Electromigration (EM) degradation evolves slowly towards failure, over a period of years. This is why EM checking methods use effective current models to represent the underlying circuit workload, which are typically constant (DC) currents over time. However, ignoring all input current variations around the mean can be risky, because low-frequency input variations can have a significant impact on EM, resulting in shorter than expected lifetimes. With the use of dark silicon and multimodal chip operation, such low-frequency changes in workload are becoming increasingly common in modern designs. Ignoring these variations can lead to false positives and must be avoided. We tackle this by developing a stochastic effective current model for the input current waveforms that is easy for users to specify and which allows stochastic estimation of the impact of input variability on the lifetime. User-provided guidance on the expected durations of various modes of operation is used to provide input current variances, which are then propagated to provide variances around the stress waveforms in the metal network, which gives a more realistic estimate of the EM lifetime. Variance propagation can be expensive for large systems, but a novel simulation-like framework will be presented that allows efficient variance propagation for large interconnect trees. This has revealed that the variance can be highly significant. Even when the standard deviation of the inputs is small, at around 20-30% of the mean, we see a 30-40% drop in the lifetimes.