# Partially Observable Markov Decision Process-Based Transmission Policy over Ka-Band Channels for Space Information Networks

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. System Model

**G**of the two-state GE channel as

## 3. Optimal Transmission Policy Based on POMDP

- (1)
- Betting aggressively: If aggressive action A is taken, then, the value function evolves as ${V}_{\beta ,A}({X}_{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{p}_{i})={p}_{i}{R}_{g}+\beta {V}_{\beta}\left(T\left({p}_{i}\right)\right)$;
- (2)
- Betting conservatively: If the conservative action C is selected, the value function evolves as ${V}_{\beta ,C}({X}_{i}={p}_{i})={R}_{b}+\beta {V}_{\beta}\left(T\left({p}_{i}\right)\right)$;
- (3)
- Betting opportunistically: If opportunistic action O is selected, the value function evolves as ${V}_{\beta ,O}({X}_{i}={p}_{i})=(1-\tau )[{p}_{i}{R}_{g}+(1-{p}_{i}){R}_{b}]+\beta {V}_{\beta}\left(T\left({p}_{i}\right)\right)$.

**Theorem**

**1.**

**Proof of Theorem**

**1.**

**Theorem**

**2.**

- (1)
- If ${R}_{b}/{R}_{g}<{\lambda}_{0}$, then the optimal transmission action is ${a}_{i}=A$ regardless of the delayed feedback CSI is ${s}_{i-1}=1$ or ${s}_{i-1}=0$;
- (2)
- If ${R}_{b}/{R}_{g}>{\lambda}_{1}$, then the optimal transmission action is ${a}_{i}=C$ regardless of the delayed feedback CSI is ${s}_{i-1}=1$ or ${s}_{i-1}=0$;
- (3)
- Finally if ${\lambda}_{0}\le {R}_{b}/{R}_{g}\le {\lambda}_{1}$, then the optimal transmission action ${a}_{i}=A$ when the delayed feedback CSI is ${s}_{i-1}=1$, and the optimal transmission action is ${a}_{i}=C$ when the delayed feedback CSI is ${s}_{i-1}=0$.

**Proof of Theorem**

**2.**

**Theorem**

**3.**

- (1)
- If ${R}_{b}/{R}_{g}<{\lambda}_{0}$, then two cases can be distinguished: if ${R}_{b}/{R}_{g}<\tau {X}_{i}/\left((1-\tau )(1-{X}_{i})\right)$, the optimal transmission action is ${a}_{i}=A$, regardless of the delayed feedback CSI being ${s}_{i-1}=1$ or ${s}_{i-1}=0$; else, if ${R}_{b}/{R}_{g}\ge \tau {X}_{i}/\left((1-\tau )(1-{X}_{i})\right)$, the optimal transmission action is ${a}_{i}=O$, regardless of the delayed feedback CSI being ${s}_{i-1}=1$ or ${s}_{i-1}=0$.
- (2)
- If ${R}_{b}/{R}_{g}>{\lambda}_{1}$, then two cases can be distinguished: if ${R}_{b}/{R}_{g}<(1-\tau ){X}_{i}/(\tau +{X}_{i}-\tau {X}_{i})$, the optimal transmission action is ${a}_{i}=O$, regardless of the delayed feedback CSI being ${s}_{i-1}=1$ or ${s}_{i-1}=0$; else, if ${R}_{b}/{R}_{g}\ge (1-\tau ){X}_{i}/(\tau +{X}_{i}-\tau {X}_{i})$, the optimal transmission action is ${a}_{i}=C$ regardless of the delayed feedback CSI being ${s}_{i-1}=1$ or ${s}_{i-1}=0$.
- (3)
- Finally if ${\lambda}_{0}\le {R}_{b}/{R}_{g}\le {\lambda}_{1}$, when the delayed feedback CSI is ${s}_{i-1}=1$ and ${X}_{i}={\lambda}_{1}$, then the optimal transmission action is ${a}_{i}=A$; when the delayed feedback CSI is ${s}_{i-1}=0$ and ${X}_{i}={\lambda}_{0}$, then the optimal transmission action is ${a}_{i}=C$.

**Proof of Theorem**

**3.**

## 4. Simulation and Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**The optimal transmission policies thresholds are determined by the SIN communication parameters based on delayed feedback CSI.

**Figure 4.**Illustration of the two-thresholds policy structure: (

**a**) ${\rho}_{2}<{\lambda}_{0}$; (

**b**) ${\rho}_{1}<{\lambda}_{0}<{\rho}_{2}<{\lambda}_{1}$; (

**c**) ${\rho}_{1}<{\lambda}_{0}<{\lambda}_{1}<{\rho}_{2}$; (

**d**) ${\lambda}_{0}<{\rho}_{1}<{\rho}_{2}<{\lambda}_{1}$; (

**e**) ${\lambda}_{0}<{\rho}_{1}<{\lambda}_{1}<{\rho}_{2}$; (

**f**) ${\lambda}_{1}<{\rho}_{1}$.

**Figure 5.**Numerical result of our POMDP-based transmission policy: (

**a**) optimality of a one-threshold policy scenario; (

**b**) optimality of a two-thresholds policy scenario.

**Figure 6.**Expected reward of adaptive transmission schemes with different setups: (

**a**) $\tau =0.4$, one-threshold policy; (

**b**) $\tau =0.1$, two-thresholds policy.

**Figure 7.**Throughput comparison of different data transmission schemes: (

**a**) Moon-to-Earth scenario; (

**b**) Mars-to-Earth scenario.

Parameters | Delayed Feedback CSI ${\mathit{s}}_{\mathit{i}-1}$ | Optimal Transmission Action ${\mathit{a}}_{\mathit{i}}$ |
---|---|---|

${R}_{b}/{R}_{g}<{\lambda}_{0}$ | ${s}_{i-1}=1/0$ | ${a}_{i}=A$ |

${\lambda}_{0}\le {R}_{b}/{R}_{g}\le {\lambda}_{1}$ | ${s}_{i-1}=1$ | ${a}_{i}=A$ |

${s}_{i-1}=0$ | ${a}_{i}=C$ | |

${R}_{b}/{R}_{g}>{\lambda}_{1}$ | ${s}_{i-1}=1/0$ | ${a}_{i}=C$ |

Conditions | Corresponding Value Functions | Closed Form Computation Expressions |
---|---|---|

$\rho <{\lambda}_{0}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{{\lambda}_{0}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{({\lambda}_{1}-\alpha \beta ){R}_{g}}{(1-\beta )(1-\alpha \beta )}$ |

${\lambda}_{0}\le \rho \le {\lambda}_{1}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{(1-\beta {\lambda}_{1}){R}_{b}+\beta {\lambda}_{0}{\lambda}_{1}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{\beta (1-{\lambda}_{1}){R}_{b}+(1-\beta +\beta {\lambda}_{0}){R}_{g}}{(1-\beta )(1-\alpha \beta )}$ |

${\lambda}_{1}<\rho $ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{{R}_{b}}{(1-\beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{{R}_{b}}{(1-\beta )}$ |

Parameters | Conditions | Delayed Feedback CSI ${\mathit{s}}_{\mathit{i}-1}$ | Optimal Action ${\mathit{a}}_{\mathit{i}}$ |
---|---|---|---|

$\frac{{R}_{b}}{{R}_{g}}<{\lambda}_{0}$ | Figure 4a: ${R}_{b}/{R}_{g}<\mathcal{A}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ${a}_{i}=A$ |

Figure 4a: ${R}_{b}/{R}_{g}<\mathcal{A}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | ||

Figure 4b: ${R}_{b}/{R}_{g}<\mathcal{A}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ||

Figure 4b: ${R}_{b}/{R}_{g}\ge \mathcal{A}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | ${a}_{i}=O$ | |

Figure 4c: ${R}_{b}/{R}_{g}\ge \mathcal{A}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ||

$\frac{{R}_{b}}{{R}_{g}}>{\lambda}_{1}$ | Figure 4c: ${R}_{b}/{R}_{g}\ge \mathcal{C}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | |

Figure 4e: ${R}_{b}/{R}_{g}\ge \mathcal{C}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ||

Figure 4e: ${R}_{b}/{R}_{g}<\mathcal{C}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | ${a}_{i}=C$ | |

Figure 4f: ${R}_{b}/{R}_{g}<\mathcal{C}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ||

Figure 4f: ${R}_{b}/{R}_{g}<\mathcal{C}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | ||

${\lambda}_{1}\le \frac{{R}_{b}}{{R}_{g}}\le {\lambda}_{1}$ | Figure 4d: ${R}_{b}/{R}_{g}<\mathcal{A}\left({\lambda}_{1}\right)$ | ${s}_{i-1}=1$ | ${a}_{i}=A$ |

Figure 4d: ${R}_{b}/{R}_{g}\ge \mathcal{C}\left({\lambda}_{0}\right)$ | ${s}_{i-1}=0$ | ${a}_{i}=C$ |

Conditions | Corresponding Functions | Closed Form Computation Expressions |
---|---|---|

${\rho}_{2}\le {\lambda}_{0}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{{\lambda}_{0}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{({\lambda}_{1}-\alpha \beta ){R}_{g}}{(1-\beta )(1-\alpha \beta )}$ |

${\rho}_{1}<{\lambda}_{0}<{\rho}_{2}\le {\lambda}_{1}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,O}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{(1-\tau )(1-\beta {\lambda}_{1}+\beta {\lambda}_{0}{\lambda}_{1}){R}_{b}+(1-\tau ){\lambda}_{0}({R}_{g}-{R}_{b})+\tau \beta {\lambda}_{0}{\lambda}_{1}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{(1-\tau )\beta (1-{\lambda}_{0})(1-{\lambda}_{1}){R}_{b}+({\lambda}_{1}-\alpha \beta -\tau \beta {\lambda}_{0}+\tau \beta {\lambda}_{0}{\lambda}_{1}){R}_{g}}{(1-\beta )(1-\alpha \beta )}$ |

${\rho}_{1}<{\lambda}_{0}<{\lambda}_{1}\le {\rho}_{2}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,O}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,O}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{(1-\beta {\lambda}_{1}){R}_{b}+\beta {\lambda}_{0}{\lambda}_{1}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{\beta (1-{\lambda}_{1}){R}_{b}+(1-\beta +\beta {\lambda}_{0}){\lambda}_{1}{R}_{g}}{(1-\beta )(1-\alpha \beta )}$ |

${\lambda}_{0}<{\rho}_{1}<{\rho}_{2}\le {\lambda}_{1}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,A}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{(1-\tau )((1-\alpha \beta ){R}_{b}+{\lambda}_{0}({R}_{g}-{R}_{b}))}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{(1-\tau )((1-{\lambda}_{1}){R}_{b}+({\lambda}_{1}-\alpha \beta ){R}_{g})}{(1-\beta )(1-\alpha \beta )}$ |

${\lambda}_{0}<{\rho}_{1}<{\lambda}_{1}\le {\rho}_{2}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,O}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{(1-\beta +\beta {\lambda}_{0}){R}_{b}+(1-\tau )\beta {\lambda}_{0}{\lambda}_{1}({R}_{g}-{R}_{b})}{(1-\beta )(1-\alpha \beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{\beta (1-\tau ){R}_{b}+(1-\tau )(1-\beta +\beta {\lambda}_{0})((1-{\lambda}_{1}){R}_{b}+{\lambda}_{1}{R}_{g})}{(1-\beta )(1-\alpha \beta )}$ |

${\lambda}_{1}\le {\rho}_{1}$ | ${V}_{\beta}\left({\lambda}_{0}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{0})$ ${V}_{\beta}\left({\lambda}_{1}\right)={V}_{\beta ,C}({X}_{i}={\lambda}_{1})$ | ${V}_{\beta}\left({\lambda}_{0}\right)=\frac{{R}_{b}}{(1-\beta )}$ ${V}_{\beta}\left({\lambda}_{1}\right)=\frac{{R}_{b}}{(1-\beta )}$ |

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**MDPI and ACS Style**

Jiao, J.; Sui, X.; Gu, S.; Wu, S.; Zhang, Q.
Partially Observable Markov Decision Process-Based Transmission Policy over Ka-Band Channels for Space Information Networks. *Entropy* **2017**, *19*, 510.
https://doi.org/10.3390/e19100510

**AMA Style**

Jiao J, Sui X, Gu S, Wu S, Zhang Q.
Partially Observable Markov Decision Process-Based Transmission Policy over Ka-Band Channels for Space Information Networks. *Entropy*. 2017; 19(10):510.
https://doi.org/10.3390/e19100510

**Chicago/Turabian Style**

Jiao, Jian, Xindong Sui, Shushi Gu, Shaohua Wu, and Qinyu Zhang.
2017. "Partially Observable Markov Decision Process-Based Transmission Policy over Ka-Band Channels for Space Information Networks" *Entropy* 19, no. 10: 510.
https://doi.org/10.3390/e19100510