ACR X HAD LTSS FROM 2 ACR Y'S WHICH CROSSED RWY AFTER ACR X HAD BEEN CLRED FOR TKOF. SYS ERROR.

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ACR X WAS IN #1 POS HOLDING SHORT OF RWY 31C ON SW SIDE ON TAXIWAY R AT MDW. ACR X WAS CLRED FOR TKOF WITH L TURN TO 250 DEG HDG FROM THAT POS BY MDW TWR. ACR X TAXIED ONTO RWY 31C INTENDING TO DO A ROLLING TKOF. AS I TURNED TO LINE UP ON RWY 31C; I NOTICED 2 ACR Y MLG'S TAXIING S ON TAXIWAY P AT A FAST RATE OF TAXI. I WAS NOT SURE THE FIRST WOULD OR EVEN COULD HOLD SHORT OF THE RWY; SO I STOPPED IN POS AT THE END OF RWY 31C. I NEVER DID ADVANCE THROTTLES FOR TKOF. BOTH ACR Y ACFT TAXIED ACROSS RWY 31C (IN FRONT OF ME) WITHOUT SLOWING DOWN. I THEN HAD COPLT CONFIRM WITH THE TWR WE WERE CLRED FOR TKOF; AND TWR REPEATED TKOF CLRNC. NORMAL TKOF AFTER THIS POINT. I AM SURE IF I HAD MADE A NORMAL ROLLING TKOF I WOULD HAVE COLLIDED WITH THE FIRST ACFT. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: THE RPTR STATED HE HAD CALLED THE TWR SUPVR AFTER ARR AT DEST AND THE SUPVR SAID THE GND CTLR HAD MISUNDERSTOOD THE COORD WITH THE LCL CTLR AND CLRED THE 2 ACFT TO CROSS THE RWY IN FRONT OF ACR X.