The siege of Tobruk lasted for 8 months. Tobruk was only relieved in December 1941, during the Crusader battles, when the German position collapsed under the British attack that had been salvaged by General Auchinleck.
The whole movement of shipping was engineered so that ships were only in port, being unloaded in the night, and then were gone by daylight. In May 1941, an average of 84 tons of supplies per day were brought into Tobruk. The goal had been to bring at least 70 tons a day, so that was surpassed. In June, they were able to increase the supplies brought in to 94 tons per day.
Naval losses were high, during the month of June, as the Australian sloop Auckland was sunk, as was the Australian destroyer Waterhen. Destroyers played a prominent part in the continuing supply operation, and they operated in pairs.
This is based on the account in Vol.II of the Official History.