The model which takes time constraints into account was originally suggested as an extension of the notion of Byzantine fault tolerance where redundancy of sharing allows robustness into the time domain (periods) and was proposed by Rafail Ostrovsky and Moti Yung in 1991.
[1] The method has been used in the areas of cryptographic protocols in secure multi-party computation and in threshold cryptosystems.
[4] In order to update the shares, the dealers (i.e., the persons who gives out the shares; and in a distributed system it is all participants one at a time) generates a new random polynomial with constant term zero and calculates for each remaining player a new ordered pair, where the x-coordinates of the old and new pairs are the same.
The dealer can change the threshold number while distributing updates, but must always remain vigilant of players keeping expired shares as in.
[5] However this is a somewhat limited view since the original methods gives the community of server the ability to be the re-sharing dealer and the regenerator of lost shares.