Journal of Machine Intelligence and Data Science (JMIDS)

Nomenclature

1. Introduction

This paper studies satellite communication channels. The first important feature of satellite communication channel is interference. Each satellite transponder represents an individual communication channel. Within a 36-MHz bandwidth channel, each transponder can handle an enormous amount of information by using different multiple-access schemes, so each channel contains many pairs of senders and receivers [1], [2]. This study assumes each pair is selfish to maximize its own performance by a specific power-allocation scheme. The interference from other pairs affects the channel performance [3]. Satellites most commonly use the C band (6/4 GHz), and the C band’s heavier use leads to more interference. Shifting satellite communication to higher frequencies is one effective way to minimize interference, but crowding and interference problems will still exist, which motivates this study to develop a technique that increases bandwidth efficiency and signal-caring capacity, and decreases interference of satellite communication subsystems. The second important feature of satellite communication is long distance, which implies the dynamic and controllable feature of channel gain has to be considered when modelling the transponder’s communication channel. The third important feature of a satellite transponder, which is interesting and studied in this paper is the existence of Stackelberg competition in this channel. The status of users in a channel are not always same. Some users called leader have priority to select their power control strategy, which implies other users called followers have to optimize with the given leader’s control strategy. Thus, it is interesting to study how the leaders take advantage of this opportunity to improve their utility, and how the followers survive better.

This paper models a transponder’s communication channel as an interference channel with aim to optimize the trade-off between transmission data rate and power consumption. Section II reviews a transponder’s communication channel and existing energy-efficient power control game models. Section III models the power-allocation optimization problem for all users in a transponder as a Stackelberg Differential Gaussian Interference Channel Game (SDGICG) based on the special properties of satellite wireless communications. Section IV and Section V derive and analyse the SDGICG model’s optimality condition, and develop numerical methods to solve for Stackelberg equilibrium and then solve the model. The numerical solution from the model provides all users in a transponder’s channel with the optimal power-allocation scheme at the Stackelberg equilibrium.

2. Preliminaries

2. 1. Satellite Wireless Communications Subsystem

A transponder is a repeater that implements a wideband communication channel that can carry many simultaneous one-to-one communication transmissions [1], so it can be modelled as a multiuser interference channel as Fig. 1 [4], [5]. This interference channel is an M-to-M network where a one-to-one correspondence exists between senders and receivers such that each sender communicates information only to its corresponding receiver [4]. This study models each pair of sender-receiver in a transponder channel as a user (a player). The interference limits the system’s performance. Interference cancellation is an option when the interference signal is sufficiently strong, but its implementation is complex, requiring prior knowledge of users’ transmission schemes is accessible by other users [5], [6]. This study assumes that each user applies power to affect the cross-coupling gain and then reduce interference without any interference cancellations.

2. 2. Static Power Control Game

Goodman and Mandayam [7] study a static energy-efficient power control game on a distributed multiple-access channel with a finite number of users, denoted by K. Each user chooses its own power control policy p_i to maximize its energy-efficiency u_i=(R_i f(SINR_i))/p_i , where R_i is the information transmission rate in bit/s for user i, and f is an efficiency function representing the block success rate, which is assumed to be sigmoidal and identical for all the users [7], [8].

It is called a static game because (a) it assumes that the users transmit data over quasi-static or block-fading channels at the same time and in the same frequency band, assuming each channel gain H_i (n) to be constant over each block. (b) Each user in the game applies a fixed power policy, once per block, to maximize its utility. However, for long-distance wireless communication such as satellite communication, channel gain varies with time, so its modulus is usually assumed to be in a compact set |H_i|²∈[η_i^min,η_i^max]. A variable power policy is expected to be designed to control channel gain. Furthermore, with assumption of complete information and rationality, the existence of Nash Equilibrium is guaranteed by Debreu-Fan-Glicksberg existence theorem [9]. The Nash Equilibria are found by solving equations . And the static power game has unique pure Nash Equilibrium, which is discussed by Yates [10], and Saraydar [11].

Besides the energy-efficient game for communication channel, there are other types of noncooperative games constructed for different utility, which are generally called Gaussian Interference Games (GIGs) [12], [13]. The water-filling algorithm also solves for Nash Equilibrium of GIG without the need for centralized control [13]. Amir Leshem applied cooperative game theory for analysing interference channels [14]. Wei Wan created a cooperative static game for a transponder’s centralized power control to maximize overall channel data transmission rate [15].

2. 3. Non-cooperative vs Stackelberg Differential Game (DG)

In non-cooperative DG models, all the competitors make decision at the same time, and their controls are combined in the same dynamics. No competitor has knowledge of the strategies of others as he decides on his own. The player i’s objective function is

(1)

, and all competitors’ controls are combined in the same dynamics:

(2)

The most interesting solution of non-cooperative DG model is Nash Equilibrium. Optimality condition for Nash Equilibrium of non-cooperative DG is a set of Differential-Algebraic Equations (DAE). Wei Wan and John Cioffi [16] set up a non-cooperative Differential Game (DG) to model the users’ competition in a transponder’s communication channel. In this game model, each user’s energy efficiency is redefined, and logistic growth is adopted to approximate the changing of channel gain under specific energy consumption. The objective function of each user is a weighted sum of energy efficiency and targeted channel gain. Then, the optimality condition for Nash Equilibrium of the model is derived. At last, an algorithm is developed to solve for Nash Equilibrium. The design of algorithm is based on a steep-descent method and optimizes all players’ objective functions simultaneously.

Stackelberg game provides a model for a system where the status of competitors is not same. Stackelberg game is played as follows in Fig. 2.

The users who select control first are leader. The leader first announces his control policy u. Then, the other users who are able to observe leader’s control u and then select their controls are followers. The followers select their control v^* by solving . In the end, the leaders select their optimal control by solve .

Furthermore, when controls are combined in a dynamic system, above game is played as Stackelberg Differential Game (SDG). The leader announces at time t=0 that he will use the control u(x(t),t) for t∈[0,T]. Then, the follower has to take the leader’s control path as given and selects his own control v(x(t),t) to minimize his objective functional

(3)

Thus, there exists a set-valued mapping:
F:U→V
by Fu={v∈V│v=argmin⁡[J₂ (u,v)] }. From Pontryagin minimum principle, if an optimal v exists, with v∈Fu, we must have a function λ:[0,T]→Rⁿ such that

(4)

For v∈Fu, the leader’s problem is to select his control u(x,t) to minimize his objective function

(5)

, which can be written explicitly as follows:

(6)

By solving above problem, the optimal control u(x,t) for the leader could be obtained, and then solving the follower’s optimal control problem to get his optimal control v(x,t).

3. Stackelberg Differential Game for a Transponder

In the process of designing power-control policy, it is assumed that some users in one satellite transponder’s channel have priority over others in terms of the order of making decision. That is there exists Stackelberg competition when allocating power among the users in this channel.

Each pair of (x_i,y_i),i∈κ represents a user in a transponder’s channel. All users choose their power-control policy before establishing communication. Each user’s communication is through N sub-frequency channels simultaneously, and each user applies independent power control policy in each sub-frequency channel. Furthermore, each user has two types of power consumption policy: the first improves its own channel gain, and the second decreases interference. Stackelberg differential game is modelled as follows.

Construction of objective function: Since the first and most interesting objective for each user in this transponder is to optimize the trade-off between the achievable data rate and energy consumption. With an assumption of no channel interference cancellation, the interference from other users is consequently noise. Then, the achievable rate for player i at time t over frequency (f¹,f²) is as follows [5], [3]:

(7)

, where approximation assumes the variables to be constant over small bands. The energy efficiency for user i,i∈κ over time [0,T] is

(8)

,which is the log transformation of ratio of information bits that are transmitted without error per unit time to the transmit power. It is to be maximized. The second goal of transponder power control is for the direct channel gain to reach a certain channel-capacity level and also to reduce the cross-coupling gain to certain level. This second objective is to minimize the following expression:

(9)

, where are weights between different objectives; are constants, and upper bounds of ; and are targeted channel-gain levels.

Construction of dynamics: Generally, belong to a compact set , and can be approximated by Kronecker’s delta function [17]. In satellite wireless communication, satellite transponders can apply energy to impact and control channel gain. The analysis in this paper assumes that the growth rate is proportional to power consumption. Thus, logistic growth with carrying capacity is adopted to approximate the dynamics of :

(10)

, where is a constant, and presents the growth rate. is the fixed constant over frequency for user , which stands for the proportion of , used by user to increase channel gain. Furthermore, when user applies to improve the channel gain , it also increases the cross-coupling gain . Furthermore, user  is able to cost power to decrease interference brought by . At last, because of threshold effects existing in channel gain, cross-coupling gain has a lower bound. Thus, the dynamics of is approximated by:

(11)

where i ≠ j, and β_i^f represents the growth rate, ξ_i^f is the lower bound of cross-coupling gain. In order to save space, x_ii(t), x_ij(t) stand for |H_ii^f(t)|², |H_ij^f(t)|², respectively. After the objective function, control, and dynamics of the system are defined, the SDGICG model (κ, {p_i^f}_{i ∈ κ}, {J_i}_{i ∈ κ}) is played according to (3)-(6).

4. Solution of SDGICG Model

There are two methods to solve SDGICG models. The first one is based on solving (6). If the control could be solved from , the following nonclassical optimal control problem is obtained:

(12)

By solving the above optimal control problem, the optimal control u(x,t) for the leader is obtained, and then the follower’s optimal control problem is solved to get his optimal control v(x,t). The obvious disadvantage of this method is: (1) the solvability of control v. (2) Deriving optimality condition for (12) is challenging, which is usually Differential Algebraic Equations and is difficult to solve [18].

The second method is iterative method. The design of this method is based on the procedure of how Stackelberg game is played in Fig. 2. In the beginning, leader and follower initiate a control  over for themselves, respectively. With their controls, the dynamics of the system is solved and then their objective functions are evaluated. Next, with this leader’s control being unchanged and embedded in the follower’s dynamics, the follower searches for an optimal control to optimize its objective function. Next, with this follower’s optimal control being embedded in the leader’s dynamics, the leader searches for its optimal control to optimize its objective function . Then, we calculate the improvement of leader’s objective function values between this step and the first step: . If the difference is small enough (), then the searching procedure stops. Otherwise, the leader’s updated optimal control is embedded in the follower’s dynamics, and follower begins to another search for its optimal control . The procedure continue until leader’s objective function values has no/little improvement. Above iterative procedure is exhibited in Fig. 3.

Based on above iterative procedure, an iterative algorithm is developed to solve the SDGICG model (Algorithm 1). In Step 2 of Algorithm 1, after follower embeds leaders’ discretized control to solve its own game model, the iterative algorithm in [16] is used and embedded. In Step 3, the iterative algorithm in [16] is also used to solve for leader’s optimal control.

Algorithm 1:

Step 1: Leader and follower initiate a random discrete control
{u_L^(f,j)(k)} and {u_F^(f,j)(k)} over t ∈ [0, T].
u_L^(f,j)(k) = u_L^(f,j)(t), t ∈ [t_k, t_(k+1)), k = 1, ⋅⋅⋅, M
u_F^(f,j)(k) = u_F^(f,j)(t), t ∈ [t_k, t_(k+1)), k = 1, ⋅⋅⋅, M

where f = 1, …, N. j stands for the j^th iteration, and set j = 0.

Then, solve the dynamics and calculate both the leader's and follower’s objective function values: J_L^j and J_F^j.

Step 2: The follower embeds the leader’s discretized control {u_L^(f,j)(k)} to solve his own game model for an optimal control {u_F^(f,j+1)(k)}, and calculates the follower’s objective function values: J_F^(j+1).

Step 3: The leader embeds the follower’s optimal control {u_F^(f,j+1)(k)} to solve his own game model for {u_L^(f,j+1)(k)} and calculates his state variable values and his objective function value J_L^(j+1).

Step 4: Check if |J_L^j - J_L^(j+1)| < ε, then stop, and output {u_L^(f,j+1)(k)} and {u_F^(f,j+1)(k)}; otherwise, set j = j + 1 and go back to Step 2.

5. Numerical Experiments

A two-player SDGICG over one sub-frequency channel is solved by Algorithm 1. The numerical experiment aims to study the effects of the cost of power on Stackelberg Equilibrium. The values of parameters are in Table 1 and Table 2.

Comparing the values of parameters in Table 1, these two players are symmetric/same except for the cost of power c_i and the order of selecting their power control policies. The relation c₁ > c₂ implies player 1’s cost of power is more expensive than player 2. And in this game, player 1 is the leader, and player 2 is the follower. Other parameters have the same values for both players. The relation of weights w₁⁽ⁱ⁾ > w₂⁽ⁱ⁾ implies both players prefer the direct channel gain to reach a certain channel-capacity level rather than reducing the cross-coupling gain to a certain level. The value of r_ii^f = 0.9 implies both players aim to reach 90% of channel-gain capacity at the end of the game, and r_ji^f = 0.3 implies they aim to reduce cross-coupling gain to 30% of channel-gain capacity at the end of the game.

TABLE 1. PARAMETERS OF OBJECTIVE FUNCTIONS

Player 1 - Leader		Player 2 - Follower
c1	6	c2	4
σ1	0.2	σ2	0.2
w1(1)	2	w1(2)	2
w2(1)	1	w2(2)	1
r11(1)	0.9	r22(2)	0.9
r21(1)	0.3	r12(2)	0.3