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The main purpose of this paper is to consider the multiple birth properties for multi-type Markov branching processes. We first construct a new multi-dimensional Markov process based on the multi-type Markov branching process, which can reveal the multiple birth characteristics. Then the joint probability distribution of multiple birth of multi-type Markov branching process until any time t is obtained by using the new process. Furthermore, the probability distribution of multiple birth until the extinction of the process is also given. [ABSTRACT FROM AUTHOR]

In this paper, we consider n-type Markov branching–immigration processes. The uniqueness criterion is first established. Then, we construct a related system of differential equations based on the branching property. Furthermore, the explicit expression of extinction probability and the mean extinction time are successfully obtained in the absorbing case by using the unique solution of the related system of differential equations and Kolmogorov forward equations. Finally, the recurrence and ergodicity criteria are given if the zero state 0 is not absorbing. [ABSTRACT FROM AUTHOR]

We consider the important decay parameter issue for a Markovian bulk-arrival and bulk-service queue which stops at reaching the first m states. The results obtained in this paper are a significant generalization of the ones obtained in Chen et al. (Queueing Syst 66:275–311, 2010). The exact value of the decay parameter is obtained. Based on the results obtained in this paper, we provide a practical method for calculating the decay parameter which is very effective. In some cases, the decay parameter can even be easily expressed explicitly. Some important key lemmas which also have their own interest are provided. The interesting and clear geometric interpretation of the decay parameter is explained. A few examples are provided to illustrate the results obtained in this paper. [ABSTRACT FROM AUTHOR]

Motivated by communication networks, we study a single-server queue with varying service rates and multiple customers' types. The difference between types of customers is defined by the profits contributed to the system. We show that the optimal trunk reservation policy exists and we compare the optimal control levels with different parameter values. Furthermore, when the optimal trunk reservation policy is more than one, we prove the uniqueness of the bias optimal policy. [ABSTRACT FROM AUTHOR]

This paper concentrates on considering the down/up crossing property of weighted Markov collision processes. The joint probability generating function of down crossing and up crossing numbers of weighted Markov collision processes until its extinction are obtained by constructing and studying a related multi-dimensional Markov chain. Hence, the joint probability distribution of down crossing and up crossing numbers and the mean numbers are obtained. [ABSTRACT FROM AUTHOR]

We consider the regularity and ergodic properties of the Branching Collision Process with Immigration (BCIP) in this paper. We establish an easy checking sufficient condition under which the Feller minimal BCIP is honest. We provide some good conditions under which the Feller minimal BCIP is positive recurrent and then establish an analytic form of the generating function of the stationary distribution. The closely associated expected hitting times are also considered. Examples and numerical calculations are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

Large deviation rates are determined for quantities associated with a Markov branching process (X t : t ≥ 0) having offspring mean m ∈ (1 , ∞) and split rate a > 0. The principal quantities examined are P (| X t + τ / X t − exp (a (m − 1) t) | > 𝜖) and P (| W − W t | > 𝜖) , where W is the almost sure limit of an appropriately normed version of X t . Modifications and conditional versions are examined. Some of this requires determination of the asymptotic behavior of harmonic moments E (X t − r). [ABSTRACT FROM AUTHOR]

This article concentrates on investigating decay properties of Markovian bulk-arriving queues withc-servers which stop immediately after hitting the state zero. The exact value of the decay parameter λCfor the caseB′c(1) > 0 is first given. Then, by using the generating functions of the correspondingq-matrix, a test functionFλ(s) is constructed. The exact value of the decay parameter λCfor the caseB′c(1) ⩽ 0 is then obtained according to the sign of. [ABSTRACT FROM PUBLISHER]

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ∈[0, λ]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ∈[0, λ]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ-transient. Some deep properties regarding λ-transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper. [ABSTRACT FROM AUTHOR]