B737 FLC BLUNDERS ONTO ACTIVE RWY AT MDW.

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WE WERE OPERATING ACFT X MDW TO ATL. WE BACKED OUT OF GATE X AND GOT OUR TAXI INSTRUCTIONS TO RWY 31C. TAXI ON TXWYS P; G TO RWY 31C CLRED TO CROSS RWY 31R. ON THE OTHER SIDE OF RWY 31C ON TXWY P; GND TOLD US TO SWITCH TO TWR BECAUSE THEY WERE GOING TO GET US RIGHT OUT. THERE WERE 2 B737'S SITTING SIDE BY SIDE SHORT OF RWY 31C. I WAS STARTING THE #1 ENG AT THE TIME IN THE EVENT WE WERE NEXT. THE CAPT I BELIEVE BECAUSE HE SAW WE COULD NOT GET AROUND THE 1ST B737 ON TXWY G; MADE A L AT TXWY K SO WE COULD GET AROUND THE B737. AS FAR AS I CAN REMEMBER THERE WERE NO RWY MARKINGS FOR RWY 31R AND THE ONES I COULD SEE WERE VERY LIGHT. AS WE ROUNDED THE CORNER AND PULLED UP NEXT TO THE B737; THE TWR ASKED US TO LET THEM KNOW IF WE WERE GOING TO DO THAT AGAIN BECAUSE WE WERE TOLD WE ENCROACHED ONTO THE END OF RWY 31R. I WOULD INCLUDE A COPY OF OUR APCH PLATE; BUT I DON'T HAVE A COPY MACHINE. BUT LOOKING AT PLATE 10-9 OF MDW THE AREA WE WERE IN WAS ALL TXWY; SO THE CAPT AND I WERE BOTH CONFUSED ON HOW WE COULD HAVE EVER ENCROACHED ON RWY 31R. AS I SAID BEFORE I DID NOT SEE ANY MARKINGS AND THE ONES WE COULD SEE WERE VERY WORN.