Conn. Gen. Stat. § 9-184c

(b) If, on November first in such year, the number of applications for justice of the peace filed with the town clerk under subsection (a) of this section exceeds the number of justices of the peace allowed under this section, (1) each such applicant who is an incumbent justice of the peace appointed under this section shall be reappointed if there are sufficient openings and (2) the town clerk shall, on or before the fifteenth business day of November, select the remaining applicants to be appointed as justices of the peace by lot in a ceremony which shall be open to the public and held on five days' public notice. At such lottery the town clerk shall determine the order of all such remaining applications for the purpose of filling future vacancies under subsection (d) of this section. If a town clerk receives a number of applications that is less than the number of justices of the peace that he is authorized to appoint under this section in any year, he shall not appoint any additional justices of the peace.

(c) Justices of the peace appointed in 1994, shall serve a term of two years beginning on the first Monday in 1995, and justices of the peace appointed in 1996 and thereafter shall serve a term of four years beginning on the first Monday in January in the succeeding year.

(d) Any vacancy in the office of any such justice of the peace shall be filled by appointment by the town clerk of an elector qualifying under subsection (a) of this section in the order determined in the lottery held under said subsection. If no such lottery is held, the vacancy shall not be filled.