A Cooperative Differential Game Model for Multiuser Rate-Based Flow Control

Abstract

Flow control is one typical resource management tool. Its objectives are to adjust the traffic in the network, and resolve data traffic congestion, maximize the utilization of available bandwidth resource and realize fairness among different sources. Taking more consideration on the interaction and the dynamic characteristics of the flow, in this paper, we propose a cooperative differential game of flow control. Our goal is to assign sources with appropriate transmission rate levels so as to the queue length in the bottleneck link is optimal for the network performance and their benefits are maximal. We derive a payment distribution mechanism that would constitute a time-consistent solution and guarantee individual rationality.

Appendix 1

Proof of Proposition 1.

Proof

Assume that $V^{i}(q_l )$ is a polynomial with respect to the state variable $q_l$. In line with Eqs. (6) and (7), we assume that

$$\begin{aligned} V^{i}(q_l )=1/2 A_{i} q_{l}^{2} +B_{i} q_{l} +C_{i}. \end{aligned}$$

(20)

$\square$

Substituting the results from (7) and (20) into (6), and upon solving (6) yields:

$$\begin{aligned}&1/2rA_i q_l^2 +rB_i q_l +rC_i =\upmu _i (\upmu _i -\alpha _i +A_i q_l+B_i )-1/2(\upmu _i -\alpha _i +A_i q_l+B_i )^{2} \\&\quad -\alpha _i (\upmu _i \!-\!\alpha _i \!-\!w_i s\!+\!A_i q_l\!+\!B_i )\!-\!\upbeta _i q_l^2 \!+\!(A_i q_l\!+\!B_i )\left[\sum _{j=1}^N {(\upmu _j \!-\!\alpha _j \!-\!w_i s\!+\!A_j q_l+B_j )} \right]\\&\quad =\upmu _i^2 \!-\!\upmu _i \alpha _i \!+\!\upmu _i A_i q_l\!+\!\upmu _i B_i \!-\!1/2(\upmu _i \!-\!\alpha _i +B_i )^{2}\!-\!(\upmu _i \!-\!\alpha _i \!+\!B_i )A_{i} q_{l} -1/2(A_i q_l)^{2} \\&\quad -\alpha _i \upmu _i \!+\!\alpha _i^2 \!+\!\alpha _i w_i s\!-\!\alpha _i A_i q_l\!-\!\alpha _i B_i \!-\!\upbeta _i q_l^2 \!+\!\sum _{j=1}^N {(\upmu _j \!-\!\alpha _j \!-\!w_i s\!+\!B_j )} A_i q_l\!+\!\sum _{j=1}^N {A_j A_i q_l^2 } \\&\quad +B_i \left[\sum _{j=1}^N {(\upmu _j -\alpha _j -w_i s+B_j )} \right]+B_i \sum _{j=1}^N {A_j q_l}. \end{aligned}$$

For the above two equations to hold, we have that

$$\begin{aligned}&A_i^2 -\left(r-2\sum _{\begin{array}{l} j=1, \\ j\ne i \\ \end{array}}^N {A_j }\right) A_i -2\upbeta _i =0, \\ rB_i&= \sum _{j=1}^N {(\upmu _j -\alpha _j -w_i s)} A_i +\sum _{\begin{array}{l} j=1, \\ j\ne i \\ \end{array}}^N {B_j } A_i +B_i \sum _{j=1}^N {A_j } \\ rC_i&= 1/2(\upmu _i -\alpha _i +B_i )^{2}+\alpha _i w_i s+B_i \left[\sum _{\begin{array}{l} j=1, \\ j\ne i \\ \end{array}}^N {(\upmu _j -\alpha _j -w_i s+B_j )} \right]. \end{aligned}$$

This completes the proof of Proposition 1.