在傳統的訊息理論中，我們通常是以傳輸通道的通道容量做為網路效能評估的依據。然而，在多重跳躍無線網路中，一個封包到達目的地之前需要透過許多中繼站幫忙傳輸，假若每次傳輸距離較長，則所需經過的中繼站即較少。所以，在評估無線網路的效能時傳輸距離也應該在裡面扮演重要的角色。有鑒於此，Gupta和Kumar在2000年提出“傳輸容量＂做為評估無線網路效能的依據。其傳輸容量的定義為網路中所有傳輸連結的傳輸速度和資料成功傳達的報酬的乘積之和。隨後，Reznik和Verdü在2002年得到高斯廣播通道的最大傳輸容量。而Gupta等人在2006年求得高斯多重存取通道的最大傳輸容量，並且利用Jindal等人在2004年證明的在相同總功率限制下高斯多重存取通道和高斯廣播通道之間的對偶關係進而推得高斯廣播通道的最大傳輸容量。最近，Cheng在2007年推廣Gupta等人在2006年時的結果。
大部分現存關於最大傳輸容量的文獻都只在所有節點（每個節點可以是發射機或是接收機）位置皆固定的無線網路環境下探討。然而，Grossglauser和Tse證明假若允許所有節點均可移動，當所有節點都移動到最佳位置時，即可增加最大傳輸容量。因此，在這篇論文中，我們假設所有節點均可移動，進而研究高斯多重存取通道的最大可能的傳輸容量以及達到此最大值的節點最佳位置。

In this thesis, we consider the transport capacity of a Gaussian multiple access channel (MAC) in a mobile communications scenario in which all the nodes are allowed to be mobile. The transport capacity was first proposed by Gupta and Kumar as a figure of merit about how effectively a wireless network operates, and is given by the summation of the ratereward products (the products of the data rates and the associated rewards for the successful transmission of the data) over all transmitter-receiver pairs. Most existing works on the maximal transport capacity are for a fixed wireless communications network in which all the nodes are located at fixed positions. However, it has been shown by Grossglauser and Tse that if mobility could be exploited to increase the maximal transport capacity. Therefore, in this thesis we investigate the maximal transport capacity of Gaussian MACs in the scenario that all the nodes are allowed to be mobile, and study the optimal positions of the mobile nodes that achieve the largest possible maximum transport capacity.

[1] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Transactions on Information Theory, vol. 46, pp. 388–404, March 2000.

[2] A. Reznik and S. Verd¨u, “On the transport capacity of a broadcast Gaussian channel,”Communications in Information and Systems, vol. 2, pp. 183–216, December 2002.

[3] G. A. Gupta, S. Toumpis, J. Sayir, and R. R. M¨uller, “On the transport capacity of Gaussian multiple access and broadcast channels,” to appear in ACM/Baltzer Wireless
Networks.

[4] F.-T. Hsu and N. Ge, “The power allocation for Gaussian multiple access and broadcast channels to achieve maximal transport capacity,” in Proceedings IEEE Asia Pa-
cific Wireless Communications Symposium (APWCS’07), Hsinchu, Taiwan, R.O.C., August 20–21, 2007.

[5] J. Cheng, “Transport capacity of Gaussian multiple access and broadcast channels with general reward and gain functions,” in Proceedings IEEE Asia Pacific Wireless Com-
munications Symposium (APWCS’07), Hsinchu, Taiwan, R.O.C., August 20–21, 2007.

[6] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge, UK: Cambridge University Press, 2004.

[7] B. Jindal, S. Vishwanath, and A. Goldsmith, “On the duality of Gaussian multipleaccess and broadcast channels,” IEEE Transactions on Information Theory, vol. 50, pp. 768–783, May 2004

[8] T. M. Cover and J. A. Thomas, Elements of Information Theory, New York, NY: John Wiley & Sons, 1991.

[9] R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advance in Computational Mathematics, vol. 5, pp. 329–359,
December 1996.

[10] M. Grossglauser and D. N. C. Tse, “Mobility increases the capacity of ad hoc wireless networks,” IEEE/ACM Transactions on Networking, vol. 10, pp. 477–486, August 2002.