Motivated by Sampath [Finite population variance estimation under LSS with multiple random starts, Commun. Statist. -Theory Methods 38 (2009), pp. 3596-3607], in this paper unbiased estimators for population variance have been developed under linear systematic sampling, balanced systematic sampling and modified systematic sampling with multiple random starts. Expressions for variances of the estimators are also developed. Detailed numerical comparative studies have been carried out to study the performances of the estimators under various systematic sampling schemes with multiple random starts and some interesting conclusions have been drawn out of the study.