This paper studies the waiting time distributions of customers in an $M/M/1/m+1$ queueing system with customer interjections and server's single vacation. The customers who enter the system are divided into regular customers and interjection customers according to whether they interject the queue or not. After entering the system, the regular customers queue up at the end of the waiting line and wait for service, while the interjection customers queue up as close as possible to the head of the waiting line to receive service. There is one server who takes single vacation in the system. By using the properties of negative exponential distribution, the phase type distribution and the Markov chain with absorbing state, the matrix expressions of waiting time distribution of customers in waiting queue position $n$, regular customers and interjection customers are derived. Further, the waiting time distributions with time $t$ are plotted.