[Tex/LaTex] Drawing a Markov chain using Tikz

tikz-pgf

I want to draw the following Markov chain using tikz, but I have a few problems that I do not know how to handle: I want the transition grid to look exactly like the picture. All outgoing edges are parallel and so are all the incoming edges. At each intersection, there must be an arrow as depicted in the picture and the labels appear above the intersections. Also, some edges are not continuous and have dots in the middle.
Markov chain - hand drawn

Below is what I have worked out so far, but it is very incomplete with lots of problems that I do not know how to fix.

\documentclass[12pt]{article}

\usepackage[landscape]{geometry}
\usepackage{tikz}
\usetikzlibrary{arrows,automata}

\begin{document}

\begin{tikzpicture}[->,>=stealth',shorten >=1pt,node
     distance=2cm,looseness=2,bend right=40]

\draw[help lines] (-1,-3) grid (16,1);

\tikzset{every state/.style={minimum size=35pt,scale=0.6}}
\node [state] (0)                    {$0$};
\node [state] (1)   [right of = 0]   {$1$};
\node [state] (2)   [right of = 1]   {$2$};
\node         (d1)  [right of = 2]   {$\cdots$};
\node [state] (km1) [right of = d1]  {$k-1$};
\node [state] (k)   [right of = km1] {$k$};
\node [state] (kp1) [right of = k]   {$k+1$};
\node         (d2)  [right of = kp1] {$\cdots$};
\node [state] (mmk) [right of = d2]  {$m-k$};
\node         (d3)  [right of = mmk] {$\cdots$};
\node [state] (mm2) [right of = d3]  {$m-2$};
\node [state] (mm1) [right of = mm2] {$m-1$};

\path (0)   edge (1)
            edge (2)
            edge (km1)
            edge [loop above] node {$\mu_0$} (0);
\path (1)   edge (2)
            edge (km1)
            edge (k)
            edge [loop above] node {$\mu_0$} (1);
\path (2)   edge (km1)
            edge (k)
            edge (kp1)
            edge [loop above] node {$\mu_0$} (2);
\path (km1) edge (k)
            edge (kp1)
            edge [loop above] node {$\mu_0$} (km1);
\path (k)   edge (kp1)
            edge [loop above] node {$\mu_0$} (k);
\path (mmk) edge (mm2)
            edge (mm1)
            edge [loop above] node {$0$} (mmk);
\path (mm2) edge (mm1)
            edge [loop above] node {$0$} (mm2);
\path (mm1) edge [loop above] node {$1$} (mm1);
\end{tikzpicture}

\end{document}

Best Answer

The trick of my solution is to use a scope with rotate=45 and then to make all links with |-. I also define three commands to factorize code. 

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{arrows}

\newcommand\reflexive[2]{% node, label
  \draw[->] (#1.100) to[reflexive] node[trans,above=-2mm] {#2} (#1.80); 
}
\newcommand\link[3]{% start node, end node, label
  \draw[edge] (#1) -- (#1 |- #2);
  \node[trans] at (#1 |- #2) {#3} (1);
  \draw[edge] (#1 |- #2) -- (#2);
}
\newcommand\linkwithdots[4]{% start node, end node, dots node, label
  \coordinate (dots1) at (#1 |- #3);
  \draw (#1) -- ([yshift=5mm]dots1);
  \draw[dotted] ([yshift=5mm]dots1) -- ([yshift=-5mm]dots1);
  \draw[edge] ([yshift=-5mm]dots1) -- (#1 |- #2);
  \node[trans] at (#1 |- #2) {#4} (1);
  \coordinate (dots2) at (#3 |- #2);
  \draw (#1 |- #2) -- ([xshift=-5mm]dots2);
  \draw[dotted] ([xshift=-5mm]dots2) -- ([xshift=5mm]dots2);
  \draw[edge] ([xshift=5mm]dots2) -- (#2);
}
\begin{document}
\begin{tikzpicture}
  \tikzset{
    >=stealth',
    node distance=1.5cm,
    state/.style={minimum size=30pt,font=\small,circle,draw},
    dots/.style={state,draw=none},
    edge/.style={->},
    trans/.style={font=\footnotesize,above=2mm},
    reflexive/.style={out=120,in=60,looseness=5,relative},
  }
  \node [state] (0)                    {$0$};
  \node [state] (1)   [right of = 0]   {$1$};
  \node [state] (2)   [right of = 1]   {$2$};
  \node [dots]  (d1)  [right of = 2]   {$\cdots$};
  \node [state] (km1) [right of = d1]  {$k-1$};
  \node [state] (k)   [right of = km1] {$k$};
  \node [state] (kp1) [right of = k]   {$k+1$};
  \node [dots]  (d2)  [right of = kp1] {$\cdots$};
  \node [state] (mmk) [right of = d2]  {$m-k$};
  \node [dots]  (d3)  [right of = mmk] {$\cdots$};
  \node [state] (mm2) [right of = d3]  {$m-2$};
  \node [state] (mm1) [right of = mm2] {$m-1$};

  \begin{scope}[rotate=45]
    \reflexive{0}{$\mu_0$}
    \reflexive{1}{$\mu_0$}
    \reflexive{2}{$\mu_0$}
    \link{0}{1}{$\mu_1$}
    \link{0}{2}{$\mu_1$}
    \link{1}{2}{$\mu_1$}
    \linkwithdots{0}{km1}{d1}{$\mu_{k-1}$}
    \linkwithdots{1}{km1}{d1}{$\mu_{k-2}$}
    \linkwithdots{2}{km1}{d1}{$\mu_{k-3}$}
  \end{scope}
\end{tikzpicture}
\end{document}

enter image description here

Now, you have just to complete with some links...

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