Approximately 4.2 million youth and young adults experience homelessness each year in the United States and lack of basic necessities puts this population at high-risk of being trafficked or exploited. Although all runaway and homeless youth (RHY) are at risk of being victims of human trafficking, certain racial, ethnic, and gender groups are disproportionately affected. Motivated by these facts, our goal is to improve equitable access to housing resources for at-risk RHY in New York City (NYC) by expanding the current housing capacity, while utilizing priority thresholds that guide decisions regarding which youth should start receiving service based on the number of beds idle in the system. Our approach involves an $M/M/N/\{K_j\}+M$ queuing model with many statistically identical servers (beds) and RHY from different demographic groups with limited patience arriving to the a large crisis and emergency shelter in NYC. The queuing model allows us to: (i) investigate the populations and demographics that are facing access barriers, (ii) project the minimum number of beds required to provide a certain global service quality level to all youth, regardless of demographic characteristics, and (iii) use priority thresholds while matching RHY with beds to promote equity. The recommendations regarding the capacity expansion and priority thresholds improves equitable access to this crisis and emergency shelter by decreasing the average number of RHY abandoning the system by 92%, with a particular reduction in the abandonment of RHY who are at high-risk of experiencing trafficking.