How to draw a dashed line from a node without connecting to any other node or a shape in tikz

The following example positions the node with $b_{11}$ below A.south and adds a dashed vector between A.south and the previously created node.

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\tikzset{state/.style={circle, fill=red,text=white}}
\begin{document}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
                semithick,scale=1.5,transform shape]
\tikzstyle{every state}=[fill=red,draw=none,text=white]

\node[state]    (A)                    {$S_1$};
\node[state]    (B) [right of=A]       {$S_2$};


\path (A) edge [loop above] node {$a_{11}$} (A)
          edge [bend left]  node {$a_{12}$} (B)
      (B) edge [loop above] node {$a_{22}$} (B)
          edge [bend left]  node {$a_{21}$} (A);

\draw[dashed, ->, inner sep=0pt] (A.south) +(-90:2em) node (tmp) {$b_{11}$}
  (A.south) -- (tmp);
\end{tikzpicture}
\end{document}

A more complicate example:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{arrows}
\tikzset{state/.style={circle, fill=red,text=white}}
\begin{document}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,auto,node distance=2.8cm,
                semithick,scale=1.5,transform shape]
\tikzstyle{every state}=[fill=red,draw=none,text=white]

\node[state]    (A)                    {$S_1$};
\node[state]    (B) [right of=A]       {$S_2$};


\path (A) edge [loop above] node {$a_{11}$} (A)
          edge [bend left]  node {$a_{12}$} (B)
      (B) edge [loop above] node {$a_{22}$} (B)
          edge [bend left]  node {$a_{21}$} (A);

\begin{scope}[densely dashed, ->, inner sep=.2em]
  \draw[shorten >=-2pt]
    (A.south) ++(-90:1.7em) node[font=\scriptsize] (tmp1) {0.4}
              ++(-90:1.8em) node[inner sep=0pt]      (tmp2) {$b_{11}$}
    (A.south) -- (tmp1) -- (tmp2);
  \draw[shorten >=4pt]
    (A.south) ++(-45:1.5em) node[font=\scriptsize] (tmp1) {0.5}
              ++(-45:2em)   node[inner sep=0pt]    (tmp2) {$b_{12}$}
    (A.south) -- (tmp1) -- (tmp2);
  \draw[shorten >=-3pt]
    (A.south) ++(-135:1.5em) node[font=\scriptsize] (tmp1) {0.1}
              ++(-135:2em) node[inner sep=0pt]      (tmp2) {$b_{10}$}
    (A.south) -- (tmp1) -- (tmp2);
\end{scope}
\end{tikzpicture}
\end{document}

Like this? The labels are not precise, you can change the labels to whatever you want.

\documentclass[border=2mm,tikz]{standalone}
\usetikzlibrary{arrows,automata,positioning}
\begin{document}

\begin{tikzpicture}[->,>=stealth',shorten >=1pt,node distance=2.8cm,
                semithick,scale=1.5,transform shape]
\tikzstyle{every state}=[fill=red,draw=none,text=white]

\node[state]    (A)                    {$S_1$};
\node[state]    (B) [right of=A]       {$S_2$};


\path[auto] (A) edge [loop above] node {$a_{11}$} (A)
          edge [bend left]  node {$a_{12}$} (B)
      (B) edge [loop above] node {$a_{22}$} (B)
          edge [bend left]  node {$a_{21}$} (A);

\node (01) at ([shift={(-3em,-4em)}]A.south) {$b12$};
\node (04) at ([shift={(0,-5em)}]A.south) {$b11$};
\node (05) at ([shift={(3em,-4em)}]A.south) {$b13$};

\draw [shorten >=0pt,dashed] (A.south) --node[pos=.7,fill=white,inner sep=1pt]{0.4} (04);
\draw [shorten >=0pt,dashed] (A.south) --node[fill=white,inner sep=1pt]{0.1} (01);
\draw [shorten >=0pt,dashed] (A.south) --node[fill=white,inner sep=1pt]{0.5} (05);

\end{tikzpicture}

\end{document}