本論文考慮多址傳送(Multicasting)使用累增性前向錯誤更正(Forward Error Correction)的模型。在此模型中，只有一位發送者但有 r^n 位接收者。此發送者使用理想的 (n, n(1-p), np) FEC編碼方式，來對n(1-p)個資料封包進行編碼，因而會多出np個多餘的封包，然而只要任意n(1-p) 個封包被某個接收者收到，就可以用來還原回原本的 n(1-p) 個資料封包。封包到這些接收者的遺失機率為q且互相獨立。為了這個模型，我們在n逼近無窮大時對極限通行速率證明了強大數法則。極限通行速率是用帶有p、q 及 r元素的方程式的唯一解來描繪其特性。

接著，我們證明如果發送者傳送夠多的封包數，所有接收者皆能收到足夠封包數(機率為 1)。這給我們的啟示是，如果發送者知道p、q 及 r元素，便可估計所需傳送的封包數，以使所有接收者皆能收到足夠封包。這是不需要接收者的回授的。如果發送者不知道封包遺失機率 q ，我們建議可採部分回授的規約，即擇定部分具有回授能力的接收者，然後利用極限通行速率的公式來估算 q。

當發送者曉得q 及 r值時，則可針對p值最佳化，使其通行速率最高。我們證明p最佳值會讓每個封包傳送相同的次數。基於此，我們發展了一個有效率的演算法，以計算最佳的p值。我們的數值結果解答了文獻上所提出的問題，即p最佳值應如何配合資料量加以選擇。

　　當多址傳送是給好幾個不同群組時，我們證明了其通行速率是由最差的那一群組所決定。這亦提供了理論認證給文獻上觀察結果。

　　我們將先前的強大數法則，延伸至具有單一共同連結的多址傳送模型(衛星模型)。對此模型，每個接收者所收到的封包數變成是空間上相關的。從各個數值上實例，我們證明空間上相關是可以增加多址傳送之通行速率。這亦吻合文獻上的觀察。

　　最後，我們解除FEC 碼具有固定比值的假設。當所有n個封包傳送完畢，但仍有接收者未收到足夠的封包時，發送者可以建立更多的多餘封包加以傳送，而不是重傳先前的n個封包。對此方案，我們調整應用先前的分析方法，對其極限通行速率證明了一項強大數法則。數值實例顯示，當r值小時，其極限通行速率與先前的極限通行速率雷同，但當r值變大時，便可超出先前的極限通行速率。

總之，我們在本論文中考慮多址傳送使用累增性前向錯誤更正的模型，對其極限通行速率證明了幾項強大數法則。這些強大數法則不僅在文獻中為許多重要的觀察提供理論證明，並且也提供可能會在未來對設計多址傳送協定造成衝擊的洞悉。

In this thesis, we consider a multicasting model that uses incremental FEC (Forward Error Correction). In this model, there is one sender and $\rd^n$ receivers. The sender uses an ideal $(n, n(1-\pone), n\pone)$ FEC code to code a group of $n (1-\pone)$ data packets with additional $n \pone$ redundant packets so that any set of $n(1- \pone)$ packets received by a receiver can be used to recover the original $n(1-\pone)$ data packets. Packets to the receivers are lost independently with probability $\ptwo$. For this model, we prove several strong laws of large numbers for the asymptotic throughput as $n \to \infty$. The asymptotic throughput is characterized by the unique solution of an equation in terms of $\pone$, $\ptwo$ and $\rd$. These strong laws not only provide theoretical justification for several important observations made in the literature, but also provide insights that might have impact on future design of multicasting protocols.

R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, ``Network Informa
{\em IEEE Transactions on Information Theory}, IT-46, pp. 1204-1216, 2000.
T. Apostol, {\em Calculus, Volume II }, John Wiley \& Sons, 1969.

M. Avriel, {\em Nonlinear Programming: Analysis and
Methods}, Prentice-Hall, 1976.

J. D. Biggins, ``The First and Last Birth Problem for
a Multitype Age-Dependent Branching Processes,'' {\em Adv. Appl.
Prob.}, Vol. 8, pp. 446-459, 1976.

J. Bucklew, {\em Large Deviation Techniques in
Decision, Simulation and Estimation.} New York, NY: J. Wiley \& Sons,
Inc., 1990.
J. Byers, M . Luby, and M. Mitzenmacher, ``Accessing multiple
%mirror sites in parallel: using Tornado codes to speed up downloads,''
%{\em Proc. of IEEE INFOCOM'99}, New York, 1999.

J. Byers, M . Luby, M. Mitzenmacher, and A. Rege, ``A
digital fountain approach to reliable distribution of bulk data,''
{\em Proc. of ACM SIGCOMM'98}, Vancouver, 1998.

H. Chen and D. D. Yao, {\em Fundamentals of Queueing Networks}, New York
C.-S. Chang, "A new ordering for stochastic majorization:
%theory and applications," {\em Adv. Appl. Prob.}, Vol. 24, pp. 604-634,
%1992.

C. S. Chang and R. D. Nelson,
"Bounds on the speedup and efficiency of partial synchronization
in parallel processing systems,"
{\em Journal of the ACM}, Vol. 42, pp. 204-231, 1995.

Y. S. Chow and H. Teicher, {\em Probability Theory:
Independence, Interchangeability, Martingales}. New York:
Springer-Verlag, 1988.

T. M. Cover and J. A. Thomas,
{\em Elements of Information Theory}.
New York: Wiley \& Sons,
1991.

A. Dembo and O. Zeitouni.
{\em Large Deviations Techniques and Applications}.
Boston: Jones and Barlett Publishers, 1992.

R. S. Ellis, ``Large deviations for a general class of
random vectors, '' {\em Ann. Probab.}, Vol. 12, pp. 1-12, 1984.

J. G\"artner, ``On large deviations from invariant
measure,'' {\em Theory Probab. Appl.}, Vol. 22, pp. 24-39, 1977.

Hiriart-Urruty, J.-B., and Lemaar\'echal, C.
{\em Convex
Analysis and Minimization Algorithms II.}
New York: Springer-Verlag, 1991.

S. Jaggi, P. A. Chou, and K. Jain,
``Low complexity algebraic multicast network codes,'' in {\em International Symposiu
2003.

M. Jung, J. Nonnenmacher, and E. W. Biersack,
``Reliable multicast via satellite: uni-directional vs.
bi-directional communication,'' in {\em Proceedings of KiVS'99},
Darmstadt, Germany.

S. K. Kasera,S. Bhattacharyya, M. Keaton, D. Kiwior, J. Kurose, D. T
S. Zabele, ``Scalable fair reliable multicast using active sevices,''
{\em IEEE Network Magazine}, Jan./Feb. 2000.

S. K. Kasera, G. Hjalmtysson, D. Towsley and
J. Kurose, ``Scalable reliable multicast using multiple multicast channels,''
{\em IEEE/ACM Transactions on Networking}, 2000.

S.K. Kasera, J. Kurose and D. Towsley, ``A comparison
of server-based and receiver-based local recovery approaches for
reliable multicast,'' {\em Proc. IEEE INFOCOM'98}, San Francisco, CA U.S.A., March 1

J.F.C. Kingman, ``The first birth problem for an age-dependent
branching process'', {\em The Annals of Probability}, Vol 3, No 5, pp. 790-801,
1975.

R. Koetter and M. Medard, ``An algebraic approach to network coding,
Transactions on Networking}, 11: 782-795, 2002.

M. Luby, ``LT codes'' in {\em The 43rd Annual IEEE Symposium on Foun
od Computer Science}, 2002.
J. Nonnenmacher, E. Biersack and D. Towsley, ``Parity-based loss
recovery for reliable multicast transmission,''
{\em Proc. ACM SIGCOMM'97}, Cannes, France, pp. 289-300, Sept. 1997.

J. Nonnenmacher, M. Lacher, M. Jung, E.W. Biersack and G. Carle,
``How bad is reliable multicast without local recovery?,''
{\em Proc. IEEE INFOCOM'98}, San Francisco, CA U.S.A., March 1998.

L. Rizzo, ``On the feasibility of software fec,''
{\em Tech. rep.,} Univ. di Pisa, Italy, Jan. 1997.

R. T. Rockafeller. {\em Convex
Analysis}. Princeton: Princeton University Press, 1970.

D. Rubenstein, S. Kasera, D. Towsley and J. Kurose,
``Improving reliable multicast using active parity encoding services (APES),''
{\em Proc. IEEE INFOCOM'99}, pp. 1248-1255, New York, U.S.A., March 1999.

P. Sanders, S. Egner, and L. Tolhuizen,
``Polynomial time algorithms for network information flow'' in {\em Proceedings of
the 15th ACM Symposium on Parallelism in Algorithms and Architectures}, 2003.

A. Shokrollahi,
``Raptor codes'' {\em Pre-print}, 2003.
A. Shwartz and A. Weiss. {\em Large Deviations
for Performance Analysis: Queues, Communication and Computing.} 1994.

D.Towsley,
``Group (Multicast) Communication in Wide-Area Networks,'' ACM
Sigmetrics 2000 Tutorial.
L. Vicisano, J. Crowcroft and L. Rizzo,
``TCP-like congestion control for layered multicast data transfer,''
{\em Proc. IEEE INFOCOM'98}, San Francisco, CA U.S.A., March 1998.