Beyond Average-Channel-Based Rate Approximations: UAV Trajectory and Scheduling Optimization With Expected Rate Consideration
This paper investigates the joint optimization of trajectory, user scheduling, and time-slot duration in unmanned aerial vehicle (UAV)-assisted wireless communication systems under minimum expected spectral efficiency (SE) constraints. Unlike most ex…
Authors: Gitae Park, Kisong Lee
1 Be yond A v erage-Channel-Based Rate Approximations: U A V T rajectory and Scheduling Optimization W ith Expected Rate Consideration Gitae Park and Kisong Lee, Senior Member , IEEE Abstract —This paper in vestigates the joint optimization of trajectory , user scheduling, and time-slot duration in unmanned aerial vehicle (U A V)-assisted wireless communication systems under minimum expected spectral efficiency (SE) constraints. Unlike most existing studies that approximate the expected SE by substituting the random channel gain with its mean value, thereby evaluating the SE at the average channel r ealization and over estimating the true expected SE due to Jensen’ s inequality , we approximate the expected SE by numerically integrating the SE o ver the channel distributions. Specifically , instead of relying on av erage-channel-based approximations, we develop a conservati ve yet tractable quadrature-based appr oximation by discretizing the associated cumulative distrib ution functions. The resulting finite-sum representation explicitly accounts for the probabilistic LoS structur e and channel fading effects, while remaining tractable for optimization. Leveraging this lower bound, we formulate a mission completion time minimization problem subject to minimum expected-SE requirements for all ground nodes. The resulting problem is a mixed-integer noncon vex optimization, which is tackled via a penalty-based block coordinate descent framework. The proposed algorithm alternately optimizes the scheduling decisions and the U A V trajectory along with adaptiv e time-slot durations, and maintains feasibility with r espect to the original expected-SE constraints by lev eraging successive conv ex approximation and quadratic transform techniques. Simulation results demonstrate that the proposed method strictly satisfies the minimum expected-SE con- straints and achieves a significantly shorter mission completion time than con ventional av erage-channel-based appr oaches, which are shown to yield infeasible or overly conservative solutions. Index T erms —Unmanned aerial vehicles, expected rate, trajec- tory design, resource allocation, con vex optimization. I . I N T RO D U C T I O N Recent advances in unmanned aerial vehicle (UA V) plat- forms have significantly expanded their role in wireless com- munications, driv en by rapid improvements in hardware capa- bility and substantial reductions in deployment cost [1], [2]. These dev elopments have positioned U A Vs as a promising architectural component for future wireless networks. Unlike con ventional terrestrial base stations, U A Vs possess inherent mobility in three-dimensional (3D) space, enabling flexible positioning that can be exploited to adapt network topology according to communication demands. By operating at ele- vated altitudes, U A Vs can mitigate the impact of obstacles such as buildings and terrain, resulting in more fav orable air- to-ground channel characteristics and improved link reliability The authors are with the Department of Information and Communica- tion Engineering, Dongguk Univ ersity , Seoul 04620, South Korea (e-mail: kslee851105@gmail.com). [3]. This mobility advantage is particularly beneficial in sce- narios where terrestrial infrastructure is una vailable, damaged, or economically infeasible. In such en vironments, U A Vs can act as aerial data collectors for ground nodes (GNs) distributed across remote or hard-to-access regions. The inherent mobility of U A Vs has prompted extensiv e research on the joint optimization of U A V trajectories and communication resources to improve wireless network per- formance [4]–[16]. Existing studies primarily focused on dif- ferent performance objecti ves, including co verage enhance- ment [4], [5], spectral efficienc y (SE) maximization [6], [7], and reliability improv ement under adverse channel conditions [8], [9]. Along this line of research, joint optimization of user scheduling, transmit po wer , and UA V trajectory was in vestigated to mitigate co-channel interference and improve system throughput [10], [11]. Cooperative UA V architectures were also explored, where base-station U A Vs coordinated with jammer U A Vs to enhance secure communications [12], [13]. More recently , U A V -assisted communication systems with GNs equipped with energy-harvesting capabilities attracted growing attention, in which U A V trajectories and communi- cation resources were jointly optimized to support wireless power transfer and data transmission [14]–[16]. A common underlying assumption in most of the aforementioned studies is the dominance of line-of-sight (LoS) air-to-ground links, under which simplified channel models were often adopted to maintain tractable optimization. Ho wev er , such assumptions are difficult to justify in practical deployment scenarios, where signal blockage caused by en vironmental obstacles is un- av oidable. This issue becomes particularly pronounced when U A Vs operate at relati vely low altitudes, as reduced elev ation angles significantly increase the likelihood of non-line-of-sight (NLoS) propagation [17]. T o address these limitations, probabilistic LoS channel models were proposed to characterize the mixed LoS/NLoS nature of air-to-ground communications by explicitly modeling the dependence of LoS probability on U A V altitude [17]. Using such models, recent studies have reexamined UA V trajectory and resource allocation problems under more realistic channel conditions [18]–[28]. In particular , some studies focused on U A V trajectory design under energy-related constraints or ob- jectiv es, jointly accounting for propulsion ener gy consumption and communication performance [18]–[20]. Others in vesti- gated time-constrained UA V operations, where trajectories and radio resources are optimized to satisfy latency or mission completion requirements [21], [22]. In [23], UA V -enabled 2 data harvesting was studied using adaptiv e offline–online trajectory design under time-varying channel conditions, while [24] considered trajectory optimization to enhance robustness against jamming attacks. More recent studies e xtended these designs to multi-U A V systems, where coordination among multiple U A Vs is exploited to improv e fairness, interference management, or overall system performance [25], [26]. Proba- bilistic LoS channel models were also adopted in U A V -enabled wireless-powered networks, where the U A V primarily serves as an energy transmitter and trajectory design is coupled with power transfer strategies to support information deliv ery [27], [28]. Despite these advances under probabilistic LoS channel models, the expected SE in many existing studies [18]– [28] is still approximated by substituting the instantaneous channel with its mean value to simplify the optimization problem. As a result, such average-channel-based approxima- tions systematically overestimate the true expected SE due to Jensen’ s inequality . While this practice is computation- ally con venient, its implications for system reliability hav e largely been ov erlooked, particularly in joint U A V trajectory and resource optimization, where inaccuracies in expected-SE characterization can distort the resulting UA V strate gy and compromise reliability . These observations motiv ate the devel- opment of a principled expected-SE modeling framework that enables reliable optimization under probabilistic propagation conditions. The contributions of our work can be summarized as follows: • W e identify a fundamental limitation of con ventional av erage-channel-based SE approximations, which sys- tematically overestimate expected SEs and may lead to unreliable system designs under stochastic channel conditions. T o the best of our knowledge, this work is the first to dev elop a conservati ve and tractable expected- SE lo wer bound based on cumulativ e distribution func- tion (CDF)-domain discretization and quadrature-based reformulation, enabling reliable constraint enforcement in UA V -assisted communications with probabilistic LoS channels. • Le veraging the proposed expected-SE characterization, we formulate a joint optimization problem of U A V trajectory , user scheduling, and time-slot duration, and dev elop a penalty-based block coordinate descent algo- rithm. By combining successiv e con vex approximation (SCA) and quadratic transform techniques, the proposed algorithm efficiently handles the coupled noncon vexities and achiev es efficient conv ergence with polynomial-time computational complexity . • The results establish that accurate expected-SE modeling is a foundational requirement for reliable U A V commu- nication design under probabilistic LoS channels. By en- suring feasibility at the modeling stage, the proposed for- mulation defines an optimization region consistent with the original expected-SE constraints, thereby enabling reliable and practically meaningful U A V communication design. The remainder of this paper is organized as follows. Section GN k GN 1 GN 2 𝜃 LoS/NLoS component s with fading Scheduling S T rajectory Q Slot-duration ∆ Fig. 1. System model for a UA V -assisted wireless communication system. II describes the system model and formulates the mission completion time minimization problem under minimum ex- pected SE constraints. Section III discusses the fundamental limitations of av erage-channel-based rate approximations and dev elops a conservati ve and tractable expected-SE lower - bound formulation. Section IV then presents the proposed joint optimization framew ork for UA V trajectory , scheduling, and time-slot duration. Section V provides simulation results and discussions, and Section VI concludes the paper . I I . S Y S T E M M O D E L A N D P RO B L E M S TA T E M E N T A. Constraints on trajectory and communication r esources As illustrated in Fig. 1, we consider a U A V -assisted wireless communication system in which a rotary-wing UA V collects data from K GNs, inde xed by k ∈ K = { 1 , 2 , . . . , K } . The U A V mission duration is discretized into N time slots, indexed by n ∈ N = { 1 , 2 , . . . , N } , where δ [ n ] denotes the duration of time slot n . The total mission completion time is thus gi ven by T = P N n =1 δ [ n ] . Assuming sufficiently small time-slot durations, the U A V position is considered constant within each slot [10], therefore, the slot length is constrained as 0 ≤ δ [ n ] ≤ δ max , ∀ n, (1) where δ max is chosen sufficiently small such that the wireless channel can be assumed to be flat fading within each time slot. For uplink data transmission, time-division multiple access (TDMA) is employed, such that at most one GN is served in each time slot. During time slot n , the UA V position is given by q [ n ] = ( x [ n ] , y [ n ] , z [ n ]) , while GN k remains fixed at w k = ( x k , y k , 0) . The U A V is required to travel from q I to q F while maintaining an operational altitude within [ H min , H max ] during mission period. Its mobility is constrained by a maximum 3D speed V max and a maximum vertical speed V z , with V max ≥ V z [23]. Accordingly , the per-slot U A V displacement is limited by δ [ n ] V max in space and δ [ n ] V z in altitude. The UA V mobility 3 is subject to the follo wing constraints: q [0] = q I , q [ N ] = q F , (2) H min ≤ [ q [ n ]] 3 ≤ H max , ∀ n, (3) q [ n ] − q [ n − 1] ≤ V max δ [ n ] , ∀ n, (4) [ q [ n ]] 3 − [ q [ n − 1]] 3 ≤ V z δ [ n ] , ∀ n. (5) Let s k [ n ] ∈ { 0 , 1 } denote the scheduling indicator for GN k at time slot n , where s k [ n ] = 1 indicates that GN k is selected for uplink transmission in slot n , and s k [ n ] = 0 otherwise. Moreov er , at most one GN can be scheduled in each time slot, which is expressed as s k [ n ] ∈ { 0 , 1 } , ∀ k , n, (6) K X k =1 s k [ n ] ≤ 1 , ∀ n. (7) B. Pr obabilistic LoS Channel Model T o accurately model the air-to-ground channel in UA V com- munications, we employ a probabilistic LoS channel model, which is particularly suitable for U A V systems. In contrast to fixed base station deployments, this model accounts for the UA V’ s ability to increase the probability of LoS links by elev ating its altitude, thereby coupling the LoS/NLoS occurrence probabilities with the U A V’ s ele vation angle. The LoS probability of the UA V –GN link is determined by an en vironment-dependent statistical model that captures the effect of building density and the UA V position. Let c k [ n ] ∈ { 0 , 1 } denote the corresponding channel state at time slot n , where c k [ n ] = 1 and c k [ n ] = 0 indicate LoS and NLoS links, respectiv ely . According to the probabilistic LoS model in [18], the LoS probability of the channel between the U A V and GN k at time slot n is represented by P ( c k [ n ] = 1) ≜ P L k [ n ] = 1 1 + A 1 e − A 2 ( θ k [ n ] − A 1 ) , (8) where A 1 and A 2 are en vironment-dependent positive con- stants. The elev ation angle between the UA V and GN k at time slot n , denoted by θ k [ n ] ∈ [0 , π 2 ] , is giv en by θ k [ n ] = 180 π arcsin [ q [ n ]] 3 ∥ q [ n ] − w k ∥ , ∀ k , n. (9) Giv en the LoS probability in (8), the corresponding NLoS probability is defined as P ( c k [ n ] = 0) ≜ P N k [ n ] = 1 − P L k [ n ] . Accordingly , the channel po wer gain between the UA V and GN k at time slot n is modeled as h k [ n ] = c k [ n ] h L k [ n ] + (1 − c k [ n ]) h N k [ n ] , (10) where h L k [ n ] and h N k [ n ] denote the channel power gain for the LoS and NLoS conditions, respecti vely , which are gi ven by h L k [ n ] = | g L k [ n ] | 2 β L ∥ q [ n ] − w k ∥ α L , (11) h N k [ n ] = ν k [ n ] | g N k [ n ] | 2 β N ∥ q [ n ] − w k ∥ α N . (12) Here, ( α L , β L ) and ( α N , β N ) denote the path-loss exponents and reference channel gains for the LoS and NLoS links, respectiv ely , with α L < α N and β L > β N [23]. The small- scale fading coefficients under LoS and NLoS conditions are denoted by g L k [ n ] and g N k [ n ] , respectiv ely , while ν k [ n ] represents the large-scale shadowing gain. The LoS small-scale fading g L k [ n ] is modeled by a Rician distribution as g L k [ n ] = r K R K R + 1 e j ϕ k [ n ] + r 1 K R + 1 ˜ g k [ n ] , (13) where K R denotes the Rician K -factor in linear scale, ϕ k [ n ] is the LoS phase shift, and ˜ g k [ n ] ∼ C N (0 , 1) represents the scattered component. In contrast, the NLoS small-scale fading is assumed to follow a standard Rayleigh distribution, i.e., g N k [ n ] ∼ C N (0 , 1) . Moreov er , the NLoS channel gain in (12) incorporates the shadowing effect through ν k [ n ] . T o ensure a unit-mean shad- owing gain, i.e., E [ ν k [ n ]] = 1 , ν k [ n ] is modeled as a bias- corrected log-normal random variable giv en by ν k [ n ] = 10 ν dB k [ n ] − ∆ dB 10 , (14) where ν dB k [ n ] ∼ N (0 , σ 2 dB ) denotes the shadowing component in the dB scale with standard deviation σ dB , and ∆ dB = ln 10 20 σ 2 dB is the bias-correction term. C. Pr oblem F ormulation Giv en the channel po wer gain in (10), the uplink SE for GN k at time slot n is r k [ n ] = log 2 1 + P S h k [ n ] σ 2 , (15) where P S denotes the constant transmit power of GNs and σ 2 is the noise power . Since the link state is either LoS or NLoS, r k [ n ] can be equiv alently written as r k [ n ] = c k [ n ] r L k [ n ] + (1 − c k [ n ]) r N k [ n ] , (16) where r L k [ n ] = log 2 1 + P S h L k [ n ] σ 2 and r N k [ n ] = log 2 1 + P S h N k [ n ] σ 2 . Under the probabilistic LoS channel model, the expected uplink SE can be expressed as follo ws [18]: E [ r k [ n ]] = P L k [ n ] E | g L k [ n ] | 2 r L k [ n ] + P N k [ n ] E ν k [ n ] , | g N k [ n ] | 2 r N k [ n ] , (17) where the expectations are tak en ov er the corresponding small- scale fading and shadowing random variables under the LoS and NLoS conditions. The resulting time-averaged uplink SE of GN k is giv en by R k = 1 T N X n =1 s k [ n ] δ [ n ] E [ r k [ n ]] , ∀ k . (18) Our objectiv e is to minimize the UA V mission completion time T = P N n =1 δ [ n ] while guaranteeing a minimum required SE R min for each GN. T o this end, we jointly optimize the scheduling variables S ≜ { s k [ n ] , ∀ k , n } , the U A V 3D trajectory Q ≜ { q [ n ] , ∀ n } , and the time-slot durations ∆ ≜ { δ [ n ] , ∀ n } . The resulting optimization problem is formulated 4 as follows. (P0): min S , Q , ∆ T subject to R k ≥ R min , ∀ k , (19) (1) − (7) . Problem (P0) is a mixed-inte ger noncon vex problem due to the binary scheduling variables S and the nonconv exity of constraint (19) with respect to the optimization variables. Moreov er , the expected SE expression in (19) requires ev aluat- ing expectations ov er random fading and shadowing variables, which leads to integral expressions that do not admit closed- form solutions. As a result, obtaining the globally optimal solution to (P0) is highly challenging. I I I . B E Y O N D A V E R AG E - C H A N N E L - B A S E D R AT E A P P RO X I M A T I O N S This section examines the limitations of av erage-channel- based approximations for expected SE modeling and in- troduces a quadrature-based lo wer-bound reformulation. The proposed approach approximates the expected SE under prob- abilistic LoS channels via a tractable finite-sum representation that is amenable to optimization. A. Limitations of A verage-Channel-Based Rate Appr oxima- tions In of fline optimization, the achie vable SE in volv es an expec- tation over random channel effects such as small-scale fading and shado wing. Ho wev er, directly handling the expected SE is challenging, as it generally admits only an integral-form expression without a closed-form representation. T o address this difficulty , most existing works [18]–[28] approximate the expected SE by substituting the random channel power gains h L k [ n ] and h N k [ n ] with their mean values, which yields an av erage-channel-based SE expression given by ¯ r k [ n ] ≜ P L k [ n ] log 2 1 + P S E | g L k [ n ] | 2 h L k [ n ] σ 2 ! + P N k [ n ] log 2 1 + P S E ν k [ n ] , | g N k [ n ] | 2 h N k [ n ] σ 2 ! . (20) Since the SE function is concav e with respect to the channel power gains, Jensen’ s inequality implies that the resulting av erage-channel-based SE provides an upper bound on the actual expected SE, i.e., E [ r k [ n ]] ≤ ¯ r k [ n ] . (21) As a consequence of Jensen’ s inequality , replacing the expected SE in constraint (19) by its av erage-channel-based approximation yields 1 T N X n =1 s k [ n ] δ [ n ] E [ r k [ n ]] ≤ 1 T N X n =1 s k [ n ] δ [ n ] ¯ r k [ n ] ≜ ¯ R k . (22) T o conservati vely satisfy the expected-SE constraint in (19), the expected SE must be replaced by a lo wer bound. Howe ver , the av erage-channel-based SE serves only as an estimate of the actual expected SE and, due to Jensen’ s inequality , pro vides an upper bound on it. As a result, e ven if the estimated SE satisfies the constraint, i.e., ¯ R k ≥ R min , the actual expected SE R k may still fall below R min , leading to violations of (19) in practice. This issue is particularly critical for mission completion time minimization problems, such as the one considered in this study , where the optimal solution typically lies near the boundary of the feasible set. Optimizing ov er such a relaxed superset therefore substantially increases the risk of violating the original expected-SE constraints. Motiv ated by this observation, we depart from the commonly adopted av erage-channel-based SE approximation and instead seek a conservati ve yet tractable reformulation of the actual expected SE. Specifically , our objecti ve is to construct a computable lower bound on E [ r k [ n ]] that can be directly embedded into the subsequent optimization frame work while faithfully capturing the impact of channel randomness. B. Quadratur e-Based Lower-Bound Reformulation T o construct such a conservati ve and tractable lower bound, we first rewrite the expected SE E [ r k [ n ]] in (17) into an explicit expectation form as E [ r k [ n ]] = P L k [ n ] E | g L k [ n ] | 2 log 2 1 + P S | g L k [ n ] | 2 β L σ 2 ∥ q [ n ] − w k ∥ α L + P N k [ n ] E ν k [ n ] , | g N k [ n ] | 2 log 2 1 + P S ν k [ n ] | g N k [ n ] | 2 β N σ 2 ∥ q [ n ] − w k ∥ α N . (23) Under LoS conditions, the small-scale fading coef ficient fol- lows a Rician distribution, and thus the corresponding power gain | g L k [ n ] | 2 follows a noncentral chi-square distribution pa- rameterized by the Rician K -factor . Under NLoS conditions, the small-scale fading coef ficient follows a Rayleigh distri- bution, such that | g N k [ n ] | 2 is exponentially distributed, while the shadowing gain ν k [ n ] is modeled as a log-normal random variable. Noting that E [ | g L k [ n ] | 2 ] = E [ | g N k [ n ] | 2 ] = E [ ν k [ n ]] = 1 and that ν k [ n ] and g N k [ n ] are statistically independent, the joint distribution associated with the NLoS term can be factorized. Consequently , (23) can be expressed in the following integral form: E [ r k [ n ]] = P L k [ n ] Z ∞ 0 f | g L k [ n ] | 2 ( x ) log 2 1 + xζ L k [ n ] dx + P N k [ n ] Z ∞ 0 Z ∞ 0 f | g N k [ n ] | 2 ( y ) f ν k [ n ] ( z ) log 2 1 + y z ζ N k [ n ] dy dz , (24) where ζ L k [ n ] = P S β L σ 2 ∥ q [ n ] − w k ∥ α L and ζ N k [ n ] = P S β N σ 2 ∥ q [ n ] − w k ∥ α N . Moreov er , f | g L k [ n ] | 2 ( · ) , f | g N k [ n ] | 2 ( · ) , and f ν k [ n ] ( · ) denote the probability density functions (PDFs) of | g L k [ n ] | 2 , | g N k [ n ] | 2 , and ν k [ n ] , respectiv ely . It is worth noting that the integral expression in (24) does not admit a closed-form solution in general, which mak es it dif ficult to incorporate directly into the optimization problem. T o overcome this challenge, we employ a numerical quadrature method to deriv e a tractable finite-sum lower bound for (24), which can be efficiently ev aluated and embedded into the proposed optimization framework. T o obtain a tractable finite-sum representation of the ex- pected SE, we discretize the random channel components in 5 both LoS and NLoS links. W e first describe the discretization procedure for the LoS small-scale fading term | g L k [ n ] | 2 . The expected SE inv olves an integral over the PDF of | g L k [ n ] | 2 . A straightforward approach would be to uniformly partition the | g L k [ n ] | 2 domain along the x -axis and approximate the integral accordingly . Howe ver , such a uniform partition may lead to large approximation errors, since the probability density of | g L k [ n ] | 2 can vary significantly across dif ferent regions. T o address this issue, we adopt a CDF-domain discretization strat- egy . Specifically , instead of uniformly partitioning the | g L k [ n ] | 2 domain along the x -axis, we uniformly partition the CDF range [0 , 1] of | g L k [ n ] | 2 into U L sub-intervals. This approach ensures that each interval contains an equal probability mass, thereby allocating finer resolution to regions where the fading realizations are more lik ely to occur . Let u ∈ { 1 , . . . , U L } index the CDF intervals, and define the u -th interv al as u − 1 U L , u U L . The representative fading po wer at the left endpoint of each interval is obtained via the in verse CDF of | g L k [ n ] | 2 , giv en by γ L u ≜ F − 1 | g L k [ n ] | 2 u − 1 U L , ∀ u, (25) where F − 1 | g L k [ n ] | 2 ( · ) denotes the in verse CDF of the noncentral chi-square distrib ution associated with the Rician f ading po wer gain. Since the SE function is monotonically increasing with respect to the fading power gain | g L k [ n ] | 2 , the SE ev aluated at γ L u serves as a lower bound for all fading realizations within the corresponding CDF interval. As a result, approximating the expected SE by ev aluating the SE function at these left-endpoint quantiles yields a conservati ve finite-sum lower bound: Z ∞ 0 f | g L k [ n ] | 2 ( x ) log 2 1 + xζ L k [ n ] dx ≥ 1 U L U L X u =1 log 2 1 + γ L u ζ L k [ n ] . (26) W ith this CDF-domain partitioning, we next extend the same discretization principle to the NLoS case, which inv olves both the small-scale fading power gain | g N k [ n ] | 2 and the shad- owing gain ν k [ n ] . Specifically , we uniformly partition the CDF range [0 , 1] of | g N k [ n ] | 2 into U N sub-intervals and that of ν k [ n ] into U ν sub-intervals. Let i ∈ { 1 , . . . , U N } and j ∈ { 1 , . . . , U ν } index the corresponding CDF intervals, defined as i − 1 U N , i U N and j − 1 U ν , j U ν , respectiv ely . The representativ e sample points associated with the left endpoints of these CDF intervals are then obtained via the in verse CDFs as γ N i ≜ F − 1 | g N k [ n ] | 2 i − 1 U N , ∀ i, (27) γ ν j ≜ F − 1 ν k [ n ] j − 1 U ν , ∀ j, (28) where F − 1 | g N k [ n ] | 2 ( · ) and F − 1 ν k [ n ] ( · ) represent the inv erse CDF of the NLoS small-scale fading power gain, which follows an exponential distrib ution, and the shadowing gain, which follows a log-normal distribution, respectiv ely . Using the discretized sample points defined in (25), (27), and (28), we construct a computable lo wer bound on the expected SE of GN k as follows. r k [ n ] = P L k [ n ] r L k [ n ] + P N k [ n ] r N k [ n ] , (29) where r L k [ n ] and r N k [ n ] are giv en by r L k [ n ] ≜ 1 U L U L X u =1 log 2 1 + P S β L γ L u σ 2 ∥ q [ n ] − w k ∥ α L , (30) r N k [ n ] ≜ 1 U N 1 U ν U N X i =1 U ν X j =1 log 2 1 + P S β N γ N i γ ν j σ 2 ∥ q [ n ] − w k ∥ α N ! . (31) W ith the tractable lo wer bound in (29), we reformulate prob- lem (P0) by replacing the expected SE E [ r k [ n ]] in constraint (19) with its conservati ve lower bound r k [ n ] , which yields the following problem: (P1): min S , Q , ∆ T subject to 1 T N X n =1 s k [ n ] δ [ n ] r k [ n ] ≥ R min , ∀ k , (32) (1) − (7) . I V . P R O P O S E D O P T I M I Z AT I O N F R A M E W O R K Although (P1) admits a tractable expected-SE representa- tion, it is still challenging to solve directly , as feasibility with respect to the minimum-SE constraint (32) must be ensured from the initial stage of the optimization. In general, finding an initial point ( S , Q , ∆ ) that satisfies the minimum-SE constraint for all GNs is dif ficult, and nai ve initializations often lead to constraint violations. T o address this issue, we adopt a penalty con ve x–concav e procedure by introducing a non- negati ve slack variable ρ ≥ 0 , which relaxes the minimum-SE constraint during the early iterations. Specifically , constraint (32) is modified as 1 T N X n =1 s k [ n ] δ [ n ] r k [ n ] ≥ R min − ρ, ∀ k. (33) By penalizing the slack variable ρ in the objectiv e function, problem (P1) is reformulated into the follo wing penalized problem: (P2): min S , Q , ∆ , ρ ≥ 0 T + η ρ subject to (1) − (7) , (33) . Here, the slack variable ρ explicitly quantifies the violation of (32) through the relaxed constraint (33). The parameter η > 0 is a penalty weight that controls the impact of the constraint- violation slack on the objectiv e v alue. By incorporating the penalty term η ρ into the objectiv e function, any violation of the minimum-SE constraint is penalized in proportion to ρ . F or small values of η , the optimization is allowed to temporarily violate the original minimum-SE constraint, which facilitates the construction of an initial feasible solution. As η increases, the penalty on ρ becomes more severe, driving ρ tow ard zero and thereby recovering feasibility with respect to the original minimum-SE constraint. Problem (P2) remains challenging to solve due to its in- herent noncon vexity . T o address this issue, we decompose 6 (P2) into two subproblems with respect to the scheduling variables and the joint U A V trajectory and time-slot duration variables, respecti vely . Each subproblem is then transformed into a con ve x optimization problem using appropriate con- ve xification techniques. These subproblems can be efficiently solved using standard con vex optimization solvers such as CVX [29], and are alternately updated via a block coordinate descent framework, as detailed in the following subsections. A. Scheduling Optimization By relaxing the binary scheduling variables s k [ n ] to the continuous interv al [0 , 1] , the scheduling subproblem for fixed Q and ∆ can be expressed as (SP1): min S , ρ ≥ 0 T + η ρ subject to 0 ≤ s k [ n ] ≤ 1 , ∀ k, n, (34) (7) , (33) . Problem (SP1) is a linear program and can be ef ficiently solved using CVX. After obtaining the relax ed solution, a binary scheduling policy can be recovered using the approach in [10], without loss of optimality . B. T rajectory and T ime-Slot Duration Optimization For fixed S , the subproblem of jointly optimizing the U A V trajectory Q and the time-slot duration ∆ is formulated as (SP2): min Q , ∆ , ρ ≥ 0 T + η ρ subject to (1) − (5) , (33) . Problem (SP2) is challenging to solve due to the nonconv exity of the minimum-SE constraint (33). In particular , one of the main difficulties arises from the LoS probability P L k [ n ] , which is expressed as a sigmoid function of the elev ation angle θ k [ n ] . Since the elev ation angle itself is gi ven by an arcsin( · ) mapping of the U A V position q [ n ] , P L k [ n ] becomes a highly nonconv ex function of the trajectory variables, making the associated constraints difficult to handle directly . T o decouple this nonlinearity , we introduce an auxiliary variable θ k [ n ] that serves as a lo wer bound on the actual elev ation angle, i.e., θ k [ n ] ≤ 180 π arcsin [ q [ n ]] 3 ∥ q [ n ] − w k ∥ , ∀ k , n. (35) Since the LoS probability P L k [ n ] is monotonically increasing with respect to the ele vation angle, substituting θ k [ n ] for θ k [ n ] yields a conservati ve lower bound: P L k [ n ] ≥ 1 1 + A 1 e − A 2 ( θ k [ n ] − A 1 ) ≜ P L k [ n ] . (36) Moreov er , since β L > β N and α N > α L , the achie vable SE under LoS conditions dominates that under NLoS conditions, i.e., r L k [ n ] ≥ r N k [ n ] . As a result, the expected SE lower bound r k [ n ] , which is a weighted combination of the LoS and NLoS SEs, is monotonically increasing with respect to P L k [ n ] . By replacing P L k [ n ] with P L k [ n ] in (29), we obtain a conservati ve lower bound of r k [ n ] : r k [ n ] ≥ P L k [ n ] r L k [ n ] + (1 − P L k [ n ]) r N k [ n ] ≜ ˆ r k [ n ] . (37) As such, ˆ r k [ n ] resolv es the inherent nonconv exity arising from the probabilistic LoS model and its coupling with the U A V trajectory . The resulting expression provides a computation- ally tractable conservati ve lo wer bound on the e xpected SE, which can be directly embedded into the subsequent trajectory optimization problem. Howe ver , constraint (35), which is newly introduced by the auxiliary variable θ k [ n ] , remains difficult to handle due to the presence of the arcsin( · ) function. T o eliminate it, we apply the sine function to both sides, which yields sin π 180 θ k [ n ] ≤ [ q [ n ]] 3 ∥ q [ n ] − w k ∥ , ∀ k , n. (38) Since sin( · ) is monotonically increasing ov er [0 , π 2 ] , the in- equality direction is preserved. Despite this transformation, constraint (38) remains non- con vex. Noting that the left-hand side of (38) is concav e with respect to θ k [ n ] , it can be upper-bounded by its first-order T aylor expansion as follows: sin π 180 θ k [ n ] ≤ sin π 180 θ prev k [ n ] + π 180 cos π 180 θ prev k [ n ] ( θ k [ n ] − θ prev k [ n ]) , (39) where θ prev k [ n ] is the v alue of θ k [ n ] at the previous SCA iteration. Meanwhile, the right-hand side of (38) has a concav e–over – con vex fractional structure. Under the conditions [ q [ n ]] 3 ≥ 0 and ∥ q [ n ] − w k ∥ > 0 , we apply the quadratic transform [30] to obtain the following concave lo wer bound: [ q [ n ]] 3 ∥ q [ n ] − w k ∥ ≥ 2 λ k [ n ] p [ q [ n ]] 3 − λ 2 k [ n ] ∥ q [ n ] − w k ∥ , (40) where λ λ λ ≜ { λ k [ n ] , ∀ k , n } is a set of auxiliary variables. Combining (39) and (40), constraint (38) can be conserva- tiv ely approximated by the following conv ex set: sin π 180 θ prev k [ n ] + π 180 cos π 180 θ prev k [ n ] ( θ k [ n ] − θ prev k [ n ]) ≤ 2 λ k [ n ] p [ q [ n ]] 3 − λ 2 k [ n ] ∥ q [ n ] − w k ∥ , ∀ k , n, (41) which guarantees the feasibility of the original constraint (35). In addition, since the right-hand side of (41) is concave in λ k [ n ] , its optimal value is obtained by differentiation as λ ∗ k [ n ] = p [ q [ n ]] 3 ∥ q [ n ] − w k ∥ , ∀ k , n. (42) The lower bound ˆ r k [ n ] in (37) inv olves both the LoS probability P L k [ n ] and its complementary term 1 − P L k [ n ] , which results in a coupled and highly noncon vex structure. In partic- ular , while P L k [ n ] follo ws a sigmoid form, its complement does not share the same functional structure, which complicates the subsequent con vexification. T o address this issue, we first establish the following lemma, which allo ws 1 − P L k [ n ] to be equiv alently expressed in a symmetric sigmoid form. 7 Lemma 1. F or the function f ( x ) = 1 − 1 1+ ae − b ( x − a ) , an equiv- alent expr ession is given by f ( x ) = 1 1+ ae − b (2( a +ln( a ) /b ) − x − a ) . Pr oof: Starting fr om the definition of f ( x ) , we obtain f ( x ) = 1 1 + e b ( x − a ) − ln a . (43) Let x s = a + ln a b , so that b ( x − a ) − ln a = b ( x − x s ) . Hence, f ( x ) = 1 1 + e b ( x − x s ) . (44) Mor eover , using bx s = ba + ln a , e b ( x − x s ) = ae − b (2 x s − x − a ) . (45) Substituting this identity into the denominator yields f ( x ) = 1 1 + ae − b (2 x s − x − a ) , (46) which completes the proof . Using Lemma 1 , 1 − P L k [ n ] can be expressed in the follo wing form that is symmetric to P L k [ n ] : 1 − P L k [ n ] = 1 1 + A 1 e − A 2 (2 x s − θ k [ n ] − A 1 ) , (47) where x s = A 1 + ln( A 1 ) A 2 . This symmetric representation enables both the LoS and NLoS components of ˆ r k [ n ] to be expressed in a unified sigmoid-based form with respect to the elevation angle. There- fore, ˆ r k [ n ] can be expressed as follows: ˆ r k [ n ] = 1 1 + A 1 e − A 2 ( θ k [ n ] − A 1 ) 1 U L × U L X u =1 log 2 1 + P S β L γ L u σ 2 ∥ q [ n ] − w k ∥ α L + 1 1 + A 1 e − A 2 (2 x s − θ k [ n ] − A 1 ) 1 U N U ν × U N X i =1 U ν X j =1 log 2 1 + P S β N γ N i γ ν j σ 2 ∥ q [ n ] − w k ∥ α N ! . (48) T o enable a tractable con vex approximation of ˆ r k [ n ] , it is necessary to characterize the conv exity of the SE terms with respect to the optimization variables. The follo wing lemma provides a useful con ve xity result that will be exploited in the subsequent reformulation. Lemma 2. F or given c ≥ 0 and α ∈ [2 , 6] , the function f ( x, y ) = 1 x log 2 (1 + c y α/ 2 ) is con vex for x > 0 and y > 0 . Pr oof: By dir ect second-order analysis [23], it can be verified that for any nonzer o vector t , t T ∇ 2 f ( x, y ) t ≥ 0 holds for x > 0 and y > 0 with c ≥ 0 and α ∈ [2 , 6] , which establishes the conve xity of f ( x, y ) . According to Lemma 2 , ˆ r k [ n ] is a con vex function with respect to the variables (1 + A 1 e − A 2 ( θ k [ n ] − A 1 ) ) , (1 + A 1 e − A 2 (2 x s − θ k [ n ] − A 1 ) ) , and ∥ q [ n ] − w k ∥ 2 . Therefore, by applying the first-order T aylor approximation of ˆ r k [ n ] with respect to these v ariables, a lower bound of ˆ r k [ n ] can be obtained as follows: ˆ r k [ n ] ≥ ˆ r prev k [ n ] − 1 U L U L X u =1 ψ L , prev k,u [ n ]( X L k [ n ] − X L , prev k [ n ]) − 1 U N U ν U N X i =1 U ν X j =1 ψ N , prev k,i,j [ n ]( X N k [ n ] − X N , prev k [ n ]) − 1 U L U L X u =1 χ L , prev k,u [ n ]( ∥ q [ n ] − w k ∥ 2 − ∥ q prev [ n ] − w k ∥ 2 ) − 1 U N U ν U N X i =1 U ν X j =1 χ N , prev k,i,j [ n ]( ∥ q [ n ] − w k ∥ 2 − ∥ q prev [ n ] − w k ∥ 2 ) ≜ ˆ r LB k [ n ] , (49) where X L k [ n ] = 1 + A 1 e − A 2 ( θ k [ n ] − A 1 ) , X L , prev k [ n ] = 1 + A 1 e − A 2 ( θ prev k [ n ] − A 1 ) , X N k [ n ] = 1 + A 1 e − A 2 (2 x s − θ k [ n ] − A 1 ) , X N , prev k [ n ] = 1 + A 1 e − A 2 (2 x s − θ prev k [ n ] − A 1 ) , and q prev [ n ] repre- sents the UA V trajectory at time slot n for the previous SCA iteration. Moreover , ˆ r prev k [ n ] , ψ L , prev k,u [ n ] , ψ N , prev k,i,j [ n ] , χ L , prev k,u [ n ] , and χ N , prev k,i,j [ n ] are defined as follo ws: ˆ r prev k [ n ] = 1 X L , prev k [ n ] U L U L X u =1 log 2 1 + Γ L u ( y prev k [ n ]) α L / 2 + 1 X N , prev k [ n ] U N U ν U N X i =1 U ν X j =1 log 2 1 + Γ N i,j ( y prev k [ n ]) α N / 2 ! , (50) ψ L , prev k,u [ n ] = 1 ( X L , prev k [ n ]) 2 log 2 1 + Γ L u ( y prev k [ n ]) α L / 2 , (51) ψ N , prev k,i,j [ n ] = 1 ( X N , prev k [ n ]) 2 log 2 1 + Γ N i,j ( y prev k [ n ]) α N / 2 ! , (52) χ L , prev k,u [ n ] = 1 X L , prev k [ n ] α L Γ L u log 2 e 2 y prev k [ n ](( y prev k [ n ]) α L / 2 + Γ L u ) , (53) χ N , prev k,i,j [ n ] = 1 X N , prev k [ n ] α N Γ N i,j log 2 e 2 y prev k [ n ](( y prev k [ n ]) α N / 2 + Γ N i,j ) , (54) where Γ L u = P S β L γ L u σ 2 , Γ N i,j = P S β N γ N i γ ν j σ 2 , and y prev k [ n ] = ∥ q prev [ n ] − w k ∥ 2 . Using (49), the minimum-SE constraint (33) can be con- servati vely enforced by replacing r k [ n ] with its lower bound ˆ r LB k [ n ] , yielding 1 T N X n =1 s k [ n ] δ [ n ] ˆ r LB k [ n ] ≥ R min − ρ, ∀ k. (55) Constraint (55) is still noncon vex due to the bilinear term δ [ n ] ˆ r LB k [ n ] . T o handle this coupling, we rewrite the product in an equiv alent concave–o ver –con vex fractional form: δ [ n ] ˆ r LB k [ n ] = ˆ r LB k [ n ] 1 /δ [ n ] , (56) where ˆ r LB k [ n ] is conca ve with respect to the trajectory vari- ables, whereas 1 /δ [ n ] is conv ex in δ [ n ] . Moreover , since ˆ r LB k [ n ] is nonnegati ve by construction, applying the quadratic transform only requires 1 /δ [ n ] > 0 . This condition can be 8 ensured by imposing the mild constraint: δ [ n ] ≥ δ min , ∀ n, (57) where δ min > 0 is a small positi ve constant. W ith (57), we apply the quadratic transform to ˆ r LB k [ n ] 1 /δ [ n ] and obtain the following concave lo wer bound of (55): 1 T N X n =1 s k [ n ] δ [ n ] ˆ r LB k [ n ] ≥ 1 T N X n =1 s k [ n ] 2 µ k [ n ] q ˆ r LB k [ n ] − µ k [ n ] 2 δ [ n ] ≜ ˆ R k , (58) where µ µ µ ≜ { µ k [ n ] , ∀ k , n } is a set of auxiliary variables. Using (58), a conserv ativ e con vex constraint for (55) is giv en by ˆ R k ≥ R min − ρ, ∀ k. (59) It is worth noting that any solution satisfying (59) also satisfies (55), and hence guarantees the original minimum-SE requirement. For fixed ( Q , ∆ ) , the right-hand side of (58) is concav e in µ k [ n ] , and the optimal auxiliary variable is also obtained in closed form as µ ∗ k [ n ] = δ [ n ] q ˆ r LB k [ n ] , ∀ k , n. (60) Using (41), (57), and (59), the noncon ve x problem (SP2) can be conservati vely approximated by the following con vex optimization problem: (SP2-1): min Q , ∆ , ρ ≥ 0 T + η ρ subject to (1) − (5) , (41) , (57) , (59) . C. Pr ocedure of Pr oposed Algorithm Both subproblems, (SP1) and (SP2-1) , are con vex with respect to their respectiv e optimization v ariables and can be efficiently solved using standard con vex optimization solvers. By alternately solving these two subproblems, the ov erall algorithm iterates until con ver gence. The detailed procedure is summarized in Algorithm 1. Algorithm 1 Proposed Algorithm 1: Set r = 0 and initialize S r , Q r , ∆ r , η r , η max , and ε > 1 2: Calculate T r = P N n =1 δ [ n ] 3: repeat 4: Update r ← r + 1 5: Update T old ← T r − 1 6: Find S r by solving (SP1) for giv en { Q r − 1 , ∆ r − 1 } 7: Update { λ λ λ r , µ µ µ r } using (42) and (60) 8: Find { Q r , ∆ r } by solving (SP2-1) for given S r 9: Update η r ← min { εη r − 1 , η max } 10: Calculate T r = P N n =1 δ [ n ] 11: until | T r − T old | < ϵ Remark 1 (Con vergence and Computational Complexity) . Algorithm 1 starts fr om an initial feasible solution { S , Q , ∆ } and employs a penalty parameter η , which is gradually in- cr eased by a factor ε > 1 until it r eaches a prescribed upper bound η max . As established in [31], ther e e xists a finite η max for which the penalty term con ver ges to zero. After η attains this upper bound, the objective value does not incr ease over successive iterations, i.e., T ( S r − 1 , Q r − 1 , ∆ r − 1 ) ≥ T ( S r , Q r , ∆ r ) . (61) Mor eover , since the objective is lower bounded by a finite con- stant [32], the pr oposed algorithm is guaranteed to conver ge. The complexity of the pr oposed algorithm is analyzed under the standard worst-case framework for interior-point methods [33], [34]. F or a pr oblem with N V variables, the computa- tional cost of each interior-point iteration scales on the or der of O ( N 3 V ) , while the total number of iterations scales on the or der of O √ N V log(1 /ϵ ) , wher e ϵ > 0 denotes the desir ed solution accuracy . T aking into account the iterative structur e of the proposed algorithm, the r esulting worst-case complexity is given by O R C ( K N ) 3 . 5 log(1 /ϵ ) , where R C is the number of iter ations r equired for con ver gence (corr espond- ing to lines 3–11 in Algorithm 1). The resulting polynomial- time complexity suggests that the pr oposed appr oach is com- putationally tractable for practical system sizes [35]. Note that the discr etization parameters { U L , U N , U ν } determine the numerical accuracy of the expected-SE appr oximation and affect the per-iteration computational cost thr ough the evaluation of constr aint (59) , b ut they do not incr ease the number of optimization variables. Consequently , they appear only as constant factors in the complexity analysis. V . S I M U L AT I O N R E S U LT S A N D D I S C U S S I O N S T ABLE I P A R A ME T E R S E T UP Description V alue Number of GNs K = 4 Number of time slots N = 160 Max/Min slot lengths { δ max , δ min } = { 0 . 5 , 10 − 5 } s Max/Min altitudes { H max , H min } = { 200 , 10 } m Max 3D/vertical speeds { V max , V z } = { 20 , V max 2 } m/s T ransmit power of GNs P S = 30 dBm Constants for LoS probability { A 1 , A 2 } = { 12 . 08 , 0 . 114 } Path-loss exponents { α L , α N } = { 2 , 2 . 7 } Reference channel gains { β L , β N } = {− 30 , − 40 } dB Noise power σ 2 = − 70 dBm Minimum required SE R min = 2.4 bps/Hz Rician K-factor K R = 15 dB Shadowing standard deviation σ dB = 10 dB Discretization parameters U L = U N = U ν = 40 Penalty parameters { η 0 , η max , ε } = { 1 , 10 5 , 1 . 5 } Con vergence threshold ϵ = 10 − 3 Number of Monte Carlo realizations 30000 The simulation parameters listed in T able I are chosen to reflect representative settings widely adopted in U A V com- munication systems [18]–[28]. T o assess the robustness of the proposed scheme against channel randomness, we con- duct Monte Carlo simulations using the original stochastic channel model. For each optimized U A V trajectory , slot-time duration, and scheduling policy , independent realizations of small-scale fading and shadowing are generated in accordance with the assumed statistical distrib utions. The corresponding 9 (a) 3D trajectory of the proposed scheme. 0 50 100 150 200 250 300 -50 0 50 100 t=20.4s R min =2.5 bps/Hz GN4 GN3 GN2 y-axis (m) x-axis (m) s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 GN1 t=2.6~3.3s t=8.1~9.3s t=13.8~14.9s t=18.4~19.9s t=5.5s t=10.5s t=15.3s R min =2 bps/Hz (b) Horizontal trajectory of the proposed scheme. (c) 3D trajectory of the A C-based scheme. 0 50 100 150 200 250 300 -50 0 50 100 t=15.8~16.6s t=9.3s t=11.4~12.2s y-axis (m) x-axis (m) s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 GN4 GN3 GN2 GN1 t=2.4~2.9s t=4.7s t=7.2~8.1s t=13.8s R min =2 bps/Hz R min =2.5 bps/Hz t=18.3s (d) Horizontal trajectory of the AC-based scheme. 0 40 80 120 160 0.0 0.5 1.0 0 40 80 120 160 0.0 0.5 1.0 Proposed Scheduling indicator, s k [n] Time slot index, n AC-based s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 Time slot index, n (e) Scheduling indicator . 0 40 80 120 160 0.0 0.2 0.4 0.6 Time slot length, [n] (s) Time slot index, n Proposed AC-based (f) T ime slot length. 0 40 80 120 160 0.0 0.2 0.4 0.6 0.8 1.0 0 40 80 120 160 0.0 0.2 0.4 0.6 0.8 1.0 s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 Proposed LoS probability, P k L [n] Time slot index, n s 1 [n]=1 s 2 [n]=1 s 3 [n]=1 s 4 [n]=1 AC-based Time slot index, n (g) LoS probability . 0 40 80 120 160 0 2 4 6 8 0 40 80 120 160 0 2 4 6 8 Proposed Time slot index, n GN1 GN2 GN3 GN4 AC-based Average SE of GNs (bps/Hz) Time slot index, n GN1 GN2 GN3 GN4 (h) A verage SE of GNs. 0 1 02 03 04 0 20 40 60 80 100 120 Value of objective and completion time Number of iterations, r Objective function (R min =2 bps/Hz) Completion time (R min =2 bps/Hz) Objective function (R min =2.5 bps/Hz) Completion time (R min =2.5 bps/Hz) (i) Con vergence behavior . Fig. 2. Comparison of trajectory and resource allocation between the proposed and A C-based schemes, and conv ergence behavior . instantaneous uplink SE is computed for each realization, and the ergodic SE is estimated by av eraging over 30 , 000 Monte Carlo realizations. In addition, the following five schemes are considered for performance comparison. 1) Pr oposed scheme: The U A V strategy ( S , Q , ∆ ) is opti- mized using Algorithm 1. 2) A vera ge-channel-based (A C-based) scheme [23], [24]: The U A V strategy is optimized based on the av erage- channel-based SE approximation in Section III-A. T o compensate for potential infeasibility caused by SE ov erestimation, a positi ve margin is iterativ ely added to the target SE in the average-channel-based optimization until the resulting design satisfies the original minimum- SE requirement R min under the actual channel model. The margin step size is set to 10 − 4 . 3) F ixed-slot-length scheme: Instead of optimizing the time-slot lengths individually , a single common slot length δ is optimized, resulting in a mission completion time of T = N δ . Consequently , the U A V strategy ( S , Q , δ ) is optimized under the same system con- straints. 4) F ixed-altitude scheme: The U A V altitude is fixed at H min , and the UA V strategy , including S , ∆ , and the horizontal trajectory , is optimized. 5) F ixed-trajectory scheme: The U A V follows a hov er-and- fly trajectory at an altitude H min , sequentially hovering at each GN location and traveling in straight lines at the maximum velocity between GNs, while optimizing the remaining variables S and ∆ . While the comparison with the A C-based scheme re veals the inherent limitations of av erage-channel-based SE approx- imations, the other baseline schemes assess the effect of successiv ely restricting individual optimization v ariables. Fig. 2 compares the U A V trajectory and resource allocation 10 obtained by the proposed scheme and the A C-based scheme. T o clearly expose the limitation of av erage-channel-based SE approximations, the A C-based results are shown without applying any margin adjustment, so that the violation of the minimum-SE constraint can be directly observed. As sho wn in Fig. 2(a) and 2(b), the U A V trajectory adapts noticeably to the required SE lev el R min . When the required SE is high (e.g., R min = 2.5 bps/Hz), the U A V approaches each GN more closely and maintains a relatively high alti- tude, particularly while traveling between GNs. This behavior increases the elev ation angle and, consequently , the LoS probability , enabling reliable rate provisioning under stringent SE requirements. In contrast, for a lo wer requirement (e.g., R min = 2 bps/Hz), the U A V prioritizes reducing the mission completion time by shortening the horizontal travel distance and limiting close approaches to each GN. T o still satisfy the minimum-SE requirement, it compensates by flying at a higher altitude, which increases the elev ation angle and the LoS probability , thereby maintaining reliable rate provisioning with reduced horizontal maneuv ering. In these figures, the color-coded trajectory segments indicate the scheduled GN at each time slot, illustrating how the scheduling decisions are coordinated with the U A V trajectory . In contrast, Fig. 2(c) and 2(d) sho w that the A C-based scheme yields a trajectory with a consistently lower altitude and a larger separation distance from each GN, compared with the proposed expected-SE-based design. This behavior stems from the average-channel-based SE approximation, which ov erestimates the true expected SE and thus enforces the minimum-SE constraint in an overly optimistic manner . As a result, the U A V does not need to sufficiently approach each GN or increase its altitude to enhance the LoS probability , leading to trajectories that remain lower and farther away from the GNs. As further confirmed in Fig. 2(e), the corresponding scheduling indicators take binary v alues of 0 or 1, demonstrat- ing that a valid TDMA-based scheduling policy is obtained and that each time slot is assigned to at most one GN. As shown in Fig. 2(f), the proposed scheme assigns longer time-slot durations when the U A V hovers above each GN, where the channel conditions are fav orable, while shorter slots are allocated during inter-GN trav el or scheduling transitions. Consistent with this beha vior , by jointly examining Fig. 2(f) and Fig. 2(g), it can be observed that longer time-slot durations tend to be assigned to time slots with higher LoS probability . In contrast, the A C-based scheme determines the time-slot lengths based on an average-channel-based SE estimate, which ov erestimates the achiev able SE. Consequently , shorter slot durations are allocated ev en when the UA V is closest to each GN. Moreov er , the A C-based scheme operates with a lower LoS probability than the proposed scheme, especially during scheduling transitions, since it does not suf ficiently increase the altitude or reduce the service distance to enhance the LoS condition. As a result, a larger portion of transmission is performed under less fa vorable channel conditions. These differences are directly reflected in the achieved av erage SE of GNs, as shown in Fig. 2(h). While the proposed scheme satisfies the minimum-SE requirement precisely at the mission completion time for all GNs, the AC-based scheme fails to meet the target R min due to its ov erly optimistic SE estimation. This result confirms that the proposed scheme yields solutions that are feasible with respect to the original minimum-SE constraint, as it explicitly accounts for the true expected SE through a conservati ve and accurate SE estima- tion. In contrast, the A C-based scheme produces infeasible solutions that violate the minimum-SE requirement due to the ov erestimation inherent in the average-channel-based SE approximation. Fig. 2(i) illustrates the con ver gence beha vior of the proposed algorithm for R min = 2 and R min = 2.5 bps/Hz, showing both the objecti ve value including the penalty term and the actual mission completion time. In the early iterations, the optimization variables are not suf ficiently refined, making it difficult to satisfy the minimum-SE constraint. As a result, the penalty term becomes activ e, causing the objective v alue to exceed the completion time. As the optimization proceeds, the minimum-SE constraint is gradually satisfied and the penalty term diminishes. Consequently , the objective value con ver ges to the mission completion time, indicating that the constraint violation has been eliminated. For both SE requirements, stable conv ergence is achiev ed within approximately 20 itera- tions. Fig. 3 provides a comprehensiv e comparison between the proposed scheme and the A C-based scheme, highlighting the fundamental limitations of av erage-channel-based SE approx- imations. In the AC-based scheme, SE ov erestimation may lead to violations of the minimum-SE requirement, and thus the target R min is iterativ ely increased by adding a margin until the resulting design satisfies the minimum-SE constraint. Fig. 3(a) compares the mission completion time of the proposed and AC-based schemes under different minimum- SE requirements R min . As shown in the figure, the proposed scheme consistently achie ves a shorter completion time than the A C-based scheme for all considered values of R min . The performance gap between the two schemes is most pronounced in the intermediate re gime of 1 . 75 ≤ R min ≤ 2 . 75 bps/Hz, where the minimum-SE requirement is neither overly stringent nor ov erly relaxed. In this intermediate regime, although NLoS transmissions can still be exploited from a mission completion time minimization perspecti ve, accurately identifying when LoS conditions must be enforced becomes critical for per- formance. The proposed scheme correctly enforces LoS con- ditions in those critical segments where NLoS transmissions are insufficient to meet the minimum-SE requirement, while allowing NLoS transmissions else where. In contrast, due to the systematic ov erestimation of achiev able SE under NLoS channels, the A C-based scheme continues to rely on NLoS transmissions ev en in segments where LoS conditions are ac- tually required, leading to mismatched trajectory and resource allocation decisions and a pronounced increase in mission completion time. When R min < 1 . 75 bps/Hz, both schemes can easily satisfy the SE requirement. In this case, the U A V follows an almost straight-line trajectory from the initial to the final location to minimize the completion time, and transmis- sions are predominantly performed under NLoS conditions, yielding only a marginal performance gap. Con versely , when R min > 2 . 75 bps/Hz, satisfying the SE requirement becomes 11 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50 60 Completion time, T (s) Required SE, R min (bps/Hz) Proposed AC-based (a) Completion time. 1.0 1.5 2.0 2.5 3.0 1.0 1.5 2.0 2.5 3.0 Estimated minimum average SE (bps/Hz) Actual minimum average SE (bps/Hz) Proposed AC-based Overestimated region (b) Overestimated region. 1E-5 1E-4 1E-3 0.01 0.1 Required SE margin 31 32 33 34 35 36 37 Proposed AC-based Completion time, T (s) 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Computation time (s) (c) Computation time. Fig. 3. Performance comparison between the proposed and AC-based schemes. challenging for both schemes, forcing most transmissions to be carried out directly above each GN. Consequently , LoS channels dominate, and the performance gap between the two schemes diminishes again in the high- R min regime. Fig. 3(b) compares the SE estimated in the optimization with the actual SE obtained via Monte Carlo simulations for both schemes. For the AC-based scheme, all operating points lie in the ov erestimation region, indicating that the achie vable SE is consistently overestimated due to the use of average- channel-based SE approximations. In contrast, the proposed scheme optimizes based on a conservati ve lo wer bound of the expected SE, resulting in estimated SE values that are consistently lower than the corresponding actual SE across all cases. This conservati ve property ensures that the minimum- SE constraint is satisfied with respect to the true expected SE, thereby guaranteeing the feasibility of the proposed scheme under the original problem formulation. Fig. 3(c) compares the mission completion time and the computation time of the proposed and A C-based schemes as functions of the SE mar gin used in the A C-based scheme to satisfy the minimum-SE constraint. The proposed scheme does not require any margin adjustment and therefore shows identical completion time and computation time for all margin values. For the A C-based scheme, when a large margin (e.g., 0.1) is applied, the optimization requires fe wer repetitions and thus results in a shorter computation time than the proposed scheme. Howe ver , such a large margin forces the A C-based optimization to target an SE le vel higher than R min to compensate for SE overestimation, which yields an overly conserv ativ e design and increases the completion time by ov er 15% compared to the proposed scheme. As the margin decreases, the completion time of the A C-based scheme gradually decreases as well. Nevertheless, it remains consistently longer than that of the proposed scheme. More- ov er , smaller margins necessitate repeated re-optimizations to eliminate violations of the minimum-SE constraint, which significantly increases the computation time compared to the proposed scheme. These results demonstrate that margin-based correction in the A C-based scheme introduces an unfav orable trade-off between performance and computational complexity , and cannot outperform the proposed scheme, which directly optimizes the trajectory and resource allocation based on an accurate characterization of the expected SE. Fig. 4 presents the performance comparison between the proposed and baseline schemes under different system param- eters. Specifically , Fig. 4(a) compares the mission completion time with respect to the U A V maximum velocity ( V max ). As V max increases, the U A V can travel faster between GNs, allowing it to spend a larger portion of time hov ering abo ve each GN for data collection. This increases the LoS probability and improv es the channel quality , enabling the minimum-SE requirement R min to be satisfied more efficiently . As a result, all schemes exhibit a decreasing completion time as V max in- creases. The fixed-trajectory scheme becomes infeasible when V max < 16 m/s. This is because it follows a predetermined hov er-and-fly trajectory at the minimum altitude H min , directly visiting and serving all GNs without allo wing trajectory or altitude adaptation. Under a low U A V speed, the prolonged inter-GN trav el time at low altitude results in insufficient SE accumulation, making it impossible to satisfy the minimum- SE requirement regardless of the mission duration. The fixed- altitude and fixed-slot-length schemes f ail to satisfy R min when V max < 12 m/s. In the fixed-altitude scheme, operating at the low altitude H min significantly reduces the LoS probability during inter-GN travel, resulting in a high proportion of NLoS transmission. As V max decreases, the time spent in these NLoS- dominated regions increases, resulting in inefficient SE and ev entually making it impossible to satisfy the minimum-SE constraint. The fixed-slot-length scheme, on the other hand, can le verage altitude adaptation to achieve relativ ely higher LoS probability during inter-GN travel. Howe ver , since a common slot length is applied to all time slots, increasing the slot duration to enable hovering abov e GNs simultaneously increases the slot duration during inter-GN trav el. Similar to the fixed-altitude case, a small V max leads to excessi ve SE loss during inter-GN trav el due to prolonged NLoS transmission, rendering the minimum-SE constraint infeasible. Although the AC-based scheme performs optimization based on an inaccurate SE approximation, it allows flexible adjustment of time-slot durations across slots as well as the 3D trajectory . This flexibility enables the A C-based scheme to outperform the fixed-slot-length scheme by allocating shorter slots during unfa vorable channel conditions, thereby highlighting the im- portance of time-slot length optimization. Nevertheless, due to 12 8 1 21 62 02 42 8 20 40 60 80 100 Completion time, T (s) Maximum flight speed, V max (m/s) Proposed AC-based Fixed-slot-length Fixed-altitude Fixed-trajectory (a) T vs. V max . 3 6 9 12 15 18 30 35 40 45 50 55 Proposed AC-based Fixed-slot-length Fixed-altitude Fixed-trajectory Completion time, T (s) Rician K-factor, K R (dB) (b) T vs. K R . Fig. 4. Performance comparison between the proposed and baseline schemes. SE overestimation, its performance remains inferior to that of the proposed scheme. Fig. 4(b) shows the performance comparison with respect to the Rician K -factor ( K R ). As K R increases, the LoS component of the wireless channel becomes more dominant, which reduces channel variability and increases the expected SE. Consequently , the mission completion time decreases for all schemes. For the A C-based scheme, howe ver , the ov erestimation of the achiev able SE leads to a tendency to minimize trajectory movement and rely more heavily on transmission under NLoS conditions in order to reduce the completion time. As a result, the AC-based scheme fails to fully e xploit the strengthened LoS component as K R increases, causing the performance gap between the proposed and A C- based schemes to become more pronounced. Since the baseline schemes achie ve performance in the order of the A C-based, fixed-slot-length, fixed-altitude, and fixed-trajectory schemes, this result indicates that system performance is increasingly influenced by flexibility in 3D trajectory design, followed by altitude optimization, time-slot length adaptation, and accurate SE approximation, in that order . Overall, the proposed scheme consistently outperforms all baseline schemes across all values of the system parameters, demonstrating the effecti veness of jointly optimizing the user scheduling, U A V 3D trajectory , time-slot durations, and the proposed effecti ve SE approxi- mation. V I . C O N C L U S I O N S This paper addressed the mission completion time min- imization problem in U A V -assisted wireless communication systems under minimum expected SE constraints. Departing from con ventional av erage-channel-based designs, we pro- posed a conservati ve and tractable expected-SE formulation that explicitly accounts for the stochastic nature of prob- abilistic LoS propagation, small-scale fading, and shadow- ing. By constructing a computable lower bound via CDF- domain discretization and quadrature-based reformulation, the proposed approach enables reliable constraint enforcement while remaining amenable to joint trajectory and resource optimization. T o solve the resulting mixed-inte ger noncon- ve x problem, we developed a penalty-based block coordinate descent frame work that alternately optimizes user schedul- ing and the UA V trajectory together with adaptive time- slot durations. The SCA and quadratic transform techniques were employed to handle the coupled noncon vexities, ensuring feasibility with respect to the original expected-SE constraints while achie ving ef ficient con vergence with polynomial-time complexity . Beyond performance impro vements, the results clearly sho w that a verage-channel-based approaches frequently yield infeasible solutions, as ov erestimation of the e xpected SE causes violations of the minimum SE requirements under the true stochastic channel conditions. In contrast, the proposed approach deri ves a conserv ativ e and accurate expected-SE lower bound that guarantees feasibility and ensures stable performance across all operating regimes. Consequently , the proposed framework not only shortens the mission completion time but also provides a rob ust and practically deplo yable design methodology for U A V communication systems based on accurate expected-SE modeling under probabilistic LoS channels. R E F E R E N C E S [1] L. Gupta, R. Jain, and G. V aszkun, “Survey of important issues in UA V communication networks, ” IEEE Commun. Surve ys Tuts. , vol. 18, no. 2, pp. 1123–1152, 2nd Quart., 2016. [2] Y . Zeng, R. Zhang, and T . J. Lim, “W ireless communications with unmanned aerial v ehicles: Opportunities and challenges, ” IEEE Commun. Mag. , vol. 54, no. 5, pp. 36–42, May 2016. [3] X. Lin, V . Y ajnanarayana, S. D. Muruganathan, S. Gao, H. Asplund, H.- L. Maattanen, M. Bergstrom, S. Euler, and Y .-P . E. W ang, “The sky is not the limit: L TE for unmanned aerial vehicles, ” IEEE Commun. Mag. , vol. 56, no. 4, pp. 204–210, Apr . 2018. [4] N. Namvar , A. Homaifar , A. Karimoddini, and B. Maham, “Heteroge- neous UA V cells: An ef fective resource allocation scheme for maximum coverage performance, ” IEEE Access , vol. 7, pp. 164708–164719, 2019. [5] S. Shakoor, Z. Kaleem, D. -T . Do, O. A. Dobre, and A. Jamalipour, “Joint optimization of U A V 3-D placement and path-loss factor for energy- efficient maximal cov erage, ” IEEE Internet Things J. , vol. 8, no. 12, pp. 9776–9786, Jun. 2021. [6] Y . Su, X. Pang, S. Chen, X. Jiang, N. Zhao, and F . R. Y u, “Spectrum and energy ef ficiency optimization in IRS-assisted UA V networks, ” IEEE T rans. Commun. , vol. 70, no. 10, pp. 6489–6502, Oct. 2022. [7] K. Heo, G. Park, and K. Lee, “Joint optimization of UA V trajectory and communication resources with complete avoidance of no-fly-zones, ” IEEE T rans. Intell. T ransp. Syst. , vol. 25, no. 10, pp. 14259–14265, Oct. 2024. 13 [8] L. Xing and B. W . Johnson, “Reliability theory and practice for unmanned aerial vehicles, ” IEEE Internet Things J. , vol. 10, no. 4, pp. 3548–3566, Feb . 2023. [9] J. Zhou, D. Tian, Y . Y an, X. Duan, and X. Shen, “Joint optimization of mobility and reliability-guaranteed air-to-ground communication for U A Vs, ” IEEE T rans. Mobile Comput. , vol. 23, no. 1, pp. 566–580, Jan. 2024. [10] Q. W u, Y . Zeng, and R. Zhang, “Joint trajectory and communication design for multi-U A V enabled wireless networks, ” IEEE T rans. W ireless Commun. , vol. 17, no. 3, pp. 2109–2121, Mar . 2018. [11] I. V aliulahi and C. Masouros, “Multi-UA V deployment for throughput maximization in the presence of co-channel interference, ” IEEE Internet Things J. , vol. 8, no. 5, pp. 3605–3618, Mar . 2021. [12] H. Lee, S. Eom, J. P ark, and I. Lee, “UA V -aided secure communications with cooperative jamming, ” IEEE Tr ans. V eh. T echnol. , vol. 67, no. 10, pp. 9385–9392, Oct. 2018. [13] K. Heo, W . Lee, and K. Lee, “U A V -assisted wireless-powered secure communications: Integration of optimization and deep learning, ” IEEE T rans. Wir eless Commun. , vol. 23, no. 9, pp. 10530–10545, Sep. 2024. [14] K. Heo, H.-H. Choi, and K. Lee, “Joint trajectory and resource optimiza- tion for UA V -assisted SWIPT systems: A comparative study of linear and nonlinear energy harvesting models, ” IEEE Internet Things J. , vol. 11, no. 24, pp. 40293–40305, Dec. 2024. [15] G. Park, K. Heo, W . Lee, and K. Lee, “UA V -assisted wireless-powered two-way communications, ” IEEE T rans. Intell. T ransp. Syst. , vol. 25, no. 3, pp. 2641–2655, Mar . 2024. [16] C. Kim, H.-H. Choi, and K. Lee, “Joint optimization of trajectory and resource allocation for multi-U A V -enabled wireless-powered communi- cation networks, ” IEEE T rans. Commun. , vol. 72, no. 9, pp. 5752–5764, Sep. 2024. [17] A. Al-Hourani, S. Kandeepan, and S. Lardner , “Optimal LAP altitude for maximum coverage, ” IEEE Wir eless Commun. Lett. , vol. 3, no. 6, pp. 569–572, Dec. 2014. [18] Y . Zeng, J. Xu, and R. Zhang, “Energy minimization for wireless communication with rotary-wing UA V , ” IEEE T rans. W ireless Commun. , vol. 18, no. 4, pp. 2329–2345, Apr . 2019. [19] A. Meng, X. Gao, Y . Zhao, and Z. Y ang, “Three-dimensional trajectory optimization for energy-constrained UA V -enabled IoT system in proba- bilistic LoS channel, ” IEEE Internet Things J. , vol. 9, no. 2, pp. 1109– 1121, Jan. 2022. [20] H. Lei, X. W u, K.-H. Park, and G. Pan, “3D trajectory design for ener gy- constrained aerial CRNs under probabilistic LoS channel, ” IEEE T rans. Cogn. Commun. Netw . , vol. 11, no. 3, pp. 1522–1534, Jun. 2025. [21] Y . Pan et al. , “Joint optimization of trajectory and resource allocation for time-constrained UA V -enabled cognitive radio networks, ” IEEE T rans. V eh. T echnol. , vol. 71, no. 5, pp. 5576–5580, May 2022. [22] K. Liu and J. Zheng, “U A V trajectory optimization for time-constrained data collection in UA V -enabled environmental monitoring systems, ” IEEE Internet Things J . , vol. 9, no. 23, pp. 24300–24314, Dec., 2022. [23] C. Y ou and R. Zhang, “Hybrid offline-online design for UA V -enabled data harvesting in probabilistic LoS channels, ” IEEE T rans. W ireless Commun. , vol. 19, no. 6, pp. 3753–3768, Jun. 2020. [24] B. Duo, Q. W u, X. Y uan, and R. Zhang, “ Anti-jamming 3D trajectory design for UA V -enabled wireless sensor networks under probabilistic LoS channel, ” IEEE T rans. V eh. T echnol. , vol. 69, no. 12, pp. 16288–16293, Dec. 2020. [25] Y . He, Y . Gan, H. Cui, and M. Guizani, “Fairness-based 3-D multi- U A V trajectory optimization in multi-UA V -assisted MEC system, ” IEEE Internet Things J . , vol. 10, no. 13, pp. 11383–11395, Jul. 2023. [26] C. Kim, H.-H. Choi, and K. Lee, “Interference coordination for multi- U A V -enabled communications under probabilistic LoS channels, ” IEEE Internet Things J . , vol. 12, no. 19, pp. 40484–40498, Oct. 2025. [27] W . Luo, Y . Shen, B. Y ang, S. W ang, and X. Guan, “Joint 3-D trajectory and resource optimization in multi-UA V -enabled IoT networks with wireless power transfer, ” IEEE Internet Things J. , vol. 8, no. 10, pp. 7833–7848, May 2021. [28] G. Park, G. Jang, W . Lee, and K. Lee, “U A V -enabled wireless-powered two-way communications under probabilistic LoS channels, ” IEEE Inter- net Things J . , vol. 13, no. 2, pp. 3188–3199, Jan. 2026. [29] M. Grant and S. Boyd. (2017). CVX: MATLAB Software for Disciplined Conve x Pro gramming, V ersion 2.1. [Online]. A vailable: http://cvxr .com/cvx. [30] K. Shen and W . Y u, “Fractional programming for communication systems–Part I: Power control and beamforming, ” IEEE T rans. Signal Pr ocess. , vol. 66, no. 10, pp. 2616–2630, May 2018. [31] Q.-D. V u, K.-G. Nguyen, and M. Juntti, “Max-min fairness for multicast multigroup multicell transmission under backhaul constraints, ” in Proc. IEEE Globecom , Dec. 2016, pp. 1–6. [32] D. P . Bertsekas, Nonlinear Progr amming. Belmont, MA, USA: Athena Scientific, 1999. [33] S. Boyd and L. V andenberghe, Con vex Optimization. Cambridge, U.K.: Cambridge Univ . Press, 2004. [34] A. Ben-T al and A. Nemirovski, Lectur es on Modern Conve x Optimiza- tion: Analysis, Algorithms, and Engineering Applications, Philadelphia, P A, USA: SIAM, 2001. [35] C. E. Leiserson, R. L. Rivest, T . H. Cormen, and C. Stein, Intr oduction to Algorithms, vol. 6. Cambridge, MA, USA: MIT Press, 2001.
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