We present a study of the joint distribution of both the state of a level-dependent quasi-birth–death (QBD) process and its associated running maximum level, at a fixed time t: more specifically, we derive expressions for the Laplace transforms of transition functions that contain this information, and the expressions we derive contain familiar constructs from the classical theory of QBD processes. Indeed, one important takeaway from our results is that the distribution of the running maximum level of a level-dependent QBD process can be studied using results that are highly analogous to the more well-established theory of level-dependent QBD processes that focuses primarily on the joint distribution of the level and phase. We also explain how our methods naturally extend to the study of level-dependent Markov processes of M/G/1 type, if we instead keep track of the running minimum level instead of the running maximum level.

We introduce two general classes of reflected autoregressive processes, INGAR+ and GAR+. Here, INGAR+ can be seen as the counterpart of INAR(1) with general thinning and reflection being imposed to keep the process non-negative; GAR+ relates to AR(1) in an analogous manner. The two processes INGAR+ and GAR+ are shown to be connected via a duality relation. We proceed by presenting a detailed analysis of the time-dependent and stationary behavior of the INGAR+ process, and then exploit the duality relation to obtain the time-dependent and stationary behavior of the GAR+ process.

We use the random-product technique from [5] to study both the steady-state and time-dependent behavior of a Markovian reentrant-line model, which is a generalization of the preemptive reentrant-line model studied in the work of Adan and Weiss [2]. Our results/observations yield additional insight into why the stationary distribution of the reentrant-line model from [2] exhibits an almost-geometric product-form structure: indeed, our generalized reentrant-line model, when stable, admits a stationary distribution with a similar product-form representation as well. Not only that, the Laplace transforms of the transition functions of our reentrant-line model also have a product-form structure if it is further assumed that both Buffers 2 and 3 are empty at time zero.

We derive factorization identities for a class of preemptive-resume queueing systems, with batch arrivals and catastrophes that, whenever they occur, eliminate multiple customers present in the system. These processes are quite general, as they can be used to approximate Lévy processes, diffusion processes, and certain types of growth‒collapse processes; thus, all of the processes mentioned above also satisfy similar factorization identities. In the Lévy case, our identities simplify to both the well-known Wiener‒Hopf factorization, and another interesting factorization of reflected Lévy processes starting at an arbitrary initial state. We also show how the ideas can be used to derive transforms for some well-known state-dependent/inhomogeneous birth‒death processes and diffusion processes.

This paper continues an investigation of the time-dependent behavior of regulated or reflecting Brownian motion (RBM). Part I focused on RBM starting at the origin; Part II focuses on RBM starting at a fixed positive state. The first two moments of RBM as functions of time are analyzed by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied in Part I. By appropriate normalization, the two monotone components can be converted into cumulative distribution functions that can be analyzed probabilistically, e.g., their moments can be calculated. Simple approximations are then developed by fitting convenient distributions to these moments. Overall, the analysis yields a better understanding of the way RBM and related stochastic flow systems approach steady state.