{eq:ocv_ls_sec6} \\
ECM & $R_{\mathrm{ref}},E_a$ & (fill) / (fill) & Arrhenius fit \eqref{eq:arrhenius_fit_sec6} \\
ECM & $R_1,C_1$ & (fill) / (fill) & Pulse relaxation \eqref{eq:relax_exp_sec6} \\
SOH coupling & $\eta_R$ & (fill) / (fill) & Resistance vs.\ SOH \eqref{eq:etaR_fit_sec6} \\
Capacity & $Q_{\mathrm{nom}},\alpha_Q$ & (fill) / (fill) & Capacity tests \eqref{eq:Qeff_fit_sec6} \\
Thermal & $C_{\mathrm{th}},hA$ & (fill) / (fill) & Cooling/heating fits \eqref{eq:cooling_exp_sec6} \\
Aging & $\lambda_{\mathrm{sei}},m,E_{\mathrm{sei}}$ & (fill) / (fill) & Log-linear fit \eqref{eq:aging_loglin_sec6} \\
Screen & $P_{\mathrm{scr},0},k_L,\gamma$ & (fill) / (fill) & Screen power fit \eqref{eq:screen_fit_sec6} \\
CPU & $P_{\mathrm{cpu},0},k_C,\eta$ & (fill) / (fill) & CPU power fit \eqref{eq:cpu_fit_sec6} \\
Network & $P_{\mathrm{net},0},k_N,\kappa$ & (fill) / (fill) & Signal penalty \eqref{eq:kappa_fit_sec6} \\
Tail & $k_{\mathrm{tail}},\tau_\uparrow,\tau_\downarrow$ & (fill) / (fill) & Burst/decay \eqref{eq:tail_decay_sec6} \\
Protection & $I_{\max,0},\rho_T,z_{\min}$ & (fill) / (fill) & Device policy / assumption \\
\hline
\end{tabular}
\end{table}