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In this paper, we consider delay-constrained wireless multi-hop ad hoc networks where a packet should be delivered to the destination within the maximum allowed delay while satisfying the target outage probability. The proposed performance metric for analyzing networks is the delay-constrained random access transport capacity (D-RATC), which quantifies the maximum end-to-end (e2e) link achievable rate per unit area of a delay-constrained network using a random access protocol. The scaling of the D-RATC is obtained for various slotted ALOHA (SA) protocols and it is shown that the SA protocol is order-optimal for delay-constrained random networks when interference control is used with an additional feature such as rate control or admission control. If interference control is not used, the SA protocol suffers from the negatively infinite scaling exponent except the case of using rate control where a finite but suboptimal scaling exponent may be achieved. Also, it is shown that multi-hop control does not affect the scaling exponent but just improves the D-RATC pre-constant.