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This should do it, may for sure be shorten some.
awk -F "[;& ]" '
/begin request/ {st[$13]=$1 " " $2}
/complete reponse/ {en[$13]=$1 " " $2}
END {
for (i in st) {
split(st[i],s1,"[-...

Here is how to get start and stop for every UID into an array and print it.
awk -F "[;& ]" '
/begin request/ {st[$13]=$1 " " $2}
/complete reponse/ {en[$13]=$1 " " $2}
END {
for (i in st) ...