ACR Y UNAUTH TKOF ON MULTIPLE RWY OP INTERSECTING RWYS HAD NMAC LTSS FROM ACR X. PLTDEV. SYS ERROR.

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I WAS WORKING LCL AT JAX TWR. I TAXIED ACR X INTO POS AND HOLD RWY 25. I THEN TAXIED ACR Y INTO POS AND HOLD RWY 31 AND TOLD HIM TFC DEPARTING RWY 25. I THEN TOLD A PRECEDING ARR TO RWY 25 TO TURN OFF AND CONTACT GND. I THEN TOLD ACR X CLRED FOR TKOF RWY 25. A FEW SECONDS LATER; THE GND CTLR ASKED ME IF I HAD CLRED THE ACR Y FOR TKOF ON RWY 31. I THEN SAW THE ACR Y ON TKOF ROLL PAST MID FIELD. ACR X WAS NOW AIRBORNE BUT HAD NOT REACHED THE END OF RWY 25. I INSTRUCTED ACR Y TO MAKE AN IMMEDIATE R TURN AFTER DEP AND ISSUED THE TFC TO HIM. HE MADE A HARD R TURN AND PASSED BEHIND ACR X. AFTER LISTENING TO THE TAPE I REALIZED I MISSED A READBACK BY THE ACR X ON HIS TKOF CLRNC. THE READBACK OF ACR X TKOF CLRNC FOR RWY 25 WAS 'ACR Y ROLLING.' AFTER CLRING ACR X FOR TKOF; I VISUALLY SCANNED BOTH ACFT AND SAW ACR Y IN POS. I THEN WATCHED ACR X DURING HIS TKOF AND DIDN'T NOTICE ACR Y ON THE ROLL UNTIL IT WAS BROUGHT TO MY ATTN BY THE GND CTLR. I BELIEVE THE SIT HAPPENED BECAUSE THE ACR Y CREW WAS NOT CONCENTRATING ON THE TWR'S INSTRUCTION. LISTEN CAREFULLY TO INSTRUCTIONS AND READBACKS!