In this paper, a network comprising wireless devices equipped with buffers transmitting deadline-constrained data packets over a slotted-ALOHA random-access channel is studied. Although communication protocols facilitating retransmissions increase reliability, packet transmission from the queue experiences delays and thus, packets with time constraints might be dropped before being successfully transmitted, while at the same time causing the queue size of the buffer to increase. Towards understanding the trade-off between reliability and delays that might lead to packet drops due to the deadline-constrained bursty traffic with retransmissions, a scenario of a wireless network utilizing a slotted-ALOHA random-access channel is investigated. The main focus is to reveal and investigate further the trade-off between the number of retransmissions and the packet deadline as a function of the arrival rate. Hence, we are able to determine numerically the optimal probability of transmissions and number of retransmissions, given the packet arrival rate and the packet deadline. The analysis of the system was done by means of discrete-time Markov chains. Two scenarios are studied: i) the collision channel model (in which a receiver can decode only when a single packet is transmitted), and ii) the case for which receivers have multi-packet reception capabilities. A performance evaluation for a user with different transmit probability and number of retransmissions is conducted, demonstrating their impact on the average drop rate and throughput, while at the same time showing that there exists a set of values, under which improved performance can be acquired.