Recurrent event data are widely encountered in clinical and observational studies. Most methods for recurrent events treat the outcome as a point process and, as such, neglect any associated event duration. This may lead to a less informative and, in some cases, a biased analysis. We introduce a correlated normal frailty model to account for the time to event and the length of stay in the event status jointly. For example, when the event is hospitalization, we can treat the time to admission and length-of-stay as two alternating recurrent events. We propose a novel approach based on penalized partial likelihood to estimate the regression parameters and variance matrix. Moreover, we develop a likelihood ratio test to assess the dependence between two recurrent processes. Simulation results demonstrate that our method provides accurate parameter estimation, with relatively fast computation times. We illustrate the methods through an analysis of hospitalizations among end-stage renal disease patients.