MD80 FLC INITIATES TKOF ABORT WHEN HEARING LAX GND CTL TWICE CLR ANOTHER ACR DOWNFIELD TO CROSS THE RWY ON WHICH THE MD80 WAS CLRED FOR TKOF.

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ACFT X WAS CLRED FOR TKOF BY LAX TWR AFTER BEING IN POS AND HOLD ON THE RWY. IMMEDIATELY AS WE BEGAN THE TKOF ROLL; WE HEARD THE TWR CLR A FLT; WHICH HAD JUST LANDED ON RWY 24R; TO CROSS RWY 24L (OUR RWY) AT TXWY AA. THE JET DID NOT ANSWER THE TWR. AFTER THE SECOND TWR CALL I ABORTED OUR TKOF (BELOW 60 KIAS) BECAUSE I DID NOT THINK IT WAS SAFE TO CONTINUE UNDER THESE CONFUSING CONDITIONS -- ANOTHER AIRPLANE BEING CLRED TO CROSS A RWY AFTER A PLANE HAS STARTED ITS TKOF ROLL AND WITHOUT POSITIVE COM BTWN THE PARTIES INVOLVED. AFTER THE SLOW SPD ABORT; WE CLRED THE RWY AND RETURNED TO TKOF AGAIN. ANYONE REVIEWING THE TAPES OF THIS INCIDENT CAN CLRLY SEE THAT THIS TWR CTLR WAS GETTING BEHIND THE EXTREME VOLUME OF INBOUND AND OUTBOUND TFC. BEFORE HE CLRED US INTO POS TO HOLD; HE STARTED TO CLR US FOR TKOF ABOUT 15 SECONDS (VERY CLOSE) TO ANOTHER ACR FLT WHICH JUST STARTED ITS TKOF ROLL ON RWY 24L. HE DID THIS WITHOUT VISUAL CONFIRMATION OF THE Z PLANE. ANYWAY; HE REALIZED THIS AND JUST CLRED US INTO POS AND HOLD. MEANWHILE PLANES WERE GETTING WITHIN 3 MI OF LNDG ON RWY 24L AND RWY 24R MAKING THIS A VERY CONGESTED TIME. ANOTHER FACTOR CONTRIBUTING TO ALL THIS IS THE MISSED RADIO CALLS AND POOR RADIO DISCIPLINE EXHIBITED THROUGHOUT THE ARPT. IN HINDSIGHT; I DO BELIEVE (I HAVE NO WAY OF KNOWING) THAT THE PLANE CLRED TO CROSS RWY 24L AT TXWY AA MIGHT HAVE SEEN THIS CONFLICT ALSO; THUS THE REASON HE NEVER ANSWERED THE TWR. HE MIGHT HAVE BEEN TAKING THE TIME TO ANALYZE A BAD CLRNC -- I HOPE SO.