MLG CREW HAD A RWY INCURSION ON TAXI OUT AT DTW.

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WE TAXIED OUT OF DTW TO RWY 22L. RWY 21L WAS CLOSED FOR CONSTRUCTION. FLTS WERE ARRIVING ON RWYS 22R AND 22L. RWY 22L IS NORMALLY A DEP ONLY RWY. ATC WAS TALKING ON 2 FREQS; CTLING BOTH RWYS. NO ACFT WERE AHEAD OF US AND WE ANTICIPATED NO DELAY. I WAS TIRED AFTER BEING ON DUTY FOR 5 DAYS; A SHORT OVERNIGHT THE PREVIOUS NIGHT; A 2 1/2 HR WAIT IN DTW AND A LARGE LUNCH. ATC ISSUED INSTRUCTIONS WHICH I THOUGHT HAD BEEN POS AND HOLD ON RWY 22L FOR OUR FLT. THE FO WAS HEADS DOWN DOUBLE-CHKING DEP FREQS AND WAS ANTICIPATING ME HOLDING SHORT. WHEN I WAS XING THE CTRLINE; HE REALIZED WE WERE ON THE RWY. HE TOLD ME WE WEREN'T CLRED INTO POS. AS I ASKED HIM TO CHK WITH TWR; I NOTICED AN ACFT ON ABOUT A 2 MI FINAL. TWR INSTRUCTED US TO CLR THE RWY AND WE CONTINUED ACROSS RWY 22L EXITING ON TXWY Z. CONTRIBUTING FACTORS INCLUDE CAPT BEING TIRED; (SINKING SPELL AFTER LUNCH) UNUSUAL RWY CONFIGN; TWR CTLR TALKING ON 2 FREQS; ALSO UNUSUAL FOR DTW. CAPT DID NOT USE HIS NORMAL VERIFICATION OF ANNOUNCING INTENTION BEFORE ENTERING RWY; ANTICIPATING NO DELAY FOR DEP. SUPPLEMENTAL INFO FOR ACN 621073: WE SWITCHED TO TWR FREQ AND I WENT 'HEADS DOWN' MOMENTARILY TO CHK THE DEP PROC. WHEN I LOOKED UP; THE ACFT HAD ALREADY CROSSED THE RWY 22L HOLD SHORT LINE. CONTRIBUTING FACTORS: 'NON STANDARD' LNDG RWY CONFIGN (RWY 22L USUALLY USED FOR DEPS ONLY); TWR CTLR WORKING MULTIPLE FREQS; FO (ME) WAS HEADS DOWN AT CRITICAL PHASE OF TAXIING; AND CAPT WAS WORKING HIS 5TH CONSECUTIVE DAY (FATIGUE?)