This paper obtains logarithmic asymptotics of moderate deviations of the stochastic process of the number of customers in a many-server queue with generally distributed inter-arrival and service times under a heavy-traffic scaling akin to the Halfin–Whitt regime. The deviation function is expressed in terms of the solution to a Fredholm equation of the second kind. A key element of the proof is the large-deviation principle in the scaling of moderate deviations for the sequential empirical process. The techniques of large-deviation convergence and idempotent processes are used extensively.

Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

The longest gap $L(t)$ up to time $t$ in a homogeneous Poisson process is the maximal time subinterval between epochs of arrival times up to time $t$; it has applications in the theory of reliability. We study the Laplace transform asymptotics for $L(t)$ as $t\rightarrow \infty$ and derive two natural and different large-deviation principles for $L(t)$ with two distinct rate functions and speeds.

We derive a large-deviation principle for super-Brownian motion with immigration, where the immigration is governed by the Lebesgue measure. We show that the speed function is t 1/2 for d = 1, t/logt for d = 2 and t for d ≥ 3, which is different from that of the occupation-time process counterpart (without immigration) and the model of random immigration.