A B737-300 CREW MISSED A DSCNT ALT CONSTRAINT BECAUSE OF DISTRS CAUSED BY TURB AND DSCNT PLANNING.

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ZJX ISSUED US A CLRNC TO DSND TO CROSS JAYJA INTXN AT FL270. I STARTED THE ACFT DOWN A LITTLE BIT EARLY BECAUSE WE WERE EXPERIENCING LIGHT TO OCCASIONAL MODERATE CHOP. BECAUSE I STARTED DOWN EARLY; I USED LESS RATE OF DSCNT THAN REQUIRED TO MAKE THE RESTR IN ORDER TO CATCH BACK UP TO THE FINAL DSCNT PROFILE. I WAS DISTR BY MULTIPLE TASK SATURATION AND DID NOT NOTICE THE ACFT GOING HIGH ON THE FMC DSCNT PROFILE UNTIL THE CAPT NOTICED IT FIRST. I INCREASED THE RATE OF DSCNT TO TRY TO MAKE THE XING RESTR. AS THE ACFT APCHED FL270; THE ALT HOLD EITHER FAILED TO ACQUIRE FL270 OR OVERSHOT FL270 AND WE DSNDED BELOW OUR ASSIGNED ALT. WHEN I SAW THE ACFT APCHING FL270 AND IT WAS NOT APPEARING TO BE IN THE PROCESS OF LEVELING OFF; I SECURED THE AUTOPLT AND MANUALLY CLBED THE ACFT BACK TO A LEVEL FL270. THE DURATION OF THE DEV WAS LESS THAN 10 SECONDS AND THE ACFT RESUMED ALT HOLD ON THE AUTOPLT WITHOUT ANY FURTHER INCIDENT. I SHOULD NOT HAVE ALLOWED OTHER COCKPIT TASKS TO DISTRACT ME FROM MY PRIMARY MISSION OF FLYING THE ACFT; AS I WAS THE PF. IN THE FUTURE; I WILL AVIATE FIRST; NAV SECOND; AND COM THIRD. THE HIGH DSCNT RATE PROBABLY CONTRIBUTED TO THE OVERSHOOT OF THE LEVELOFF ALT; BUT IF I HAD STAYED ON THE DSCNT PROFILE; THIS WOULD NOT HAVE BEEN AN ISSUE. SUPPLEMENTAL INFO FROM ACN 649390: THE FREQ WAS CONGESTED SO WE COULDN'T TALK TO CTR. NOTHING WAS SAID BY ATC.