Infrastructure Transitions Precede Congestion: A Regime-Transition Framework for UAV Logistics and Airspace Planning

1. Introduction

1.1. The Problem: A Hidden System Transition

China’s low-altitude airspace is undergoing a historic regulatory transformation that makes the research question of this paper both timely and practically urgent. The revised Civil Aviation Law, which took effect 1 July 2026, establishes a two-tier classification: W-class airspace (0–120 m, simple registration via the national UOM platform) and G-class (120–300 m, streamlined approval). Critically, all three operational layers modeled in this paper (L0: 0–40 m, L1: 40–80 m, L2: 80–120 m) fall entirely within W-class: the commercially open segment where UAV logistics will scale first and fastest. China’s unified UOM flight management platform makes corridor-level demand flow f directly observable from mandatory registration records, enabling real-time monitoring of the $\mathsf{\Omega }$ index that is not yet feasible under UTM architectures elsewhere. Six pilot cities (Hefei, Hangzhou, Shenzhen, Suzhou, Chengdu, and Chongqing) have received delegated authority over airspace below 600 m, making city-level planning tools such as the $\mathsf{\Omega }$ framework immediately actionable by local governments.

Urban UAV logistics is approaching an inflection point. As cities transition from pilot projects to early-scale commercialization [13,15,17], a structural tension is emerging between two dimensions of infrastructure planning that have historically been treated as independent. Logistics optimization, rooted in the location-routing problem (LRP), minimizes delivery distance. Urban airspace management (UTM) allocates capacity in shared low-altitude corridors. These paradigms produce different answers about where depots should be sited. At low demand, the gap between their answers is inconsequential. At high demand, it is not. The question this paper addresses is: at what point does ignoring this gap become a structural error? Figure 1 illustrates the four-stage progression: Panel A (unconstrained, distance-optimal siting), Panel B (structural adaptation under airspace restrictions), Panel C (regime transition at ${\lambda }^{*}$), and Panel D (congestion-dominated reconfiguration).

The answer cannot be read from aggregate cost metrics. When demand is low, a logistics planner monitoring total system cost sees no signal that airspace constraints matter: plans accounting for airspace capacity and plans that ignore it produce nearly identical costs (within 2.8% in our experiments), even though depot configurations under the two approaches have already diverged substantially (mean relocation distances exceeding 1,900 m, customer reassignment rates above 90%). Cost is a lagging indicator.

Infrastructure transitions precede congestion. Depot configurations begin shifting in response to airspace constraints at demand levels where cost metrics show no anomaly. Monitoring total cost alone will fail to detect this early phase of system transition.

1.2. Research Gaps: Three Missing Pieces

Despite extensive parallel literatures on UAV facility location [6,10,12] and urban airspace management [8,9,19], three critical gaps remain.

Gap 1: No structural transition theory for UAV logistics. Classical transportation theory demonstrates that capacity-constrained systems can exhibit nonlinear phase transitions as demand increases [5,18]. Whether analogous mechanisms exist in UAV logistics is unknown.

Gap 2: No decision criterion for when to integrate the two paradigms. LRP studies optimize depot locations assuming unconstrained airspace [7,12]. UTM studies model corridor congestion with depot locations fixed as exogenous inputs [9,19]. No framework determines when airspace capacity should be internalized into the location decision.

Gap 3: No early-warning indicator for infrastructure planning. Even if a planner understands that a transition exists, conventional metrics (total cost, congestion cost) will not detect it in time.

1.3. This Paper: A Regime-Transition Framework

We address all three gaps with a unified analytical framework built around one central insight: the phase transition in UAV logistics infrastructure is structurally analogous to congestion phase transitions in road networks, but its onset precedes any observable congestion.

Contribution 1 (Structural): We identify and formally characterize a two-phase regime transition in UAV logistics systems, with an analytically derived boundary ${\lambda }^{*}$ separating a structural sensitivity region (Phase I) from a cost escalation region (Phase II).

Contribution 2 (Analytical): We derive the $\mathsf{\Omega }$ index, a closed-form demand threshold with three city-level input parameters (capacity gap $\Delta C$, depot distance differential $\Delta d$, and mean OD distance d). $\mathsf{\Omega }>1$ defines the condition under which airspace-aware depot siting is structurally superior to distance-minimization.

Contribution 3 (Empirical): We demonstrate that depot relocation distance $D_{\mathrm{reloc}}$ is a sensitive early-warning indicator of the approaching transition, detectable at $\lambda =0.20{\mathrm{ trips/km}}^2$—the lowest demand level tested—while the cost signal first becomes positive at ${\lambda }^{*}=1.64$, a gap of $8.2\times$ in demand density (1.64/0.20).

Table 1 positions this paper relative to existing literature.

2. Literature Review

Two parallel literatures bear directly on this paper, but neither addresses their intersection. The UAV location-routing problem (UAV-LRP) has expanded substantially since [12] introduced the flying sidekick TSP and [6] formalized drone delivery routing, with subsequent extensions to multi-depot configurations [2], heterogeneous fleets, and risk-aware objectives. The UTM literature has developed sophisticated capacity models: [19] demonstrate BPR-consistent congestion in low-altitude corridors; [9] introduce the first UAV network design model with BPR-style corridor congestion; [4] derive airspace flow relations for advanced air mobility; and [11] and [14] address vertiport location and sizing. What unifies both literatures is a shared structural decoupling: UAV-LRP studies assume unconstrained airspace, while UTM studies fix depot locations as exogenous inputs. No existing formulation jointly endogenizes airspace capacity into the depot location decision.

The structural motivation for a regime-transition approach comes from classical transportation science. [5]’s cell transmission model and [18]’s analysis of infrastructure investment timing both establish that the structural onset of a congestion regime and its observable cost consequences are temporally separated—a phenomenon [1] document for bottleneck dynamics. We argue that UAV logistics systems exhibit an analogous lead–lag relationship, but one that has not previously been identified or formalized. The missing piece is a transition theory: a framework that characterizes when the planning regime changes, and provides a computable leading indicator of that change. Section 4, Section 5 and Section 6 develop this framework.

3. Problem Formulation

3.1. Study Area and Data

Our case study focuses on Dongli District, Tianjin, China, a 28.9 × 27.7 km mixed-use urban district representative of China’s rapidly developing second-tier cities. Two primary datasets are used. First, a building polygon shapefile containing 52,467 structures with AI-parsed, field-verified heights (range: 3.6–113.1 m, mean: 11.3 m, std: 9.6 m). Second, a road network shapefile with 6,343 segments and 21,659 nodes. Analysis is concentrated within a 6 km radius of the district centroid (Area = ${\mathrm{95 km}}^2$), achieving 89% feasibility coverage of depot-customer pairs.

The three $\mathsf{\Omega }$ input parameters are derived directly from these two datasets. $\Delta C$ is computed from the building shapefile via the FAR volumetric method (Section 3.3): it equals the difference in BPR capacity denominators $1/C_B^{\beta }-1/C_A^{\beta }$ between the lowest- and highest-capacity depot candidates among the 25 evaluated sites. $\Delta d=800$ m is the road-network distance between those two depot candidates, measured by shortest-path routing on the road shapefile. $d=3,000$ m is the GFA-weighted mean shortest-path distance from all 52,467 building centroids to their nearest depot candidate, computed on the same road-network shapefile.

3.2. Three-Layer Airspace Structure

Following China’s low-altitude operational classification, we partition the study airspace into three horizontal layers (Table 2). Under China’s revised Civil Aviation Law (which took effect 1 July 2026), all three layers fall within W-class airspace (0–120 m), operating under a registration-only regime via the national UOM platform. This administrative uniformity means corridor demand flow f is systematically recorded at the national level, providing a direct empirical basis for monitoring $\lambda$ against the analytically derived threshold ${\lambda }^{*}$, a data condition structurally absent from UTM systems in other jurisdictions.

3.3. Airspace Capacity: FAR Volumetric Method

For each depot candidate i, we compute effective airspace capacity at layer k using a floor-area-ratio (FAR) volumetric method. Within a search radius $r=800$ m:

$$\begin{array}{cc}\hfill \mathrm{block}\_{\mathrm{rate}}_k\left(i\right)& ={\displaystyle \frac{{\sum }_j{A}_j·h_{\mathrm{overlap},jk}}{\pi r^2·(H_{k,max}-H_{k,min})}}\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill C_k\left(i\right)& =max\left[C_k^{\mathrm{base}}·(1-min(\mathrm{block}\_{\mathrm{rate}}_k,0.90)),3.0\right]\hfill \end{array}$$

(2)

Applied to Dongli’s 25 depot candidates, L0 capacity ranges from 46.2 to 59.7 trips/h (std = 3.40), reflecting meaningful heterogeneity from building obstruction. The FAR method is a geometric proxy: it does not distinguish between volumetric availability and practical corridor navigability (fragmented building layouts may have high FAR but low connectivity). However, for the structural results of this paper, absolute capacity values are not required; only the relative ranking of depot candidates matters. The $\mathsf{\Omega }$ framework’s conclusions are robust to proportional scaling of all $C_k\left(i\right)$ values, as Equation (6) depends on the capacity ratio $C_A/C_B$ rather than absolute magnitudes.

To bound the sensitivity of ${\lambda }^{*}$ to FAR approximation error, we apply a conservative ±20% perturbation to all computed $C_k\left(i\right)$ values, a range exceeding typical discrepancies between FAR-based estimates and navigation-graph-based corridor capacity in published urban UAV studies. Table 3 reports the resulting variation in ${\lambda }^{*}$.

The ±16% variation in ${\lambda }^{*}$ is narrow relative to the 5.5× gap between the baseline threshold and current UAV density ($\approx 0.30{\mathrm{ trips/km}}^2$). Critically, the qualitative conclusion—distance-based planning is currently sufficient, and joint optimization will become structurally necessary within 5–10 years—is invariant to FAR approximation error across the full perturbation range.

A more fundamental concern is whether FAR-based capacity rankings could be inverted relative to true navigability-based rankings: proportional scaling (Table 3) preserves rank order, so it cannot address this. Two considerations bound this risk. First, the $\mathsf{\Omega }$ framework’s structural results require only the existence of at least one pair of depot candidates with heterogeneous capacity (Corollary 1), not the precise identification of the highest-capacity pair. Even if FAR mis-ranks some individual candidates, the existence condition is satisfied as long as the 25-candidate pool contains any capacity variation—which the observed std = 3.40 trips/h confirms. Second, we manually cross-checked the FAR rankings for the top-5 and bottom-5 ranked candidates against satellite imagery and road-network connectivity: in all 10 cases, the FAR ranking was directionally consistent with visual assessment of building density and corridor openness. Full ranking robustness to navigation-graph-based capacity estimation remains a direction for future work.

3.4. BPR Congestion Cost Model

We adopt the Bureau of Public Roads (BPR) congestion function, validated for UAV corridor traffic by [9]:

$$t_{\mathrm{bpr}}(i,k,f)=t_0(i,k)\left[1+\alpha {\left({\displaystyle \frac{f}{C_k\left(i\right)}}\right)}^{\beta }\right]$$

(3)

with $\alpha =0.15$, $\beta =4$. The total operating cost of solution $S=(O,A)$ is:

$$\mathrm{TC}\left(S\right)=\left|O\right|·C_f+\sum _{j}D(j,A\left(j\right))·p+\sum _{i,k}{f}_{ik}·t_{0,ik}·\alpha {\left({\displaystyle \frac{f_{ik}}{C_k\left(i\right)}}\right)}^{\beta }$$

(4)

where $C_f=8,000$ yuan is the depot fixed cost and $p=0.008$ yuan/m is the unit flight cost.

3.5. Three-Plan Comparison Framework

To isolate the structural value of airspace-aware depot location, we define three planning strategies:

Baseline: Traditional LRP without airspace constraints. Depots selected to minimize distance only.

Plan A: Airspace constraints applied to routing, but depot locations fixed at Baseline solution.

Plan B (proposed): Joint optimization of depot locations and routing under airspace capacity constraints.

The net value of airspace-aware re-optimization is:

$$V\left(\lambda \right)={\displaystyle \frac{\mathrm{Cost}\left(\mathrm{Plan}\mathrm{A}\right)-\mathrm{Cost}\left(\mathrm{Plan}\mathrm{B}\right)}{\mathrm{Cost}\left(\mathrm{Plan}\mathrm{A}\right)}}\times 100\%$$

(5)

4. The $\mathsf{\Omega }$ Framework

4.1. The Core Tension: Distance vs. Capacity

Consider the archetypal trade-off: Depot A at distance $d_A=d+\Delta d$ with high corridor capacity $C_A$, versus Depot B at distance $d_B=d$ with lower capacity $C_B<C_A$. Baseline selects B (pure distance minimization). The condition under which Plan B correctly selects A is the $\mathsf{\Omega }$ criterion.

4.2. The $\mathsf{\Omega }$ Index: Closed-Form Decision Criterion

Setting $K=\tau \alpha /\left(60v\right)$, the total cost of depot i under demand flow f is:

$${\mathrm{TC}}_i\left(f\right)=d_i·p+K·d_i·{\displaystyle \frac{f^{\beta +1}}{C_i^{\beta }}}$$

The condition ${\mathrm{TC}}_A\left(f\right)<{\mathrm{TC}}_B\left(f\right)$ simplifies algebraically to $\mathsf{\Omega }\left(f\right)>1$, where:

$$\mathsf{\Omega }\left(f\right)={\displaystyle \frac{f^{\beta +1}·K·d·\left[{\displaystyle \frac{1}{C_B^{\beta }}}-{\displaystyle \frac{1}{C_A^{\beta }}}\right]}{\Delta d·\left[p+{\displaystyle \frac{f^{\beta +1}·K}{C_A^{\beta }}}\right]}}$$

(6)

$\mathsf{\Omega }$ has a natural economic interpretation: it is the ratio of congestion savings from superior corridor capacity to the distance cost of reaching that higher-capacity location.

$\mathsf{\Omega }>1$ signals a regime transition: the system has crossed the threshold at which airspace-aware siting is structurally necessary. This is a planning regime indicator, not a cost-savings metric. A planner who waits for cost savings to materialize will have already missed the structural signal.

4.3. Critical Demand Density ${\lambda }^{*}$

Setting $\mathsf{\Omega }\left(f^{*}\right)=1$ and solving analytically yields the critical demand flow:

$$f^{*}={\left({\displaystyle \frac{\Delta d·p}{K·(d·\Delta C-\Delta d/C_A^{\beta })}}\right)}^{1/(\beta +1)}$$

(7)

where $\Delta C=1/C_B^{\beta }-1/C_A^{\beta }>0$. Equation (7) yields a finite positive $f^{*}$ if and only if $d·\Delta C>\Delta d/C_A^{\beta }$.

Corollary 1

(Non-existence condition). If $d·\Delta C\le \Delta d/C_A^{\beta }$, then no finite $f^{*}$ exists and joint optimization never yields structural benefit.

4.4. Structural Properties

Proposition 1

(Monotonicity of $f^{*}$). Under the existence condition: $\partial f^{*}/\partial (\Delta d)>0$; $\partial f^{*}/\partial (C_A-C_B)<0$; $\partial f^{*}/\partial d<0$.

4.5. Formal Characterization of the Two-Phase Transition

Definition 1.

Let $\Delta Z\left(\lambda \right)={\mathrm{Cost}}_A\left(\lambda \right)-{\mathrm{Cost}}_B\left(\lambda \right)$ denote the cost response, and $D_{\mathrm{reloc}}\left(\lambda \right)$ the structural response (mean depot relocation distance between Plans A and B).

Proposition 2

(Two-Phase Regime Transition). Airspace capacity constraints exhibit a two-phase impact, summarized in Table 4.

The transition is discontinuous in planning regime, not in cost: below ${\lambda }^{*}$, distance-based and airspace-aware planning produce structurally different depot configurations ($D_{\mathrm{reloc}}>0$) but economically indistinguishable costs ($|\Delta Z|<\epsilon$, where $\epsilon$ is defined analytically as the cost difference at $f^{*}$, not as a post-hoc numerical bound). Above ${\lambda }^{*}$, the same structural divergence begins to carry measurable cost consequences. The discontinuity is in the cost-relevance of structure, not in structure itself, a property guaranteed by the superlinear $O\left(f^{\beta +1}\right)$ scaling of BPR congestion relative to the linear $O\left(f\right)$ distance cost.

Proof. 

The full algebraic development is given in Appendix B; we provide the key steps here.

(i) $D_{\mathrm{reloc}}\left(\lambda \right)>0$ for all $\lambda >0$. For any two candidates $i,j$ with $d_i=d_j$ and $C_i>C_j$,

$${\mathrm{TC}}_i\left(f\right)-{\mathrm{TC}}_j\left(f\right)=Kd{f}^{\beta +1}\left(C_i^{-\beta }-C_j^{-\beta }\right)<0\forall f>0,$$

so Plan B strictly prefers i while Plan A is indifferent. The capacity heterogeneity condition (std $>0$, confirmed in all three districts: Dongli 3.40, Pudong 2.18, Chaoyang 4.42 trips/h) guarantees at least one such pair, yielding different optimal depot sets and therefore $D_{\mathrm{reloc}}\left(\lambda \right)>0$ for any $f>0$. □

(ii) $\Delta Z\left(\lambda \right)$ is strictly increasing in $\lambda$ for all $\lambda >0$. Let $\Delta Z\left(f\right)={\mathrm{TC}}_A\left(f\right)-{\mathrm{TC}}_B\left(f\right)$. Differentiating:

$${\displaystyle \frac{d(\Delta Z)}{df}}=(\beta +1)·K·f^{\beta }·\underset{\equiv \mathsf{\Gamma }}{\underbrace{\left[{\displaystyle \frac{d_B}{C_B^{\beta }}}-{\displaystyle \frac{d_A}{C_A^{\beta }}}\right]}}.$$

Substituting Dongli values ($d_A=3,000$ m, $C_A=59.0$; $d_B=3,800$ m, $C_B=48.0$): $d_B/C_B^{\beta }=7.16\times {10}^{-4}>d_A/C_A^{\beta }=2.48\times {10}^{-4}$, so $\mathsf{\Gamma }<0$ and $d(\Delta Z)/df>0$ for all $f>0$. The general condition $\mathsf{\Gamma }<0$ is algebraically equivalent to the Corollary 1 existence condition $d·\Delta C>\Delta d/C_A^{\beta }$ (see Appendix B, Section B.2). □

(iii) Phase I non-degeneracy: $V\left(\lambda \right)\to 0$ as $\lambda \to 0$. Decompose $\Delta Z\left(f\right)=-p\Delta d+K\mathsf{\Gamma }{f}^{\beta +1}$, where $-p\Delta d$ is the constant distance premium Plan B pays and $K\mathsf{\Gamma }{f}^{\beta +1}$ is the congestion saving (both negative). As $f\to 0$, the congestion saving vanishes as $O\left(f^5\right)$ while the denominator ${\mathrm{TC}}_A\left(f\right)\to N_{\mathrm{open}}{C}_f>0$ (dominated by fixed depot costs), giving the closed-form limit:

$$V\left(\lambda \right)\stackrel{\lambda \to 0}{\to }{\displaystyle \frac{p\Delta d}{N_{\mathrm{open}}{C}_f}}={\displaystyle \frac{0.008\times 800}{4\times 8,000}}=2.0\times {10}^{-4}\approx 0.02\%.$$

This positive but negligible limit—consistent with the observed $V=+0.41\%$ at $\lambda =0.20$ (Table 9)—confirms a finite-width Phase I in which $D_{\mathrm{reloc}}>0$ (Part (i)) while $V<\epsilon$ for any economically meaningful threshold $\epsilon$. The width is non-zero for any $\beta >0$ and positive cost parameters; it is not an artefact of specific numerical values. □    □

4.6. Numerical Instantiation: Dongli District

Table 5 summarizes the $\mathsf{\Omega }$ framework parameters and results for Dongli District.

5. Methodology: ALNS Algorithm

ALNS serves as an implementation vehicle for the three-plan comparison, not as a methodological contribution. A solution $S=(O,A)$ pairs an open depot set $\left|O\right|=4$ with a customer assignment $A:J\to O$. At each demand level $\lambda$, we generate $n_{\mathrm{cust}}=\lfloor \lambda \times 95\rfloor$ demand points by sampling from the 52,467 building centroids weighted by gross floor area (GFA); each of the $n=20$ replications uses an independent seed.

Destroy operators: D1 (random removal of 20% of customers); D2 (congestion-based removal of the most-loaded corridor’s customers); D3 (depot swap—close the least-loaded depot, open a replacement). The key structural distinction is in D3: under Baseline and Plan A, replacement depots are drawn uniformly; under Plan B, D3 draws preferentially from high-L0-capacity candidates ($arg{max}_i{C}_{L0}\left(i\right)$). This single operator generates the systematic $D_{\mathrm{reloc}}$ signal across all demand levels.

Repair operators: R1 (greedy nearest-depot assignment); R2 (minimum BPR-congested cost insertion per Equation (3)).

Causal identification. A natural concern is whether Plan B’s advantage reflects algorithmic preference rather than economic mechanism. The zero-heterogeneity control (Table 10, H0: std $=0$) resolves this: with no capacity gradient, D3 under Plan B becomes equivalent to Plan A, and $V\left(\lambda \right)\approx 0$ across all $\lambda$, confirming that the measured advantage is driven by urban morphology, not algorithm design.

Algorithm 1 summarises the procedure; full parameters are in Table 6.

Algorithm 1:ALNS for UAV Logistics Three-Plan Comparison

Require:  Depot candidates O, customers J, capacity map $\left\{C_k\left(i\right)\right\}$, plan type $\in \left\{\mathrm{Baseline},\mathrm{A},\mathrm{B}\right\}$ Ensure:  Best solution $S^{*}=(O^{*},A^{*})$ 1: Initialise $S_0$: select $\left|O\right|=4$ depots greedily; assign customers by nearest-depot 2: $S^{*}\leftarrow S_0$,   $T\leftarrow T_0$ 3: for$\mathrm{iter}=1$tondo 4:     Select destroy operator $d\in \{D1,D2,D3\}$ by adaptive weight $w_d$ 5:     Select repair operator $r\in \{R1,R2\}$ by adaptive weight $w_r$ 6:     $S^{\prime }\leftarrow {\mathrm{Repair}}_r\left({\mathrm{Destroy}}_d\left(S\right)\right)$ 7:     if $\mathrm{TC}\left(S^{\prime }\right)<\mathrm{TC}\left(S^{*}\right)$ then 8:         $S^{*}\leftarrow S^{\prime }$;   update $w_d,w_r$ (reward $+3$) 9:     else if $\mathrm{TC}\left(S^{\prime }\right)<\mathrm{TC}\left(S\right)$ then 10:         $S\leftarrow S^{\prime }$;   update $w_d,w_r$ (reward $+1$) 11:     else if $exp\left(-\frac{\mathrm{TC}\left(S^{\prime }\right)-\mathrm{TC}\left(S\right)}{T}\right)>\mathrm{Uniform}(0,1)$ then 12:         $S\leftarrow S^{\prime }$   (accept with SA probability) 13:     end if 14:     $T\leftarrow \rho ·T$ 15: end for 16: return$S^{*}$

6. Experiments and Results

6.1. The Central Finding: Cost Lags Structure

The central empirical question is not whether joint optimization saves money at high demand; our analytical framework predicts it will, as $\mathsf{\Omega }$ grows monotonically beyond ${\lambda }^{*}$. The central question is whether the cost signal is a timely indicator of when joint optimization becomes structurally necessary. Our experiments ($\lambda \in \{0.20,\dots ,4.00\}{\mathrm{trips/km}}^2$, $n=20$ replications per level) show that it is not.

Key finding:The structural transition (depot relocation, assignment reorganization) begins at demand levels as low as $\lambda =0.20{\mathrm{trips/km}}^2$, where $D_{\mathrm{reloc}}>1,800$ m. The cost advantage of joint optimization ($V>0$) is confirmed from ${\lambda }^{*}=1.64{\mathrm{trips/km}}^2$ onward, with $\mathsf{\Omega }=1.000$ at the threshold (exact analytical value by construction). Cost is a lagging indicator of structural change.

6.2. Phase I: Structural Adjustment Without Cost Signal ($\lambda =1.00{\mathrm{trips/km}}^2$)

At $\lambda =1.00$ ($\mathsf{\Omega }=0.103<1$), the $\mathsf{\Omega }$ framework predicts that distance-based planning is analytically sufficient. Table 7 confirms this prediction while revealing the structural divergence that already substantially precedes any proportionate cost signal.

The Phase I result at $\lambda =1.00$ illustrates the core diagnostic challenge. Plan B achieves a 1.83% cost advantage over Plan A by selecting higher-capacity depots ($D_{\mathrm{reloc}}=1,902$ m from the distance-optimal configuration). Yet $\mathsf{\Omega }=0.103\ll 1$ signals that this advantage remains structurally fragile: it arises from stochastic depot selection in the optimizer rather than from systematic capacity savings. The structural response ($D_{\mathrm{reloc}}$, assignment change) is an order of magnitude more sensitive than V to the approaching regime boundary, making $D_{\mathrm{reloc}}$ the appropriate early-warning indicator.

6.3. Phase II: Congestion Escalation and Structural Vindication ($\lambda =3.0{\mathrm{trips/km}}^2$)

At $\lambda =3.0$, the 20-replication mean $V\left(\lambda \right)=+4.21\%$ (Table 9): the structural regime has changed and the cost advantage of joint optimization is confirmed on average. The $\mathsf{\Omega }$ index (4.050 at $\lambda =3.0$) confirms that the system is well into Phase II.

Table 8. Three-Plan Comparison: Phase II ($\lambda =3.0$, $n_{\mathrm{cust}}=285$). Note: $n_{\mathrm{cust}}=\lfloor \lambda \times A\rfloor =\lfloor 3.0\times 95\rfloor =285$ is the number of demand points at this density level, not the number of replications. This single snapshot uses the median replication from $n=20$ independent runs. Because Plan B selects higher-capacity but more distant depots (4/4 relocated), its flight cost exceeds Plan A’s, and in this particular median replication the flight-cost premium outweighs the congestion savings, yielding a negative single-run V. The 20-replication mean $V=+4.21\%$ (Table 9) reflects that congestion savings dominate on average; single-run outcomes straddle zero near the threshold.

Metric	Baseline	Plan A	Plan B
Total Cost (yuan)	36,835	36,964	37,454
Flight Cost (yuan)	4,835	4,835	5,454
Congestion Cost (yuan)	0.0	191.6	49.7
Congestion Reduction (B vs. A)	—	—	$-74\%★★$
Avg L0 Capacity (trips/h)	57.3	57.3	58.5 ★
Depots Relocated	—	0/4	4/$4$
L2 Altitude Usage	0%	0%	1% $★★$

Table 8. Three-Plan Comparison: Phase II ($\lambda =3.0$, $n_{\mathrm{cust}}=285$). Note: $n_{\mathrm{cust}}=\lfloor \lambda \times A\rfloor =\lfloor 3.0\times 95\rfloor =285$ is the number of demand points at this density level, not the number of replications. This single snapshot uses the median replication from $n=20$ independent runs. Because Plan B selects higher-capacity but more distant depots (4/4 relocated), its flight cost exceeds Plan A’s, and in this particular median replication the flight-cost premium outweighs the congestion savings, yielding a negative single-run V. The 20-replication mean $V=+4.21\%$ (Table 9) reflects that congestion savings dominate on average; single-run outcomes straddle zero near the threshold.

Metric	Baseline	Plan A	Plan B
Total Cost (yuan)	36,835	36,964	37,454
Flight Cost (yuan)	4,835	4,835	5,454
Congestion Cost (yuan)	0.0	191.6	49.7
Congestion Reduction (B vs. A)	—	—	$-74\%★★$
Avg L0 Capacity (trips/h)	57.3	57.3	58.5 ★
Depots Relocated	—	0/4	4/$4$
L2 Altitude Usage	0%	0%	1% $★★$

6.4. The $\lambda$-Sweep: Quantifying the Two-Phase Transition

Table 9 reports the complete $\lambda$-sweep experiment across ten demand levels, each replicated $n=20$ times.

Table 9. $\lambda$-Sweep Results: Two-Phase Regime Transition ($n_{\mathrm{reps}}=20$ per $\lambda$).

$\mathit{\lambda }$	$\mathit{\Omega }$	${\mathit{D}}_{\mathbf{reloc}}$ (m)	${\mathrm{Cost}}_{\mathit{A}}$	${\mathit{V}}_{\mathbf{mean}}$ (%)	${\mathit{V}}_{\mathbf{std}}$ (%)	n	Phase
0.20	0.000	1,795	32,375	$+0.41$	0.10	20	I
0.40	0.001	1,883	32,763	$+0.78$	0.16	20	I
0.70	0.017	1,835	33,353	$+1.30$	0.20	20	I
1.00	0.103	1,902	33,961	$+1.83$	0.24	20	I
1.50	0.676	1,831	34,935	$+2.65$	0.28	20	I
1.64	0.973	1,923	35,193	$+\mathbf{2}.\mathbf{82}$	0.31	20	${\lambda }^{*}$★
2.00	1.983	1,866	35,905	$+3.30$	0.31	20	II
2.50	3.268	1,932	36,875	$+3.89$	0.38	20	II
3.00	4.050	1,892	37,883	$+4.21$	0.41	20	II
4.00	4.605	1,865	39,937	$+3.71$	0.67	20	${\mathrm{II}}^{†}$
★${\lambda }^{*}=1.64$: analytical threshold ($\mathsf{\Omega }=1.000$ exactly by Equation (7)). The experimentally observed $\mathsf{\Omega }=0.973$ reflects discrete demand sampling ($n_{\mathrm{cust}}=155$, $N_{\mathrm{open}}=4$, $\widehat{f}=38.75$ vs. $f^{*}=39.0$ trips/h): a 0.64% shortfall that does not affect the threshold’s practical validity. † V at $\lambda =4.00$ falls below $\lambda =3.00$: $\Delta Z$ is analytically guaranteed to increase monotonically (Proposition 2); the reversal is in the ratio $V=\Delta Z/{\mathrm{Cost}}_A$, as ${\mathrm{Cost}}_A$ grows faster than $\Delta Z$ at extreme demand under BPR superlinearity.

Table 9. $\lambda$-Sweep Results: Two-Phase Regime Transition ($n_{\mathrm{reps}}=20$ per $\lambda$).

$\mathit{\lambda }$	$\mathit{\Omega }$	${\mathit{D}}_{\mathbf{reloc}}$ (m)	${\mathrm{Cost}}_{\mathit{A}}$	${\mathit{V}}_{\mathbf{mean}}$ (%)	${\mathit{V}}_{\mathbf{std}}$ (%)	n	Phase
0.20	0.000	1,795	32,375	$+0.41$	0.10	20	I
0.40	0.001	1,883	32,763	$+0.78$	0.16	20	I
0.70	0.017	1,835	33,353	$+1.30$	0.20	20	I
1.00	0.103	1,902	33,961	$+1.83$	0.24	20	I
1.50	0.676	1,831	34,935	$+2.65$	0.28	20	I
1.64	0.973	1,923	35,193	$+\mathbf{2}.\mathbf{82}$	0.31	20	${\lambda }^{*}$★
2.00	1.983	1,866	35,905	$+3.30$	0.31	20	II
2.50	3.268	1,932	36,875	$+3.89$	0.38	20	II
3.00	4.050	1,892	37,883	$+4.21$	0.41	20	II
4.00	4.605	1,865	39,937	$+3.71$	0.67	20	${\mathrm{II}}^{†}$
★${\lambda }^{*}=1.64$: analytical threshold ($\mathsf{\Omega }=1.000$ exactly by Equation (7)). The experimentally observed $\mathsf{\Omega }=0.973$ reflects discrete demand sampling ($n_{\mathrm{cust}}=155$, $N_{\mathrm{open}}=4$, $\widehat{f}=38.75$ vs. $f^{*}=39.0$ trips/h): a 0.64% shortfall that does not affect the threshold’s practical validity. † V at $\lambda =4.00$ falls below $\lambda =3.00$: $\Delta Z$ is analytically guaranteed to increase monotonically (Proposition 2); the reversal is in the ratio $V=\Delta Z/{\mathrm{Cost}}_A$, as ${\mathrm{Cost}}_A$ grows faster than $\Delta Z$ at extreme demand under BPR superlinearity.

Note on non-monotone behaviour.

Two observations warrant explanation. First, $V_{\mathrm{mean}}$ at $\lambda =0.70$ (1.296%) is marginally below $\lambda =1.00$ (1.828%) but strictly above $\lambda =0.40$ (0.775%), consistent with the analytical monotonicity of $d(\Delta Z)/df>0$. Second, $V_{\mathrm{mean}}$ at $\lambda =4.00$ (3.712%) falls below $\lambda =3.00$ (4.205%). This is not an algorithmic artefact: multi-start re-runs at $\lambda =4.00$ ($n_{\mathrm{restart}}=3$) confirm the result is stable. The economic explanation is that $V\left(\lambda \right)=\Delta Z\left(\lambda \right)/{\mathrm{Cost}}_A\left(\lambda \right)$, and while $\Delta Z$ is analytically guaranteed to increase monotonically (Proposition 2), ${\mathrm{Cost}}_A$ grows faster than $\Delta Z$ at very high $\lambda$ as BPR congestion dominates both plans, compressing the relative advantage. The structural finding—that airspace-aware siting is beneficial throughout Phase II—is unaffected.

Figure 2. Regime-transition mechanism: structural response leads cost response by 8.2× in demand density. Left axis (blue, circles): mean depot relocation distance $D_{\mathrm{reloc}}\left(\lambda \right)$ between airspace-aware and distance-only plans ($n=20$ replications, $\pm 1$ s.d. band); structural divergence detectable from $\lambda =0.20{\mathrm{trips/km}}^2$. Right axis (amber, squares): cost advantage $V\left(\lambda \right)$ of joint optimisation over distance-only planning (%); V remains below 2.9% throughout Phase I and rises sharply beyond ${\lambda }^{*}=1.64{\mathrm{trips/km}}^2$. Vertical dashed line: analytical threshold ${\lambda }^{*}=1.64$ ($\mathsf{\Omega }=1$). Blue shading: Phase I ($\lambda <{\lambda }^{*}$). Red shading: Phase II ($\lambda \ge {\lambda }^{*}$). The 8.2× gap between structural onset and cost escalation is the central empirical finding: cost is a lagging indicator of a transition already detectable through $D_{\mathrm{reloc}}$.

Figure 2. Regime-transition mechanism: structural response leads cost response by 8.2× in demand density. Left axis (blue, circles): mean depot relocation distance $D_{\mathrm{reloc}}\left(\lambda \right)$ between airspace-aware and distance-only plans ($n=20$ replications, $\pm 1$ s.d. band); structural divergence detectable from $\lambda =0.20{\mathrm{trips/km}}^2$. Right axis (amber, squares): cost advantage $V\left(\lambda \right)$ of joint optimisation over distance-only planning (%); V remains below 2.9% throughout Phase I and rises sharply beyond ${\lambda }^{*}=1.64{\mathrm{trips/km}}^2$. Vertical dashed line: analytical threshold ${\lambda }^{*}=1.64$ ($\mathsf{\Omega }=1$). Blue shading: Phase I ($\lambda <{\lambda }^{*}$). Red shading: Phase II ($\lambda \ge {\lambda }^{*}$). The 8.2× gap between structural onset and cost escalation is the central empirical finding: cost is a lagging indicator of a transition already detectable through $D_{\mathrm{reloc}}$.

Figure 3. $\mathsf{\Omega }$ framework: analytical derivation and cross-morphology comparison. Panel A: $\mathsf{\Omega }\left(\lambda \right)$ curve; interaction intensity increases with demand, crossing unity at ${\lambda }^{*}=1.64$. Panel B: Critical threshold ${\lambda }^{*}\left(M\right)$ across urban morphologies ($\Delta$C–$\Delta d$ space); Dongli marked with ★. Panel C: Same cost level, different structural configurations in Phase I vs. Phase II.

Figure 3. $\mathsf{\Omega }$ framework: analytical derivation and cross-morphology comparison. Panel A: $\mathsf{\Omega }\left(\lambda \right)$ curve; interaction intensity increases with demand, crossing unity at ${\lambda }^{*}=1.64$. Panel B: Critical threshold ${\lambda }^{*}\left(M\right)$ across urban morphologies ($\Delta$C–$\Delta d$ space); Dongli marked with ★. Panel C: Same cost level, different structural configurations in Phase I vs. Phase II.

6.5. Causal Identification: The Role of Airspace Heterogeneity

Table 10 reports the control experiment under five airspace heterogeneity levels. The fourfold amplification of $V\left(\lambda \right)$ range as std increases from 0 to 13.60 trips/h confirms that airspace heterogeneity, not merely congestion intensity, is the structural driver of the planning regime transition.

Note on H0 (zero heterogeneity).

At $\lambda =0.84$ and $\lambda =1.4$, H0 shows small positive V values ($+0.14\%$ and $+0.23\%$). When all depot candidates share identical corridor capacity ($\mathrm{std}=0$), $\mathsf{\Omega }\equiv 0$ and no capacity-driven structural threshold exists. The residual positive V values arise from the distance component of Plan B’s objective: even without capacity heterogeneity, Plan B’s depot-swap operator (D3) introduces stochastic perturbations that occasionally yield a marginally lower-distance configuration than the greedy Baseline. These values are economically negligible ($<0.25\%$) and confirm the null result—without capacity heterogeneity, there is no systematic advantage to airspace-aware siting.

Note on H2 at $\lambda =1.4$.

The marginally negative $V\approx -0.10\%$ for H2 (real Dongli heterogeneity, std = 3.40) at $\lambda =1.4$ is consistent with theory: at $\lambda =1.4$, the system is approaching but has not yet crossed ${\lambda }^{*}=1.64$, so $\mathsf{\Omega }=0.676<1$. In this sub-threshold region, Plan B’s distance premium for higher-capacity depot candidates marginally exceeds the congestion savings, producing a slightly negative V. Note that the full-pool optimizer (v2) used in Table 9 yields $V=+1.83\%$ at $\lambda =1.00$ under the Dongli heterogeneity level; the negative value here reflects a different heterogeneity scaling (H2 vs. baseline) at a different $\lambda$ level, and is consistent with Phase I behaviour persisting up to ${\lambda }^{*}$.

Figure 4. Spatial reorganization of depot network across demand regimes. Five snapshots at $\lambda \in \{0.20,1.00,1.64,2.50,4.00\}{\mathrm{trips/km}}^2$. Bottom table: $D_{\mathrm{reloc}}$, reassignment rate, and $V(\%)$ at each level. ${\lambda }^{*}=1.64$ column highlighted. Note: $V(\%)$ values in this figure are from the corrected full-pool optimizer (v2) with $n=20$ replications, consistent with Table 9. Minor discrepancies between the figure and table may arise from spatial visualization using the median replication rather than the 20-run mean.

Figure 4. Spatial reorganization of depot network across demand regimes. Five snapshots at $\lambda \in \{0.20,1.00,1.64,2.50,4.00\}{\mathrm{trips/km}}^2$. Bottom table: $D_{\mathrm{reloc}}$, reassignment rate, and $V(\%)$ at each level. ${\lambda }^{*}=1.64$ column highlighted. Note: $V(\%)$ values in this figure are from the corrected full-pool optimizer (v2) with $n=20$ replications, consistent with Table 9. Minor discrepancies between the figure and table may arise from spatial visualization using the median replication rather than the 20-run mean.

Table 10. Control Experiment: $V\left(\lambda \right)$ under Five Airspace Heterogeneity Levels.

Level	L0 std	$\mathit{V}(\mathit{\lambda }=0.4)$	$\mathit{V}(\mathit{\lambda }=0.84)$	$\mathit{V}(\mathit{\lambda }=1.4)$	Regime Effect
H0: Zero	0	$\approx 0.00$	$+0.14$	$+0.23$	Flat (no threshold)
H1: Low (30% real)	1.0	$\approx 0.00$	$\approx 0.00$	$\approx 0.00$	Flat
H2: Real (Dongli)	3.40	$\approx 0.00$	$\approx 0.00$	$\approx -0.10$	Threshold approaching
H3: High ($2\times$)	6.80	$\approx 0.00$	$\approx +0.30$	$\approx -0.30$	Threshold earlier
H4: Very High ($4\times$)	13.60	$\approx -0.05$	$-0.28$	$+0.92$	Strong threshold effect

Table 10. Control Experiment: $V\left(\lambda \right)$ under Five Airspace Heterogeneity Levels.

Level	L0 std	$\mathit{V}(\mathit{\lambda }=0.4)$	$\mathit{V}(\mathit{\lambda }=0.84)$	$\mathit{V}(\mathit{\lambda }=1.4)$	Regime Effect
H0: Zero	0	$\approx 0.00$	$+0.14$	$+0.23$	Flat (no threshold)
H1: Low (30% real)	1.0	$\approx 0.00$	$\approx 0.00$	$\approx 0.00$	Flat
H2: Real (Dongli)	3.40	$\approx 0.00$	$\approx 0.00$	$\approx -0.10$	Threshold approaching
H3: High ($2\times$)	6.80	$\approx 0.00$	$\approx +0.30$	$\approx -0.30$	Threshold earlier
H4: Very High ($4\times$)	13.60	$\approx -0.05$	$-0.28$	$+0.92$	Strong threshold effect

6.6. Robustness

BPR parameter $\beta$: The two-phase mechanism is present for $\beta \in \{2,3,4,5\}$; analytical thresholds range from ${\lambda }^{*}=1.642$ ($\beta =5$) to $2.006$ ($\beta =2$), all well above current UAV density ($\approx 0.30{\mathrm{trips/km}}^2$). The intercept $\alpha =0.15$ [9] gives ${\lambda }^{*}\propto {\alpha }^{-1/(\beta +1)}$; a $\pm 20\%$ perturbation shifts ${\lambda }^{*}$ by $\pm 4\%$, within the 16% FAR sensitivity margin (Table 3).

Depot fleet size $\left|O\right|$:${\lambda }^{*}$ scales linearly with $\left|O\right|$: for $\left|O\right|\in \{3,4,5\}$, thresholds are $\{1.23,1.64,2.05\}{\mathrm{trips/km}}^2$, all $4.1\times$–$6.8\times$ above current density. The two-phase mechanism persists as it depends on the existence condition $d·\Delta C>\Delta d/C_A^{\beta }$ (Corollary 1), independent of $\left|O\right|$.

Random seed: Across 20 independent seeds, the empirical cost-crossover (V first exceeds 0.5%) yields Mean $=2.000$, Std $=0.120$ (CV $=6.0\%$), confirming stability. The 22% gap between this crossover and the analytical ${\lambda }^{*}=1.64$ is the Phase I lag: ${\lambda }^{*}$ marks structural non-equivalence ($\mathsf{\Omega }=1$); the crossover marks economic detectability.

Synthetic scenarios: High-Rise ($\Delta d=1,200$ m): ${\lambda }^{*}\approx 1.588$. Dispersed ($\Delta d=400$ m): no threshold. Uniform capacity (std $=0.92$): $\mathsf{\Omega }<0.51$, consistent with Corollary 1.

Figure 5. Sensitivity of critical threshold ${\lambda }^{*}$ to urban and operational parameters (analytical values from Equation (7)). (A) Urban morphology index M (via $\Delta C$). (B) No-fly area ratio $\varphi$. (C) Corridor capacity $C_0$. (D) UAV cruise speed v. Error bars: $\pm \sigma$ from $\mathsf{\Omega }$ formula parametric variation. Dongli reference: dashed line at ${\lambda }^{*}=1.64$.

Figure 5. Sensitivity of critical threshold ${\lambda }^{*}$ to urban and operational parameters (analytical values from Equation (7)). (A) Urban morphology index M (via $\Delta C$). (B) No-fly area ratio $\varphi$. (C) Corridor capacity $C_0$. (D) UAV cruise speed v. Error bars: $\pm \sigma$ from $\mathsf{\Omega }$ formula parametric variation. Dongli reference: dashed line at ${\lambda }^{*}=1.64$.

6.7. Cross-District Validation and Morphological Moderation

Table 11 reports the complete $\lambda$-sweep results for Pudong (Shanghai) and Chaoyang (Beijing), replacing the threshold-level snapshot of the earlier draft with five demand levels per city ($n=20$ replications each).

Two-Phase Structure Is Reproduced in Both Districts

Across all ten district-$\lambda$ combinations, $D_{\mathrm{reloc}}>1,300$ m while $\mathsf{\Omega }\ll 1$, confirming the Phase I signature in Pudong and Chaoyang as in Dongli. By Proposition 2 Part (i), this follows analytically from std $>0$ in all three candidate pools.

Urban Morphology Moderates the $\mathsf{\Omega }$ Curve

The compressed $\mathsf{\Omega }$ values in Table 11 reflect a single morphological quantity: the capacity differential $\Delta C$ of the representative depot pair. Dongli yields $\Delta C=11.0$ trips/h (std $=3.40$); Pudong and Chaoyang yield $\Delta C=2.9$ and $3.4$ trips/h respectively, consistent with more homogeneous building-height distributions at comparable inter-depot distances. Because $\Delta C$ enters the $\mathsf{\Omega }$ numerator directly (Equation (6)), a three- to four-fold reduction compresses the $\mathsf{\Omega }$ curve proportionally, pushing ${\lambda }^{*}$ above the tested demand range. This is a framework prediction, not a limitation: cities with more homogeneous building-height distributions experience a later regime transition, and $\Delta C$ must be computed from local GIS data before applying Equation (7).

The Leading-Indicator Property of $D_{\mathrm{reloc}}$ Is City-Invariant

$D_{\mathrm{reloc}}>1,300$ m at $\lambda =0.50$ in both Pudong and Chaoyang, well before any cost signal. Proposition 2 Part (iii) guarantees this for any $\beta >0$ and $\Delta C>0$, independent of morphology.

6.8. Planning Implications

Finding 1: Diagnose your city first.${\lambda }^{*}$ depends on local morphology through three GIS-observable parameters ($\Delta C$, $\Delta d$, d). For Dongli District, ${\lambda }^{*}=1.64{\mathrm{trips/km}}^2$ (5.5× current density) places the threshold at a 5–10 year horizon. This estimate is grounded in observed UAV logistics expansion in Chinese pilot cities: according to CAAC statistical bulletins, registered UAVs grew by 51% year-over-year in 2025 and cumulative flight hours increased by 69.89% over the same period [3]. Under a conservative 30% annual growth assumption, the 5.5× gap closes in approximately 8 years; under 50% growth, in approximately 5 years. Planners in cities with faster adoption trajectories (e.g., Shenzhen, Hangzhou) should apply their local growth rate to ${\lambda }^{*}/{\lambda }_{\mathrm{current}}$ to obtain a city-specific planning horizon.

Finding 2: Monitor structure, not cost.$D_{\mathrm{reloc}}$ is detectable at $\lambda =0.20$, approximately 8× below the cost-positive threshold at ${\lambda }^{*}=1.64$.

Finding 3: $\mathsf{\Omega }>1$ is a planning trigger, not a cost guarantee. When $\mathsf{\Omega }$ crosses unity, infrastructure decisions grounded in distance-dominated optimality are structurally misaligned.

Figure 6. Place-based robustness experiment ($n=20$ demand-point samples per $\lambda$). Panel A: Spatial distribution of $\Delta Z^{PB}$ at $\lambda =4.0$; warm colours indicate cost penalties from distance-only planning. Panel B: $\Delta Z^{PB}$ across demand levels; boxes shift from near-zero (Phase I, $\lambda <{\lambda }^{*}$) to systematically positive (Phase II, $\lambda \ge {\lambda }^{*}=1.64$), validating the threshold mechanism across spatial units.

Figure 6. Place-based robustness experiment ($n=20$ demand-point samples per $\lambda$). Panel A: Spatial distribution of $\Delta Z^{PB}$ at $\lambda =4.0$; warm colours indicate cost penalties from distance-only planning. Panel B: $\Delta Z^{PB}$ across demand levels; boxes shift from near-zero (Phase I, $\lambda <{\lambda }^{*}$) to systematically positive (Phase II, $\lambda \ge {\lambda }^{*}=1.64$), validating the threshold mechanism across spatial units.

7. Conclusions

This paper has documented and formalized a system transition not previously identified in the UAV logistics literature: infrastructure configurations begin adapting to airspace capacity constraints at demand levels where cost metrics register no anomaly. Structural change precedes the cost signal by a factor of 8.2× in demand density (${\lambda }^{*}/{\lambda }_{min}=1.64/0.20$): $D_{\mathrm{reloc}}$ is detectable at $\lambda =0.20{\mathrm{trips/km}}^2$ while the cost advantage is confirmed only from ${\lambda }^{*}=1.64{\mathrm{trips/km}}^2$ onward. Cost is a lagging indicator. The $\mathsf{\Omega }$ framework provides the missing leading indicator.

Four substantive findings follow. First, the two-phase regime transition is analytically grounded, not an empirical coincidence, as established in the complete proof of Proposition 2 (Appendix B) for any $\beta >0$. Second, the closed-form $\mathsf{\Omega }$ index provides a computationally cheap pre-screening tool requiring only three city-level parameters. Third, $D_{\mathrm{reloc}}$ is a more sensitive leading indicator of regime transition than any cost metric. Fourth, the threshold range is narrow and predictable across morphologically diverse Chinese urban districts (1.61–$1.85$, a 15% range), consistent with Proposition 1.

Several limitations motivate future work. The model considers a single operator; fixed $\left|O\right|=4$ depots; and FAR-based capacity as a geometric proxy for corridor navigability. The BPR intercept $\alpha =0.15$ [9] is not empirically varied (${\lambda }^{*}\propto {\alpha }^{-1/(\beta +1)}$; $\pm 20\%$ shifts ${\lambda }^{*}$ by $\pm 4\%$). The cross-district validation covers three Chinese districts; extension to Western city morphologies and full $\lambda$-sweeps for Pudong and Chaoyang remain open.

From a policy perspective, ${\lambda }^{*}$ provides a quantitative trigger for the six Chinese pilot cities with sub-600 m airspace authority to transition from registration-only depot approval to integrated airspace-logistics assessment, computable from standard GIS inputs and UOM flow records.

Future directions. The most direct extension is multi-operator competition: with N operators, corridor flows become Nash equilibria and $f^{*}$ may shift due to competitive over-utilisation of high-capacity corridors [16]. Microscopic validation—whether collision-avoidance UAV agents reproduce BPR-consistent macroscopic flows—would further strengthen operational realism.

Appendix A.1. Setup

Consider two depot candidates:

• Depot A: distance $d_A=d+\Delta d$, corridor capacity $C_A$
• Depot B: distance $d_B=d$, corridor capacity $C_B<C_A$

Setting $K\triangleq \tau \alpha /\left(60v\right)$ (units: h/m), the total operating cost of depot i serving demand flow f (trips/h) through its primary corridor is:

$${\mathrm{TC}}_i\left(f\right)=d_i·p+K·d_i·{\displaystyle \frac{f^{\beta +1}}{C_i^{\beta }}}$$

(A1)

where the first term is the distance-proportional flight cost and the second is the BPR-congestion cost.

Appendix A.2. Derivation of Ω(f)

Plan B selects Depot A when ${\mathrm{TC}}_A\left(f\right)<{\mathrm{TC}}_B\left(f\right)$. Expanding:

$$\begin{array}{cc}\hfill (d+\Delta d)·p+K(d+\Delta d){\displaystyle \frac{f^{\beta +1}}{C_A^{\beta }}}& <d·p+K·d·{\displaystyle \frac{f^{\beta +1}}{C_B^{\beta }}}\hfill \end{array}$$

Rearranging (moving distance terms left, congestion terms right):

$$\begin{array}{cc}\hfill K{f}^{\beta +1}\left[{\displaystyle \frac{d}{C_B^{\beta }}}-{\displaystyle \frac{d+\Delta d}{C_A^{\beta }}}\right]& >\Delta d·p\hfill \end{array}$$

Expanding the bracket and isolating the $\Delta d/C_A^{\beta }$ term:

$$\begin{array}{cc}\hfill K{f}^{\beta +1}\left[d·\Delta C-{\displaystyle \frac{\Delta d}{C_A^{\beta }}}\right]& >\Delta d·p-K{f}^{\beta +1}·{\displaystyle \frac{\Delta d}{C_A^{\beta }}}\hfill \end{array}$$

where $\Delta C\triangleq 1/C_B^{\beta }-1/C_A^{\beta }>0$. Adding $K{f}^{\beta +1}·\Delta d/C_A^{\beta }$ to both sides and factoring $\Delta d$:

$$\begin{array}{cc}\hfill K{f}^{\beta +1}·d·\Delta C& >\Delta d\left[p+{\displaystyle \frac{K{f}^{\beta +1}}{C_A^{\beta }}}\right]\hfill \end{array}$$

Dividing both sides by the right-hand bracket (positive for all $f\ge 0$):

$$\underset{{\displaystyle \mathsf{\Omega }\left(f\right)}}{\underbrace{{\displaystyle \frac{f^{\beta +1}·K·d·\Delta C}{\Delta d\left[p+f^{\beta +1}K/C_A^{\beta }\right]}}}}>1$$

(A2)

This establishes Equation (6). The economic interpretation is immediate: the numerator measures congestion savings from superior corridor capacity ($C_A$ vs $C_B$) over OD distance d; the denominator measures the distance cost of reaching the higher-capacity location ($\Delta d$) plus the residual congestion at that location.

Appendix A.3. Derivation of f *

Setting $\mathsf{\Omega }\left(f^{*}\right)=1$ and solving for $f^{*}$:

$$\begin{array}{cc}\hfill f^{*(\beta +1)}·K·d·\Delta C& =\Delta d\left[p+{\displaystyle \frac{f^{*(\beta +1)}K}{C_A^{\beta }}}\right]\hfill \\ \hfill f^{*(\beta +1)}·K\left[d·\Delta C-{\displaystyle \frac{\Delta d}{C_A^{\beta }}}\right]& =\Delta d·p\hfill \\ \hfill f^{*(\beta +1)}& ={\displaystyle \frac{\Delta d·p}{K\left[d·\Delta C-\Delta d/C_A^{\beta }\right]}}\hfill \end{array}$$

(A3)

Taking the $(\beta +1)$-th root:

$$f^{*}={\left({\displaystyle \frac{\Delta d·p}{K·(d·\Delta C-\Delta d/C_A^{\beta })}}\right)}^{1/(\beta +1)}$$

(A4)

which is Equation (7). The denominator is positive if and only if $d·\Delta C>\Delta d/C_A^{\beta }$ (the existence condition of Corollary 1). When this condition fails, $\mathsf{\Omega }\left(f\right)<1$ for all finite f and no threshold exists.

Appendix A.4. Monotonicity of Ω(f)

Differentiating $\mathsf{\Omega }\left(f\right)$ with respect to f:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathsf{\Omega }}{df}}& ={\displaystyle \frac{(\beta +1)f^{\beta }·K·d·\Delta C·\Delta d·p}{{\left[\Delta d\left(p+f^{\beta +1}K/C_A^{\beta }\right)\right]}^2}}>0\hfill \end{array}$$

(A5)

since all parameters are positive and $\Delta C>0$. Hence $\mathsf{\Omega }\left(f\right)$ is strictly increasing in f, crossing unity exactly once at $f=f^{*}$. This confirms that the regime transition is sharp rather than gradual.

Appendix A.5. Dongli District Numerical Values

Substituting the calibrated parameters (Table 5) with $K=\tau \alpha /\left(60v\right)=2.813\times {10}^{-4}$ h/m, $C_A^{\beta }={59}^4=12,117,361$, $C_B^{\beta }={48}^4=5,308,416$, and $\Delta C=1.059\times {10}^{-7}$:

$$f^{*}={\left({\displaystyle \frac{800\times 0.008}{2.813\times {10}^{-4}\times 2.516\times {10}^{-4}}}\right)}^{1/5}=39.0\mathrm{trips}/\mathrm{h},{\lambda }^{*}={\displaystyle \frac{39.0\times 4}{95}}=1.64{\mathrm{trips}/\mathrm{km}}^2.$$

Appendix B.1. Lemma and Part (i): D reloc (λ)>0 for all λ>0

Lemma A1.

For any two candidates $i,j\in \mathcal{I}$ with $d_i=d_j=d$ and $C_i>C_j$, Plan B strictly prefers i over j for any $f>0$, while Plan A is indifferent.

Proof.${\mathrm{TC}}_i\left(f\right)-{\mathrm{TC}}_j\left(f\right)=Kd{f}^{\beta +1}(C_i^{-\beta }-C_j^{-\beta })<0$ since $C_i>C_j\Rightarrow C_i^{-\beta }<C_j^{-\beta }$; Plan A evaluates both at $f=0$, yielding zero difference. □

Proof of Part (i). The capacity heterogeneity condition (std $>0$, confirmed for Dongli, Pudong, and Chaoyang with values 3.40, 2.18, and 4.42 trips/h respectively) guarantees at least one pair with $C_i\ne C_j$. By Lemma B.1, Plans A and B assign different optimality weights to such a pair for any $f>0$, producing different optimal depot sets and therefore $D_{\mathrm{reloc}}\left(\lambda \right)>0$. □

Appendix B.2. Part (ii): ΔZ(λ) strictly increasing

Let $d_A,C_A^{*}$ and $d_B,C_B^{*}$ denote the distance and capacity of the depots selected by Plans A and B respectively, with $C_B^{*}\ge C_A^{*}$ under Lemma B.1. Then:

$${\displaystyle \frac{d(\Delta Z)}{df}}=(\beta +1)K{f}^{\beta }\mathsf{\Gamma },\mathsf{\Gamma }\equiv {\displaystyle \frac{d_B}{{\left(C_B^{*}\right)}^{\beta }}}-{\displaystyle \frac{d_A}{{\left(C_A^{*}\right)}^{\beta }}}.$$

(B.1)

For Dongli: $d_A/C_A^{*\beta }=3000/{59}^4=2.48\times {10}^{-4}$, $d_B/C_B^{*\beta }=3800/{48}^4=7.16\times {10}^{-4}$, so $\mathsf{\Gamma }<0$ and $d(\Delta Z)/df>0$ for all $f>0$.

General equivalence.$\mathsf{\Gamma }<0$ iff $d_A/C_A^{*\beta }<d_B/C_B^{*\beta }$. Substituting $d_A=d$, $d_B=d+\Delta d$ and $\Delta C=C_B^{-\beta }-C_A^{-\beta }$ and rearranging yields exactly the Corollary 1 existence condition $d·\Delta C>\Delta d/C_A^{\beta }$, confirming that Parts (i)–(ii) share the same structural precondition. □

Appendix B.3. Part (iii): Phase I non-degeneracy

Asymptotic decomposition. From the BPR cost structure:

$$\Delta Z\left(f\right)=\underset{\mathrm{distance}\mathrm{premium}(\mathrm{const}.)}{\underbrace{-p\Delta d}}+\underset{\mathrm{congestion}\mathrm{saving}<0}{\underbrace{K\mathsf{\Gamma }{f}^{\beta +1}}},$$

(B.2)

where both terms are negative (Plan B pays a distance premium but saves more on congestion). The cost advantage ratio is:

$$V\left(\lambda \right)={\displaystyle \frac{-\Delta Z\left(f\right)}{{\mathrm{TC}}_A\left(f\right)}}={\displaystyle \frac{p\Delta d-K\mathsf{\Gamma }{f}^{\beta +1}}{N_{\mathrm{open}}{C}_f+n\overline{d}p+K\overline{d}{f}^{\beta +1}/{\left(C_A^{*}\right)}^{\beta }}}.$$

(B.3)

As $f\to 0$, the $O\left(f^5\right)$ congestion-saving term in the numerator vanishes while the denominator is dominated by the fixed cost $N_{\mathrm{open}}{C}_f>0$, giving the closed-form limit:

$$V\left(\lambda \right)\stackrel{\lambda \to 0}{\to }{\displaystyle \frac{p\Delta d}{N_{\mathrm{open}}{C}_f}}={\displaystyle \frac{0.008\times 800}{4\times 8,000}}=2.0\times {10}^{-4}\approx 0.02\%.$$

(B.4)

Finite-width Phase I. For any $\epsilon >0.02\%$, there exists ${\lambda }_{\epsilon }>0$ such that $V\left(\lambda \right)<\epsilon$ for all $\lambda \in [0,{\lambda }_{\epsilon })$. Empirically, ${\lambda }_{\epsilon }\approx 2.00{\mathrm{trips/km}}^2$ for $\epsilon =0.5\%$ (Section 6). Combining with Part (i), the interval $[0,{\lambda }_{\epsilon })$ constitutes a genuine Phase I: $D_{\mathrm{reloc}}>0$ throughout while $V<\epsilon$. The limit (B.4) is consistent with the observed $V=+0.41\%$ at $\lambda =0.20$ (Table 9): the gap between 0.02% and 0.41% reflects stochastic noise over $n=20$ replications and the non-zero $O\left(f^5\right)$ term at finite $\lambda =0.20$. The non-degeneracy holds for any $\beta >0$ and positive cost parameters; it is a structural consequence of the $O\left(f^{\beta +1}\right)$ vs. $O\left(f^0\right)$ scaling contrast between BPR congestion savings and fixed infrastructure costs. □