Packet Loss Analysis Under Doppler Effects and Reinforcement-Learning-Based Resource Allocation for Direct-to-Satellite LoRaWAN

I. Introduction

• DtS-IoT LoRaWAN research lacked an analytical framework jointly accounting for Doppler impairments and random resource allocation in PLR evaluation.The introduction also notes that much prior performance analysis relied on simulations or experiments.Source: §I · p. 2
• The proposed analysis models time- and frequency-domain collisions alongside Doppler shift and rate, and uses the resulting PLR to guide PPO and DQN resource allocation.Agents assign each ED's spreading factor, coding rate, bandwidth, and first packet start time.Source: §I · p. 2

II. System Model

• The system model places randomly and uniformly distributed EDs in a coverage area and has them transmit to a LoRaWAN gateway installed on a LEO satellite.An ED can transmit only while visible to the satellite.Source: §II · p. 2, B. Satellite Parameters · p. 3

A. LoRaWAN Parameters

• For the analytical setup, EDs randomly select spreading factor, coding rate, and channel, while transmit power, bandwidth, and payload size are shared across devices.The text says these shared-parameter assumptions simplify the derivation and do not affect its generality.Source: A. LoRaWAN Parameters · p. 3

B. Satellite Parameters

• Satellite geometry determines each ED's visibility interval and thus its transmission opportunity; devices near the footprint center trajectory can have the longest visibility.Some edge-located EDs may have too little visibility time to transmit even one packet.Source: B. Satellite Parameters · p. 3

1) Visibility distance dn:

• Visibility distance is the distance the satellite travels while it remains visible to an ED.The section calculates this distance from the ED's position relative to the satellite footprint.Source: 1) Visibility distance dn · p. 3

2) Transmission time parameters:

• With a 1% duty-cycle constraint, an ED can transmit up to floor(0.01tv,n/ToAn) packets during its satellite visibility duration.The section defines the visibility start, duration, and end times as transmission-time parameters.Source: 2) Transmission time parameters · pp. 3–4

3) Doppler shift:

• Doppler shift depends on the time-varying central angle and its minimum during visibility; EDs near the satellite ground-track are more vulnerable to Doppler-shift packet loss.The text explains that their smaller minimum central angle can produce a larger Doppler-shift range, especially near the start and end of visibility.Source: 3) Doppler shift · p. 4

III. Packet Loss Ratio Derivation

• The total PLR is divided into losses from Doppler shift, Doppler rate, and packet collision.For Doppler-shift loss, the model assumes an ED transmits its packets consecutively from the start of satellite visibility.Source: §III · p. 4, A. Packet loss due to Doppler shift · p. 4

A. Packet loss due to Doppler shift

• The Doppler-shift loss probability is averaged over packet start time, time-on-air, packet count, and satellite visibility duration.The derivation uses a Doppler-shift CDF and a joint distribution for packet start time and visibility duration, which are not independent.Source: A. Packet loss due to Doppler shift · pp. 4–5
• Increasing bandwidth reduces Doppler-shift packet loss, and the derived PLR expression has no analytical solution and must be solved numerically.The integral's lower limit reflects the 1% duty-cycle condition that at least one packet can be transmitted.Source: A. Packet loss due to Doppler shift · p. 5

B. Packet loss due to Doppler rate

• Doppler-rate packet loss is defined by whether the change in Doppler shift across a packet's time-on-air exceeds a bandwidth- and spreading-factor-dependent threshold.The probability is evaluated for an arbitrary packet and ED.Source: B. Packet loss due to Doppler rate · p. 5
• The Doppler-rate probability derivation reduces to a 16th-degree polynomial inequality whose roots must be obtained numerically.The derivation constructs valid intervals in the y-coordinate from the polynomial's nonnegative regions.Source: B. Packet loss due to Doppler rate · p. 5
• Doppler-rate PLR decreases with bandwidth and increases with spreading factor.Source: B. Packet loss due to Doppler rate · p. 6

• Collision loss is the event that at least one packet interferes with the desired packet in both frequency and time domains.Source: C. Packet loss due to collision · p. 6
• The vulnerable time interval spans from the desired packet’s start minus the interferer’s airtime to the desired packet’s end.An interfering packet collides in time if its start falls within that interval; collision probability averages over possible interfering start times and accounts for same-channel probability 1/NCH.Source: C. Packet loss due to collision · p. 6
• Longer vulnerable periods increase collision-induced PLR; larger ToA, associated with higher SF and lower CR, therefore raises collision susceptibility.Source: C. Packet loss due to collision · p. 7
• At sufficiently large bandwidth, Doppler effects become negligible and PLR approaches collision loss, but at practical bandwidths Doppler rate remains significant while Doppler shift can be neglected.Source: C. Packet loss due to collision · p. 7

• The objective is to minimize total PLR by assigning each ED’s SF, BW, CR, and packet transmission start times.Source: §IV · p. 7
• The joint allocation problem is mixed-integer and non-convex, couples ED decisions through collision loss, and must be resolved as visibility and collision conditions change.Source: §IV · p. 7
• The authors use RL because instance-specific optimization may not scale to large networks or time-constrained onboard allocation; RL offers constant amortized assignment cost and generalizes beyond training network sizes.Source: §IV · p. 7

• The environment includes ED positions, visibility windows, and each ED’s Doppler shift and Doppler-rate profiles throughout its window.Source: 1) Environment block · p. 7
• For a candidate schedule, the environment evaluates each packet’s outcome using the system’s loss mechanisms and a collision component calculated from q(τki, ToAi).Source: 1) Environment block · pp. 7–8
• The closed-form collision model assumes consecutive transmissions beginning at satellite visibility, unlike RL-scheduled transmissions, so exact policy-dependent collision PLR is not analytically available.Source: 1) Environment block · p. 8
• The reward retains q(τki, ToAi) as a computationally efficient collision surrogate because it captures packet-duration dependence without modeling adaptive schedules exactly.Source: 1) Environment block · p. 8

• The state uses visibility-window duration, the window’s position within the pass, and channel load to represent packet count, Doppler exposure, and congestion.Source: 2) State, Action, and Reward · p. 8
• Each ED’s action selects an SF, CR, and BW combination plus a discrete first-packet timing bin; subsequent packets are scheduled consecutively.Source: 2) State, Action, and Reward · pp. 8–9
• The reward is the mean success probability of the packets scheduled for that ED.Source: 2) State, Action, and Reward · p. 9
• A larger γ values future rewards more strongly because scheduling one ED affects the collision environment and rewards of later EDs; this work sets γ = 0.95.The passage associates this setting with an effective planning horizon of approximately 20 scheduling decisions.Source: 2) State, Action, and Reward · p. 9

3) Agent:

• PPO and DQN use the same environment, state, reward, and discrete action space, enabling comparison of policy-based and value-based approaches.PPO learns a masked categorical policy, while DQN estimates action Q-values.Source: 3) Agent · p. 9

4) Trained Policy:

• After training converges, the policy parameters are saved as a frozen network snapshot for later deployment.Each deployment run applies the saved policy to a new network realization.Source: 4) Trained Policy · p. 9, 5) New instance · p. 9

6) Trained policy in deployment:

• At deployment, EDs are scheduled sequentially using deterministic action selection, and a satellite pass takes under one second to schedule for the considered N values.Each ED requires one forward pass through the trained network.Source: 6) Trained policy in deployment · p. 9

7) Assignment:

• Per-ED actions are concatenated into the network-wide SF, BW, CR, and packet-time assignment without further optimization.The combination index maps to an SF, CR, and BW tuple; packet starts follow the assigned times and duty-cycle spacing.Source: 7) Assignment · p. 9

8) Evaluate:

• Evaluation checks Doppler conditions at actual transmission times and counts time-overlapping same-channel transmissions as collisions; the reported PLR uses actual packet losses.The controller assumes schedule coordination and a sufficiently accurate shared time reference, but does not account for signaling overhead.Source: 8) Evaluate · p. 10

A. Analytical Derivation Evaluation

• The analytical Doppler-shift PLR matches Monte Carlo results and is nearly negligible at LoRaWAN bandwidths of 125–500 kHz.The shift-related PLR decreases as bandwidth increases.Source: A. Analytical Derivation Evaluation · p. 10, Figure 5 · p. 10
• Doppler-rate PLR is significant at practical bandwidths, with greater network degradation at lower bandwidths.The analytical expression is also validated against simulation.Source: A. Analytical Derivation Evaluation · p. 11, Figure 6 · p. 11
• Random resource allocation causes congestion beyond about 4,500 EDs at an acceptable PLR threshold of 10^-1; combined Doppler-rate effects further worsen degradation.The combined PLR trend follows the collision-related trend.Source: A. Analytical Derivation Evaluation · p. 11, Figure 7 · p. 11, Figure 8 · p. 11

B. RL-based PLR Improvement Results

• Each ED’s action selects an SF, CR, and BW combination plus packet-burst timing; ToA-based feasibility constraints restrict the choices available at different distances.Allocations are trained on independent realizations and evaluated on previously unseen networks, typically averaging results over five held-out networks.Source: B. RL-based PLR Improvement Results · p. 11
• For both algorithms, converged reward—mean per-ED packet success probability—decreases from approximately 0.92 at N = 100 to 0.59 at N = 1000, reflecting increasing unavoidable packet collisions.Source: B. RL-based PLR Improvement Results · p. 11
• Distance-based action masking prevents assignments with ToA shorter than the distance permits, while both agents mostly select feasible, lower-index, shorter-ToA combinations.Farther EDs have fewer feasible combinations because received SNR decreases with closest-approach distance.Source: B. RL-based PLR Improvement Results · p. 12, Figure 10 · p. 12
• PPO and DQN reduce PLR across the evaluated network sizes under the 1% duty-cycle scenario, while their advantage narrows in the U.S. 400 ms-per-20 s occupancy scenario.At larger U.S.-scenario network sizes, PPO approaches random allocation while DQN retains a modest gain.Source: B. RL-based PLR Improvement Results · p. 12
• Both RL schemes transmit more total packets than random allocation, with the learned preference for lower ToA reducing collision probability and increasing available transmission time.The conclusion reports the gains for 800 EDs as 106.7% for PPO and 88.0% for DQN.Source: B. RL-based PLR Improvement Results · pp. 12–13, §VI · p. 13, Figure 12 · p. 13

VI. Conclusion

• At a target PLR of 3 × 10^-2, PPO and DQN support approximately 650 and 690 EDs, versus 100 under random allocation; for 800 EDs, transmitted packets increase by 106.7% and 88.0%, respectively.The comparison covers the single-pass packet total for PPO and DQN relative to random allocation.Source: Figure 11 · p. 12, §VI · p. 13, Figure 12 · p. 13

Appendix A Proof of lemma 1

• The conditional probability is evaluated for a given x and then averaged over X’s marginal distribution to obtain the final expression and complete Lemma 1.The derivation first obtains the probability involving the transformed y variable.Source: Appendix A · p. 13
• The joint PDF of visibility start and duration has singular components in Areas I and III and a two-dimensional continuous density in Area II.In Area II, the start time is related to the visibility duration and position; in Area III, their sum is fixed.Source: Appendix A · pp. 13–14
• For the Area II density, the derivation maps coordinates to visibility duration and start time, summing the two x-coordinate branches that map to the same point.The Jacobian magnitudes for both branches are equal.Source: Appendix A · p. 14