  Irregular repetition slotted Aloha (IRSA) has shown significant advantages as
a modern technique for uncoordinated random access with massive number of users
due to its capability of achieving theoretically a throughput of $1$ packet per
slot. When the receiver has also the multi-packet reception of multi-user (MUD)
detection property, by applying successive interference cancellation, IRSA also
obtains very low packet loss probabilities at low traffic loads, but is unable
in general to achieve a normalized throughput close to the $1$. In this paper,
we reconsider the case of IRSA with $k$-MUD receivers and derive the general
density evolution equations for the non-asymptotic analysis of the packet loss
rate, for arbitrary frame lengths and two variants of the first slot used for
transmission. Next, using the potential function, we give new capacity bounds
on the capacity of the system, showing the threshold arrival rate for zero
decoding error probability. Our numerical results illustrate performance in
terms of throughput and average delay for $k$-MUD IRSA with finite memory at
the receiver, and also with bounded maximum delay.
