MULTIPLE RWY OP INTERSECTING RWYS. ACR X TKOF HAD LTSS; NMAC WITH SMA Y. SYS ERROR.

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RWYS 20L; 20C; 20R DESIGNATED ACTIVE. ACFT (ACR) FREQUENTLY REQUEST RWYS 13 OR 31 DURING THE SUMMER DUE TO ADDED AVAILABLE LENGTH. CALLED READY RWY 31; WAS TAXIED INTO POS TO HOLD FOR TFC LNDG RWY 20R (INTERSECTING RWY). SMA Y WAS ALSO ON FINAL TO RWY 20C (FLT PATH OVERFLIES RWY 31). CLRED TO LAND; ABOUT 1 MI BEHIND THE RWY 20R ARR; AS DETERMINED FROM BRITE RADAR DISPLAY. SMA Y WAS NOT IN SIGHT DUE TO BRIGHT; HAZY SKY CONDITIONS. MY CTL PLAN WAS TO DEPART ACR X OFF RWY 31 AS SOON AS THE RWY 20R ARR CLRED THE DOWNFIELD INTXN. THIS WOULD HAVE ACR X THROUGH THE ARR PATH OF SMA Y PRIOR TO SMA Y REACHING THE INTXN OF RWY 31. I MADE NO ATTEMPT TO DETERMINE SMA Y'S POS VISUALLY PRIOR TO CLRING ACR X FOR TKOF; EXPECTING FROM THE LAST KNOWN POS OF SMA Y (AS DETERMINED EARLIER FROM THE BRITE) THAT THE APPLICABLE SEPARATION WOULD EXIST. I LOOKED DOWN TO MY PODIUM TO ATTEND TO SOME STRIP MARKING AND TO LOG SOME ARRS AND DEPS ON THE APPROPRIATE FACILITY FORM. WHEN I LOOKED BACK UP; I WAS STARTLED TO SEE SMA Y NEARLY OVER-FLYING ACR X ON TKOF ROLL. THERE WAS NO TIME TO ISSUE ALTERNATE INSTRUCTIONS; AND THE OPS CONTINUED WITH NO FURTHER INCIDENT. ONLY ADDED AWARENESS OF SMA Y PROGRESS TOWARD HIS RWY; AND DETERMINING HIS POS VISUALLY PRIOR TO DEPARTING ACR X WOULD HAVE ALLEVIATED THIS SIT. SUPPLEMENTAL INFO FROM ACN 245147: ACR X WAS CLRED FOR TKOF RWY 31 AT BNA. DURING TKOF ROLL (AT APPROX 95 KTS); MY COPLT CALLED OUT 'WATCH OUT! LOOK AT THAT!' THERE WAS AN SMA ON SHORT FINAL TO A PERPENDICULAR RWY ABOUT 100 FT ABOVE US; DSNDING. I DETERMINED WE WOULD NOT COLLIDE AND CONTINUED THE TKOF. THAT LNDG TFC WAS NOT ISSUED TO ME AT ALL. HAD I KNOWN ABOUT THE SMA; I WOULD HAVE DELAYED OUR TKOF. THE SMA WAS ON ANOTHER TWR (BNA) FREQ. I BELIEVE THE CTLR SHOULD HAVE ISSUED US THAT TFC.