QoS-equilibrium slot allocation for beam hopping in broadband satellite communication systems

Abstract

In broadband satellite communication systems, beam hopping (BH) is more and more welcome to compensate the low-power-efficiency wide beam and the low-spectrum-efficiency multiple spot beams simultaneously. As a flexible coverage method, BH demands efficient hopping strategy to reach the desired quality of service (QoS). In this paper, we consider the QoS equilibrium, in particular the delay fairness, among different cells illuminated by one hopping beam through designing the hopping strategy on the condition of capacity satisfaction. Due to the physical limitation that hopping speed is much slower than packet arrival speed, the ideal first-come-first-serve packet scheduling is not realizable, which compels us to pursuit the optimal slot scheduling instead. In this paper, we consider a long-term delay fairness problem by per-period slot allocation for BH, using instantaneous and statistical information. To deal with the complex stochastic programming problem, we introduce the time-sharing principle to uncouple the variables and adopt stochastic gradient theory to deduce the suboptimal close-form solution. Our contributions are in three folds: firstly, it is the first time to consider the delay fairness not the traditional capacity fairness for BH; secondly, a general per-period decision scheme is proposed for the unique slot allocation in the perspective of queuing; thirdly, a suboptimal close-form solution for the per-period slot allocation is obtained, which could also cope with burst traffic cases. In simulation, we verify our method’s advantage in terms of delay fairness comparing with the existing similar works.

Appendices

Appendix 1: Proof of Theorem 1

Without loss of generality, we define ${{\mathop {t}\limits ^{\rightharpoonup }}^*} = \left( {t_1^ *,t_2^ * ,\ldots ,t_M^ *} \right)$ as the optimal solution for minimizing the reference function under constraint, i.e.

$$\begin{aligned} {{\mathop {t}\limits ^{\rightharpoonup }}^*}= arg\min {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T). \end{aligned}$$

(28)

subjecting to $\sum \nolimits _{i = 1}^M {t_i^ * /T} = 1$. Since ${{\mathop {t}\limits ^{\rightharpoonup }}^*}$ is the minimum value, for any $\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}$, we have that

$$\begin{aligned} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) \ge {\fancyscript{G}}({{\mathop {t}\limits ^{\rightharpoonup }}^*}/T). \end{aligned}$$

(29)

Substituting this $\mathop {t}\limits ^{\rightharpoonup }$ into the shifted function, we have that

$$\begin{aligned} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } ) > {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T). \end{aligned}$$

(30)

Therefore, we have

$$\begin{aligned} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } ) > {\fancyscript{G}}({{\mathop {t}\limits ^{\rightharpoonup }}^*}/T). \end{aligned}$$

(31)

for any given $\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}$. This is to say,

$$\begin{aligned} \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } ) > {\fancyscript{G}}({{\mathop {t}\limits ^{\rightharpoonup }}^ * }/T). \end{aligned}$$

(32)

This completes the proof.

Appendix 2: Proof of Lemma 1

At first, it can be easily known that

$$\begin{aligned} \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) + \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } ) \le \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} \left[ {{\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) + {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } )} \right] . \end{aligned}$$

(33)

Combining with the Theorem 1 which says $\mathop {\hbox {min}}\nolimits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) <\mathop {\hbox {min}}\nolimits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } )$, we know

$$\begin{aligned} \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} \mathrm{{2}}{\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) \le \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} \left[ {{\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) + {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } )} \right] , \end{aligned}$$

(34)

which proves the left part of (18). Then, we define a new function $\varphi (\mathop {t}\limits ^{\rightharpoonup } /T) = \sum \nolimits _{i = 1}^M {\frac{{{A_i}}}{{{t_i}/T \times {\mu _i}}}} + \sum \nolimits _{i = M + 1}^{2M} {\frac{{{A_i}}}{{{t_i}/T \times {\mu _i} - {\lambda _i}}}}$ with $2M$ variables, i.e. $\mathop {t}\limits ^{\rightharpoonup } = ({t_1},{t_2},\ldots ,{t_M},{t_{M + 1}},\ldots ,{t_{2M}})$, where ${\mu _i} = {\mu _{i + M}}, {A_i} = {A_{i + M}}, {\lambda _i} = {\lambda _{i + M}}$.

In addition, the candidate variable vector set is set as ${\mathfrak{R}}' = \left\{ {({t_1},{t_2},\ldots ,{t_{2M}})|({t_1},{t_2},\ldots ,{t_M}) \in {\mathfrak{R}},{t_i} = {t_{i + M}},\forall i} \right\}$. Learned from the proof of the Theorem 1, we know that the conclusion of Theorem 1 could also be used to the new function, i.e.

$$\begin{aligned} \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}'} \varphi (\mathop {t}\limits ^{\rightharpoonup } /T) < \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}'} \, \,\varphi (\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {q}\limits ^{\rightharpoonup }), \hbox {where} \mathop {q}\limits ^{\rightharpoonup } = (\mathop {p}\limits ^{\rightharpoonup },\underbrace{0,0,\ldots ,0}_M). \end{aligned}$$

(35)

This proves the right part of the (18):

$$\begin{aligned} \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} \left[ {{\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T) + {\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } )} \right] < \mathop {\min }\limits _{\mathop {t}\limits ^{\rightharpoonup } \in {\mathfrak{R}}} \mathrm{{2}}{\fancyscript{G}}(\mathop {t}\limits ^{\rightharpoonup } /T - \mathop {p}\limits ^{\rightharpoonup } ). \end{aligned}$$

(36)

Till now, the proof is completed.