Mfcttrf

Below is my MWE. My only problem is, I need the tikzpicture to be pagewidth (textwidth) and not overflow to the right. I tried adjustbox, fbox and resizebox.

documentclass[
a4paper, % Papierformat A4
12pt, % Schrift 12-Punkt
headsepline, % mit Linie unter der Kopfzeile
numbers=noenddot, % Nummern ohne Punkt am Ende
bibliography=totoc, % Literaturverzeichnis mit Nummer im 
Inhaltsverzeichnis (``TO Table Of Contents'')
index=totoc, % Index mit Nummer im Inhaltsverzeichnis
fleqn, % Formeln werden linksbündig statt zentriert 
angeordnet
headings=normal % Etwas kleinere Überschriften
]scrbook

usepackagetikz
usetikzlibrarymatrix, positioning,chains
usetikzlibrarycalc,trees,positioning,arrows,chains,shapes.geometric,%
decorations.pathreplacing,decorations.pathmorphing,shapes,%
matrix,shapes.symbols

usepackagecapt-of


tikzstylerounddiagram = [
%tikzset
>=stealth',
punktchain/.style=
rectangle, 
rounded corners, 
% fill=black!10,
draw=black, very thick,
text width=10em, 
minimum height=3em, 
text centered, 
on chain,
line/.style=draw, thick, <-,
element/.style=
tape,
top color=white,
bottom color=blue!50!black!60!,
minimum width=8em,
draw=blue!40!black!90, very thick,
text width=10em, 
minimum height=3.5em, 
text centered, 
on chain,
every join/.style=->, thick,shorten >=1pt,
decoration=brace,
tuborg/.style=decorate,
tubnode/.style=midway, right=2pt,
]%


begindocument
beginfigure

 begintikzpicture
 [style=rounddiagram, node distance=.8cm, start chain=going right]
 node[punktchain, join]FairRun;
 node[punktchain, join]Set I/O;
 node[punktchain, join]CbmStsFindClusters;
 node[punktchain, join]CbmStsFindHits;
 endtikzpicture
 endfigure
enddocument

Any help is appreciated.

You can always compute the maximal width the nodes should have to fill the full page precisely. We need 4*(space taken by a node)+3*(node distance)=textwidth, where space taken by a node=(text width) + 2*(inner sep)+2*(outer sep). Note that tikzstyle is deprecated.

documentclass[
a4paper, % Papierformat A4
12pt, % Schrift 12-Punkt
headsepline, % mit Linie unter der Kopfzeile
numbers=noenddot, % Nummern ohne Punkt am Ende
bibliography=totoc, % Literaturverzeichnis mit Nummer im 
Inhaltsverzeichnis (``TO Table Of Contents'')
index=totoc, % Index mit Nummer im Inhaltsverzeichnis
fleqn, % Formeln werden linksbündig statt zentriert 
angeordnet
headings=normal % Etwas kleinere Überschriften
]scrbook

usepackagetikz
usetikzlibrarymatrix, positioning,chains
usetikzlibrarycalc,trees,positioning,arrows,chains,shapes.geometric,%
decorations.pathreplacing,decorations.pathmorphing,shapes,%
matrix,shapes.symbols

usepackagecapt-of


tikzsetrounddiagram/.style=>=stealth',
punktchain/.style=
rectangle, 
rounded corners, 
% fill=black!10,
draw=black, very thick,
text width=10em, 
minimum height=3em, 
text centered, 
on chain,
% line/.style=draw, thick, <-,
% element/.style=
% tape,
% top color=white,
% bottom color=blue!50!black!60!,
% minimum width=8em,
% draw=blue!40!black!90, very thick,
% text width=10em, 
% minimum height=3.5em, 
% text centered, 
% on chain,
every join/.style=->, thick,shorten >=1pt,
% decoration=brace,
% tuborg/.style=decorate,
% tubnode/.style=midway, right=2pt,



begindocument
beginfigure

begintikzpicture[rounddiagram, node distance=.8cm, start chain=going right]
 pgfmathsetmacromynodewidth(textwidth-3*0.8cm)/4-4pt
 beginscope[punktchain/.append style=text width=mynodewidth pt,inner sep=1pt,outer
 sep=1pt]
 node[punktchain, join]FairRun;
 node[punktchain, join]Set I/O;
 node[punktchain, join]CbmStsFindClusters;
 node[punktchain, join]CbmStsFindHits;
 endscope
endtikzpicture
endfigure
enddocument

The output suggests that you may either want to write the text of the third node in two lines, have different widths for different nodes or to make them smaller by using scale=(something <1) in the nodes.

There are two approaches:

As @Zarko mentioned, you can change the font and reduce the size of the text width. The second approach is to leave everything the same and add transform canvas=xshift = 0.8cm to tikzpicture options.

documentclass[
a4paper, % Papierformat A4
12pt, % Schrift 12-Punkt
headsepline, % mit Linie unter der Kopfzeile
numbers=noenddot, % Nummern ohne Punkt am Ende
bibliography=totoc, % Literaturverzeichnis mit Nummer im 
Inhaltsverzeichnis (``TO Table Of Contents'')
index=totoc, % Index mit Nummer im Inhaltsverzeichnis
fleqn, % Formeln werden linksbündig statt zentriert 
angeordnet
headings=normal % Etwas kleinere Überschriften
]scrbook

usepackagetikz
usetikzlibrarymatrix, positioning,chains
usetikzlibrarycalc,trees,positioning,arrows,chains,shapes.geometric,%
decorations.pathreplacing,decorations.pathmorphing,shapes,%
matrix,shapes.symbols

usepackagecapt-of


tikzstylerounddiagram = [
%tikzset
>=stealth',
punktchain/.style=
rectangle, 
rounded corners, 
% fill=black!10,
draw=black, very thick,
text width=7.5em, 
minimum height=3em, 
text centered, 
font = footnotesize,
on chain,
line/.style=draw, thick, <-,
element/.style=
tape,
top color=white,
bottom color=blue!50!black!60!,
minimum width=8em,
draw=blue!40!black!90, very thick,
text width=10em, 
minimum height=3.5em, 
text centered, 
on chain,
every join/.style=->, thick,shorten >=1pt,
decoration=brace,
tuborg/.style=decorate,
tubnode/.style=midway, right=2pt,
]%


begindocument
 beginfigure
 begintikzpicture
 [style=rounddiagram, node distance=.8cm, start chain=going right]
 node[punktchain, join]FairRun;
 node[punktchain, join]Set I/O;
 node[punktchain, join]CbmStsFindClusters;
 node[punktchain, join]CbmStsFindHits;
 endtikzpicture
 endfigure

 beginfigure
 begintikzpicture
 [style=rounddiagram, node distance=.8cm, start chain=going right, transform canvas=xshift = 0.8cm]
 node[punktchain, join, text width=10em, font =normalsize]FairRun;
 node[punktchain, join, text width=10em, font =normalsize]Set I/O;
 node[punktchain, join, text width=10em, font =normalsize]CbmStsFindClusters;
 node[punktchain, join, text width=10em, font =normalsize]CbmStsFindHits;
 endtikzpicture
 endfigure
enddocument

i would change text width with minimum width which i would reduce to smaller size, for example to 7em. also I would all doubled TikZ libraries and use tikzset instead of tikzstyle:

(red lines indicate text borders)

documentclass[
a4paper, % Papierformat A4
12pt, % Schrift 12-Punkt
headsepline, % mit Linie unter der Kopfzeile
numbers=noenddot, % Nummern ohne Punkt am Ende
bibliography=totoc, % Literaturverzeichnis mit Nummer im
Inhaltsverzeichnis (``TO Table Of Contents'')
index=totoc, % Index mit Nummer im Inhaltsverzeichnis
fleqn, % Formeln werden linksbündig statt zentriert
angeordnet
headings=normal % Etwas kleinere Überschriften
]scrbook

usepackagetikz
usetikzlibraryarrows,
 calc, chains,
 decorations.pathreplacing, decorations.pathmorphing,
 matrix,
 positioning,
 shapes.geometric, shapes.symbols,
 trees
 

usepackagecapt-of


tikzsetrounddiagram/.style = 
>=stealth',
punktchain/.style = rectangle, rounded corners, draw, very thick,
 minimum width=7em, % <--- changed from "text width"
 minimum height=5ex,
 align=center, on chain,
 line/.style = draw, thick, <-,
 element/.style = tape, draw=blue!40!black!90, very thick,
 top color=white, bottom color=blue!50!black!60!,
 text width=10em, minimum height=7ex,
 align=center, on chain,
every join/.style = ->, thick,shorten >=1pt,
 decoration = brace,
 tuborg/.style = decorate,
 tubnode/.style = midway, right=2pt,


%---------------- show page layout. don't use in a real document!
usepackageshowframe
renewcommandShowFrameLinethickness0.15pt
renewcommand*ShowFrameColorcolorred
%---------------------------------------------------------------%


begindocument
beginfigure
 centering
 begintikzpicture[style=rounddiagram,
node distance=5mm,
start chain=going right,
punktchain/.append style = join
 ]
node[punktchain] FairRun;
node[punktchain] Set I/O;
node[punktchain] CbmStsFindClusters;
node[punktchain] CbmStsFindHits;
endtikzpicture
 endfigure
enddocument