On the Sum of Inverse Gamma RVs and its Application to Wireless Communications Networks

I. INTRODUCTION

• Existing exact shadowing-sum results can require computationally demanding multivariate special functions, while approximations may lack consistent accuracy across distribution regions and parameter settings.The paper identifies sums of inverse-gamma variables as unexplored and exact, easily computable sum statistics as desirable.Source: §I · p. 1
• The paper proposes a framework for sums of independent, not necessarily identically distributed inverse-gamma random variables and derives exact, easy-to-evaluate closed-form PDF and CDF expressions.It also assesses computational efficiency, investigates shaping-parameter quantization, and demonstrates the framework in two application scenarios.Source: §I · p. 1
• The framework analyzes signal-to-interference-ratio statistics in UAV-assisted wireless communications.The paper also studies carrier aggregation in a high-speed railway communication system.Source: §I · p. 1

II. STATISTICS OF THE SUM OF IG RVS

• A half-step shaping-parameter change has limited CDF impact across broad parameter ranges, with larger differences confined to specific operating regimes.The localized differences occur at moderate K and relatively small shaping parameters for −10 dB, and intermediate K and shaping parameters for −5 dB.Source: Figure 1 · p. 2, §II · p. 2
• Across tested configurations, t_NILT/t_closed remains above one, indicating that the closed-form finite-sum evaluation is typically faster than numerical inverse Laplace transformation.The benchmark evaluates the PDF of two inverse-gamma variables over selected x and shaping-parameter values.Source: §II · pp. 2–3, Table I · p. 3

• The two application settings are a UAV-assisted network and a high-speed railway communication system.Source: §III · p. 3
• Outage probability is the probability that received SIR, or SNR in noise-limited scenarios, falls below a target threshold.Source: §III · p. 3
• Coverage probability is complementary to outage probability: Pcov = 1 − Pout.Source: §III · p. 3

A. UAV-Assisted Wireless Network

• A ground user receives its desired signal from a serving UAV and co-channel interference from K neighboring UAVs.Source: A. UAV-Assisted Wireless Network · p. 3
• The model averages out fast fading and represents each link’s shadowing factor with an inverse-gamma distribution.Source: A. UAV-Assisted Wireless Network · p. 3
• The outage CDF is evaluated from the desired-signal distribution and the distribution of total interfering signals.Source: A. UAV-Assisted Wireless Network · p. 3
• Although the CDF expression involves an improper integral over (0, ∞), its smooth, rapidly decaying integrand supports accurate conventional numerical integration.Source: A. UAV-Assisted Wireless Network · p. 3

1) Asymptotic Results:

• For serving-BS transmit power Pm < 20 dBm, the upper incomplete gamma function in the CDF expression is approximated to obtain a simpler closed form.Source: 1) Asymptotic Results · p. 3
• The approximation and integral evaluation yield a closed-form expression for coverage probability.Source: 1) Asymptotic Results · p. 3
• The channel reliability slope depends only on the shadowing-severity coefficient.Source: 1) Asymptotic Results · p. 3

2) Numerical Results:

• In the UAV scenarios, performance degrades as the number of interfering UAVs increases, while the i.n.d. shadowing case is shown against transmit power.Source: 2) Numerical Results · p. 4, Figure 2 · p. 4
• Increasing αD reduces the desired-link shadowing mean and average useful received power, lowering coverage probability at the same transmit power.Source: 2) Numerical Results · p. 4
• Analytical and Monte Carlo results agree excellently, with small deviations only in the very low-outage regime.Source: 2) Numerical Results · p. 4
• Below 10 dBm, the asymptotic approximation is accurate and practically useful across the considered operating points.Source: 2) Numerical Results · p. 4

B. Coverage Analysis for HSR Communication Networks

• Backhaul analysis focuses on service quality because it dominates the HSR scenario.The model places a TAP above the train’s lead car and treats passenger-to-TAP and TAP-to-BS transmission as a two-hop path.Source: B. Coverage Analysis for HSR… · p. 4
• High-demand track sections, such as those near stations, use two non-contiguous carriers, while ordinary segments use f1 alone.The carrier pair is f1 and f2.Source: B. Coverage Analysis for HSR… · p. 4
• BSs beside the track use outward-facing directional antennas to form narrow strip-shaped cells, with carrier-specific Hata-based path-loss models for different terrain conditions.The BSs are uniformly spaced beside the track.Source: B. Coverage Analysis for HSR… · p. 4
• With carrier aggregation, the output SNR sums the two backhaul-link SNRs; without it, only the f1 link contributes.The model represents aggregation with η2 = 1 and single-carrier operation with η2 = 0.Source: B. Coverage Analysis for HSR… · p. 5

1) Asymptotic Results:

• For higher transmit powers, the parabolic cylinder function is asymptotically approximated for higher Wk.The approximation is the leading-order truncation, intended to capture dominant behavior in the large-argument regime while preserving analytical tractability.Source: 1) Asymptotic Results · p. 5
• The leading-order truncation of the parabolic cylinder function’s asymptotic expansion supplies the approximation used in this regime.The passage describes it as sufficient to capture dominant behavior at large arguments.Source: 1) Asymptotic Results · p. 5
• Using this approximation and mathematical simplifications, the output-SNR CDF reduces to a closed-form expression.The stated condition is higher transmit power, corresponding to higher Wk.Source: 1) Asymptotic Results · p. 5

2) Numerical Results:

• Across the HSR scenarios, outage probability decreases as transmit power increases; viaduct has the smallest outage and cutting the highest.The comparison is for r = 1 km and γth = 0 dB.Source: 2) Numerical Results · p. 5, Figure 4 · p. 5
• The asymptotic outage curves match the exact results excellently, particularly at higher transmit power levels.The figure compares outage probability versus transmit power across HSR propagation scenarios.Source: 2) Numerical Results · p. 5, Figure 4 · p. 5
• At PT = 40 dBm and edge CP = 3 · 10−1, carrier aggregation extends the cell radius by almost 0.5 km.At large cell radii, edge CP approaches zero for both systems; at small radii and higher transmit power, their coverage can be comparable.Source: 2) Numerical Results · p. 5, Figure 5 · p. 5
• Carrier aggregation consistently outperforms operation without aggregation across scenarios, with the edge-coverage gap widening at moderate radii.At large radii, edge CP approaches zero for both systems.Source: 2) Numerical Results · p. 5, Figure 5 · p. 5

IV. CONCLUSION

• The framework gives exact, numerically robust closed-form PDF and CDF expressions for sums of i.n.d. inverse-gamma RVs without multi-fold integration or slowly convergent series.The conclusion also notes computational efficiency and that asymptotic approximations provide additional insights.Source: §IV · pp. 5–6
• In UAV-assisted networks, the analysis yields tractable outage expressions and examines the effects of interfering-UAV count and i.n.d. versus i.i.d. shadowing.The conclusion also describes HSR coverage dependence on propagation conditions and gains from aggregation, particularly for moderate cell radii.Source: §IV · p. 6
• The framework supports reliable outage and coverage evaluation over broad parameter ranges, and its practical value is demonstrated in UAV-assisted and HSR scenarios.The conclusion describes these as representative scenarios.Source: §IV · pp. 5–6
• Future work could study sums of correlated inverse-gamma RVs, relevant to spatially correlated shadowing across UAVs and along UAV trajectories.The conclusion suggests this analysis could be applied to UAV-assisted communications.Source: §IV · p. 6

• For αk = nk + 1/2, the proof starts from the Laplace transform of Ik in (3) and uses [12, eq. (8.468)].Source: APPENDIX · p. 6
• Because the Ik are independent, the Laplace transform of their sum I is obtained from their individual transforms.Source: APPENDIX · p. 6
• The derivation expresses each term as MI(s) = As^B exp(−C√s), with B = 1, then obtains the PDF and CDF through inverse Laplace transforms.The coefficient A is related to the multiple sum.Source: APPENDIX · p. 6
• Substituting the corresponding A, B, and C values into (19a) and (19b) yields equations (5) and (6), completing the proof.Source: APPENDIX · p. 6