This thesis consists of three essays in service operations and supply chains. The first essay is on managing stochastic inventory systems with scarce resources. We study an inventory system where a firm is subject to an allowance (a limit) on either the amount of input it can use or the amount of output it can produce over a specified compliance period. With such an allowance constraint, the quantity produced in one period affects the quantity that can be produced in future periods. We formulate the problem as a stochastic dynamic program with a two-dimensional state space. Using a novel extended state-space analysis, we reduce the problem into one that is single-dimensional and easier to analyze. We show the optimal policy for this modified version and then use it to characterize the structure of the optimal policy for the original problem. We also consider an extended version of the problem where the firm decides the allowance amount at the beginning of the compliance period.The second essay is on service systems with finite and heterogeneous customer arrivals. We analyze a queueing system where a finite number of customer arrivals occur over a period of time. Customer inter-arrival times and service times are heterogeneous. Using an embedded Markov chain approach, we analytically characterize various performance measures of interest, including the expected waiting time of a specific customer, the expected waiting time of an arbitrary customer, and the expected completion time of all customers. Through numerical experiments, we examine the effect of heterogeneity in inter-arrival and service times. We derive managerial insights and discuss implications for settings where inter-arrival and service time features can be induced. We also validate the numerical results using a fluid approximation that yields closed form expressions.The third essay is on service systems with appointment-driven arrivals, non-punctual customers, and no-shows. We consider settings where a finite number of customers arrive to a service system based on appointments. However, customers are not necessarily punctual and may also not show up altogether. We develop an exact analytical approach to obtain various performance measures related to customer waiting time.