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In this letter, we investigate the state estimation problem in Mobile Edge Computing with unreliable communications. A finite-time convergent estimator is proposed, aiming to restore the unmeasurable states within a finite time length. We apply the proposed estimator in two practical scenarios to evaluate its performance. The numerical results show that the proposed estimator is able to quickly make accurate estimations once the interrupted connection resumes. The results also suggest that the effect of short-term data loss is easier to remedy than that of long-term data loss, which shed light on relay deployment and future investigations in this field.