We might never know the answer to this question (so little historical information about these sages have been preserved), but the reason I’m asking this is because I’m trying to figure out how the different schools of Shammai and Hillel came to be. If Shammai was older than Hillel then I was thinking about reconstructing the following scenario: Shammai, most likely situated in Israel at the time (Shammai’s place of origin is not known for sure), was a very influential Rabbi attracting a lot of students. Hillel comes up from Babylon, and gains ascendency over the family of Betheira and is appointed Nasi (Hillel may have been more suitable for the job than Shammai, either because of his personality, or because he was from the davidic line). Hillel starts to attract a school of his own, but Shammai is not ready to yield his rabbinical power to Hillel, thus two schools are formed, although Hillel promotes Shammai within the Sanhedrin and appoints him av bes din, i.e., second in command to him. Shammai still remains the de facto rabinnical authority in Israel as is evidenced from Shabbos 17a (“Hillel sat submissively before Shammai”), only later does the school of Hillel gain ascendency over Shammai.
How likely is such a scenario? Is there any evidence that Shammai was older than Hillel?