Hanjun Zhang

Markovian bulk-arrival and bulk-service queues with state-dependent control

We study a modified Markovian bulk-arrival and bulk-service queue incorporating state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated and the relationship with our queueing model is examined and exploited. Equilibrium behaviour is studied and the probability generating function of the equilibrium distribution is obtained. Queue length behaviour is also examined and the Laplace transform of the queue length distribution is presented. The important questions regarding hitting time and busy period distributions are answered in detail and the Laplace transforms of these distributions are presented. Further properties including expectations of hitting times and busy period are also explored.

Uniqueness criteria for continuous-time Markov chains with general transition structures

We derive necessary and sufficient conditions for the existence of bounded or summable solutions to systems of linear equations associated with Markov chains. This substantially extends a famous result of G. E. H. Reuter, which provides a convenient means of checking various uniqueness criteria for birth-death processes. Our result allows chains with much more general transition structures to be accommodated. One application is to give a new proof of an important result of M. F. Chen concerning upwardly skip-free processes. We then use our generalization of Reuters lemma to prove new results for downwardly skip-free chains, such as the Markov branching process and several of its many generalizations. This permits us to establish uniqueness criteria for several models, including the general birth, death, and catastrophe process, extended branching processes, and asymptotic birth-death processes, the latter being neither upwardly skip-free nor downwardly skip-free.

Strong ergodicity of monotone transition functions

By revealing close links among strong ergodicity, monotone, and the Feller-Reuter-Riley (FRR) transition functions, we prove that a monotone ergodic transition function is strongly ergodic if and only if it is not FRR. An easy to check criterion for a Feller minimal monotone chain to be strongly ergodic is then obtained. We further prove that a non-minimal ergodic monotone chain is always strongly ergodic. The applications of our results are illustrated using birth-and-death processes and branching processes.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Feller Transition Functions, Resolvent Decomposition Theorems, and their Application in Unstable Denumerable Markov Processes

This paper surveys the recent progresses made in the field of unstable denumerable Markov processes. Emphases are laid upon methodology and applications. The important tools of Feller transition functions and Resolvent Decomposition Theorems are highlighted. Their applications particularly in unstable denumerable Markov processes with a single instantaneous state and Markov branching processes are illustrated.

Convergence Property of Standard Transition Functions

A standard transition function P = (P ij (t)) is called ergodic (positive recurrent) if there exists a probability measure π = (π i ; i ∈ E) such that

The collision branching process

\mathop{{\lim }}\limits_{{t \to \infty }} {{p}_{i}}_{j}(t) = {{\pi }_{j}} > 0,\forall i \in E

Existence and Uniqueness of "Q"-Processes with a Given Finite "μ"-Invariant Measure

(0.1) The aim of this paper is to discuss the convergence problem in (0.1). We shall study four special types of convergence: the so-called strong ergodicity, uniform polynomial convergence, L 2-exponential ergodicity and exponential ergodicity. Our main interest is always to characterize these properties in terms of the q-matrix.