  Security and privacy are major concerns in modern communication networks. In
recent years, the information theory of covert communications, where the very
presence of the communication is undetectable to a watchful and determined
adversary, has been of great interest. This emerging body of work has focused
on additive white Gaussian noise (AWGN), discrete memoryless channels (DMCs),
and optical channels. In contrast, our recent work introduced the
information-theoretic limits for covert communications over packet channels
whose packet timings are governed by a Poisson point process. However, actual
network packet arrival times do not generally conform to the Poisson process
assumption, and thus here we consider the extension of our work to timing
channels characterized by more general renewal processes of rate $\lambda$. We
consider two scenarios. In the first scenario, the source of the packets on the
channel cannot be authenticated by Willie, and therefore Alice can insert
packets into the channel. We show that if the total number of transmitted
packets by Jack is $N$, Alice can covertly insert
$\mathcal{O}\left(\sqrt{N}\right)$ packets and, if she transmits more, she will
be detected by Willie. In the second scenario, packets are authenticated by
Willie but we assume that Alice and Bob share a secret key; hence, Alice alters
the timings of the packets according to a pre-shared codebook with Bob to send
information to him over a $G/M/1$ queue with service rate $\mu>\lambda$. We
show that Alice can covertly and reliably transmit $\mathcal{O}(N)$ bits to Bob
when the total number of packets sent from Jack to Steve is $N$.
