mjhjmtu

I would like to draw the following figure:

(the shape of blocks are not important). I began by writing the code:

begintikzpicture[node distance=5cm]
tikzstyleplace=[circle,thick,draw=gray!75,fill=gray!20,minimum size=2cm]
beginscope
node [place] (s1c) $U$;
node [place, right of=s1c] (s2c) $S$;
draw[densely dotted, thick,->,shorten >=1pt] (s2c) to (s1c);
draw[thick,->,shorten >=1pt] (s2c) to [out=0,in=90,loop,looseness=4.8] (s2c);
endscope
endtikzpicture

which produces the figure

My problem is edges names which I can't to draw (i.e. a,b,c and so on). Any suggestions please?

an alternative ... (considering only your code snippet). for labels are used quotes library

edit:
now considered your image, but for nodes are used circles. for their positioning is used syntax from positioning library.

documentclass[tikz, margin=3.141592mm]standalone
usetikzlibraryarrows.meta,
 positioning,
 quotes

begindocument
 begintikzpicture[
 node distance = 4cm and 4cm,
 place/.style = circle, draw=gray!75, thick, fill=gray!20, minimum size=1cm,
 arr/.style = thick, shorten >=1pt, -Straight Barb[length=5pt,width=5pt],
every edge quotes/.style = fill=white, font=footnotesize,
 inner sep=2pt, anchor=center,
 bend angle = 15
 ]
node [place] (n1) $A$;
node [place, above right=of n1] (n2) $B$;
node [place, below right=of n2] (n3) $C$;
draw[arr, densely dotted] (n2) edge [bend left, "b"] (n3)
 (n3) to [bend left, "c"] (n2); 
draw[arr] (n1) edge [bend left=30, "d"] (n2) 
 (n1) edge [bend left, "f"] (n2) 
 (n2) edge [bend left, "g"] (n1)
 (n2) edge [bend left=30, "h"] (n1) 
 (n3) edge [bend left=30, "d"] (n2)
 (n2) to [in=120, out=60,loop, "a"] ();
 endtikzpicture
enddocument

Here is a code that produces something that resembles your screen shot.

documentclass[tikz,border=3.14mm]standalone
begindocument
usetikzlibrarypositioning,shadings
tikzsetbox/.style=draw,top color=gray!20,bottom
color=gray!20,middle color=white,minimum width=2.5cm,minimum height=1.5cm
begintikzpicture[font=sffamily]
node[box] (A) A;
node[box,below left=1cm and 3cm of A] (B) B;
node[box,below right=1cm and 3cm of A] (C) C;
draw[-latex] (A.60) to[out=90,in=90,looseness=2] 
node[midway,fill=white]a (A.120);
draw[-latex,dashed] (A.0) to[out=00,in=120,looseness=0.8] 
node[midway,fill=white]b (C.90);
draw[-latex,dashed] (C.160) to[out=180,in=-45,looseness=0.8] 
node[midway,fill=white]c (A.-40);
draw[-latex] (C.190) to[out=180,in=-60,looseness=0.8] 
node[midway,fill=white]d (A.-70);
draw[-latex] (B.110) to[out=60,in=180,looseness=1] 
node[midway,fill=white]e (A.160);
draw[-latex] (B.70) to[out=45,in=180,looseness=1] 
node[midway,fill=white]f (A.200);
draw[-latex] (A.-140) to[out=-135,in=0,looseness=1] 
node[midway,fill=white]g (B.20);
draw[-latex] (A.-110) to[out=-120,in=0,looseness=1] 
node[midway,fill=white]h (B.-10);
endtikzpicture 
enddocument