DC9 LNDG CLE RWY 24R CROSSED RWY 24L WITHOUT CLRNC PROMPTING DEP ABORT.

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I WAS THE PNF FOR OUR THIRD AND LAST LEG FOR THE DAY FROM MSP TO CLE AND WAS ASSIGNED A VISUAL APCH TO RWY 24R. I KNEW THIS WAS A RELATIVELY NEW RWY AND THIS WAS TO BE MY FIRST LNDG ON THIS SIDE. AS WE WERE EXPECTING THIS APCH AS PER ATIS; WE HAD BRIEFED THE APCH AND TAXI RTE IN NOTING THAT WITH A NORMAL LNDG ROLLOUT WE COULD EXPECT TO TURN OFF AT TXWY K AND THAT WOULD BE THE EXPECTED TXWY TO TAKE US TO OUR RAMP FOR THE ACARS ASSIGNED GATE. WE ALSO OBSERVED WE WOULD HAVE TO CROSS BOTH RWYS 24C AND 24L. I NOTICED THAT TXWY K WAS NOT JUST A STRAIGHT LINE ACROSS THE RWYS DUE TO SOME OFFSETS INDICATED ON THE ARPT DIAGRAM. DURING THE FINAL PORTION OF THE APCH; I NOTICED AN ACFT WHICH APPEARED TO BE TAXIING TOWARD THE TKOF END OF RWY 24L AND MADE A MENTAL NOTE THAT WE WOULD PROBABLY HAVE TO HOLD SHORT OF RWY 24L. AFTER A NORMAL LNDG AND ROLLOUT; I TOOK CTL OF THE ACFT AT AROUND 60 KTS AND SLOWED FOR THE TURN ONTO TXWY K; FOLLOWING THE CURVED LEAD-OFF TXWY LINES. AS WE WERE TURNING AND CLEANING UP; I OBSERVED THAT THE TAXI LIGHT ONLY ILLUMINATED AHEAD OF THE ACFT TO THE CROWN OF RWY 24C AND THAT THERE WERE NO TAXI LINES DESIGNATING THE TXWY; ONLY MORE CURVED LEAD-IN LINES FOR RWY 24C. ABOUT THIS TIME; TWR ISSUED WHAT I HEARD TO BE 'CLRED TO CROSS AND STAY ON MY FREQ' AS IS A COMMON PRACTICE DURING NON-PEAK TIMES. I WAS TRYING TO INTERPRET THE CURVED TAXI LINES AND AS WE NEARED THE CROWN; THE NOSE TAXI LIGHT ILLUMINATED FURTHER ACROSS AND I SAW THE CURVED LEAD IN TAXI LINES FOR COMING FROM THE OTHER DIRECTION AND WAS TRYING TO SEE WHERE THEY INTERSECTED; WHICH WOULD INDICATE WHERE TXWY K CROSSED BTWN RWYS 24C AND 24L. ANOTHER SET OF CURVED LEAD-IN LINES CAME INTO VIEW; SO I AGAIN ATTEMPTED TO JUDGE WHERE THE LINES WOULD INTERSECT AND SAW HOLD SHORT LINE FOR WHAT WAS NOW RWY 24L COMING FROM THE OTHER DIRECTION. ABOUT THIS TIME; TWR CAME UP AND ASKED US IF WE WERE HOLDING SHORT OF RWY 24L. THE FO WAS ABOUT TO ANSWER IN THE AFFIRMATIVE AND I TOLD HER NO; WE WERE NOW STOPPED AT THE HOLD SHORT LINE ON THE OTHER SIDE OR RWY 24L. I HEARD TWR XMIT IN THE BLIND (BLIND FOR US) TO AN ACFT ON ANOTHER FREQ WHO HAD ABORTED THEIR TKOF EARLY THANKFULLY; BUT; MY HEART SANK NOT BELIEVING WHAT HAD JUST HAPPENED. IF THEY WERE THERE; I NEVER NOTICED A HOLD SHORT LINE FOR EITHER RWYS 24C OR 24L COMING FROM OUR DIRECTION. I BELIEVE IF I HAD NOTICED A HOLD SHORT LINE FOR RWY 24L; I WOULD HAVE AT LEAST LOOKED DOWN THE RWY BEFORE XING AS USUAL; BUT ESPECIALLY AS I EXPECTED TO HOLD SHORT DUE TO THE OTHER ACFT I HAD THOUGHT OBSERVED TAXIING OUT. I ALWAYS TURN ON THE NACELLE LIGHTS XING RWYS DAY OR NIGHT SO WE'RE SEEN AND HOPEFULLY THIS HELPED PREVENT WHAT COULD HAVE BEEN DISASTROUS. CONTRIBUTING FACTORS FOR ME WERE NOT HEARING AND AFFIRMING THE HOLD SHORT INSTRUCTIONS FOR WY 24L; CONFUSION AS TO TXWY LOCATION AND IDENT; AND RWY HOLD SHORT IDENT FROM OUR DIRECTION. IF THE HOLD SHORT LINES ARE THERE; I WAS SO CONFUSED AS TO TXWY IDENT TO THE POINT WHERE I WAS DISTR FROM THE BIG PICTURE AS TO WHERE WE WERE WITH RESPECT TO RWY 24L. YOU WOULDN'T THINK SUCH A SHORT TAXI WOULD BE SUCH A BIG PROB. FATIGUE: I WAS ASSIGNED THE TRIP THE DAY BEFORE THE DEP DAY AROUND XA30 AND DEADHEADED TO DETROIT THE NEXT MORNING ON THE XD30 FLT FROM MEM; WHICH REQUIRED ME TO AWAKEN AT XA30. AFTER A RELEASE TIME OF XA24; MY FIRST LAYOVER OF XA41 IN RIC WAS INTERRUPTED SEVERAL TIMES THROUGHOUT THE NIGHT WITH FIRE TRUCK SIRENS WHICH SEEMED TO ORIGINATE FROM A COUPLE OF BLOCKS AWAY. I WOKE 2 HRS BEFORE THE ALARM CLOCK FOR NO APPARENT REASON; TOOK A SHOWER AND WENT BACK TO SLEEP UNTIL THE ALARM CLOCK AND WAKE UP CALL WENT OFF AT XC00. I THEN DRESSED AND WENT TO THE HOTEL RESTAURANT FOR BREAKFAST. OUR DUTY TIME ON THE DAY OF THIS OCCURRENCE WAS XF26; WHICH COMMENCED AT XD05 EST AND HAD A SCHEDULED RELEASE OF XN31 EST. WE HAD XA46 GND TIME IN DTW AFTER OUR FIRST LEG. DTW-MSP WAS OUR NEXT LEG AND WAS REMARKABLE ONLY IN THAT WE WERE LATE GETTING IN DUE TO TAXI-IN DELAYS AND A LAST MIN GATE CHANGETHAT ONLY GAVE US 45 MINS (VICE THE SCHEDULED 59 MINS) TO GET FROM GATE Y TO GATE X. I DIDN'T REALLY FEEL DEFICIENT IN REST AS I DIDN'T FEEL THE LEAST BIT SLEEPY. BUT I ALSO WONDER IF I WAS AS SHARP AS I COULD HAVE BEEN. THE WX HAD BEEN GOOD ALL DAY AND NO ACFT MAINT PROBS WERE ENCOUNTERED. NOR WERE THERE ANY MELS TO DISTRACT FROM ROUTINE PROCS.