Access to primary care has a direct impact on morbidity and mortality, and is strongly influenced by indirect waiting time: the delay between the requested and allotted appointment day. Our models describe the heterogeneous appointment seeking patterns of a primary care patient panel using stochastic processes parameterized to reflect the diversity of primary care visit rates in the US. For capacity planning, we estimate the distribution of daily appointments, and show that the distribution variability can be reduced by heuristics that use patient flexibility regarding the day of the appointment. For delays, we demonstrate that in a first-come, first-served system, patients who need the most frequent appointments suffer the greatest delays, motivating the need to reserve slots for high-visit patient classes. To further understand the inequity in delay, we model the primary care appointment system as a Discrete-Time Markov Chain. We derive an analytical expression for delay in terms of the patient’s probability of daily visit. We show that conditions for monotone mapping of the probability of visit to delay are intractable and give numerical results that support monotonicity. In our last chapter, we expand our scope to include specialty care networks. Using patient-level longitudinal data from the Medical Expenditure Panel Survey, we model the sequence of appointments with multiple specialty types and the time intervals between such appointments as a Markov Renewal Process (MRP). We use comorbidity count to model patient heterogeneity and extract the MRP parameters for each patient subgroup. Next, we adapt the steady state results to provide an analytical expression of the expected appointment fill-rate by specialty and patient subgroups. Our analytical results demonstrate that patients with higher comorbidity count typically have a lower fill-rate because of shorter lead time between appointments thereby necessitating either overtime or reserved slots to ensure timely access. We further simulate appointment seeking patterns of a nationally representative panel of patients in the specialty network and estimate the distribution of daily appointment requests for each specialty. Similar to the primary care case, we show that heuristics that leverage patient flexibility regarding the day of the appointment can reduce variability in appointment requests for each specialty.