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

Abstract

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. A. Chen ( )

Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

“…In the following we shall use u(λ) and v(λ) to denote the smallest positive root and the largest negative root of B(s) + λs 2 = 0, respectively. Note that for λ < 0, we have used u(λ) and v(λ) to denote these two roots of B(s) + λs 2 = 0, see Chen et al [8]. [8], one would find, interestingly, that the two functions u(λ) and v(λ) defined here can be viewed as extensions of the two corresponding functions, bearing the same title in [8], with the domain extended from (−∞, 0) to (−∞, λ C ).…”

“…Note that for λ < 0, we have used u(λ) and v(λ) to denote these two roots of B(s) + λs 2 = 0, see Chen et al [8]. [8], one would find, interestingly, that the two functions u(λ) and v(λ) defined here can be viewed as extensions of the two corresponding functions, bearing the same title in [8], with the domain extended from (−∞, 0) to (−∞, λ C ). Hence they inherit the relevant properties stated in Lemma 3 in Chen et al [8].…”

“…The queueing models with both bulk-arrival and bulk-service incorporating with state-dependent control have also been attracted much interest. In particular, very recently, Chen et al [8] discussed an interesting model where state-dependent control can happen only when no customer is at waiting list except possibly a customer in service.…”

“…In fact, even calculating the exact value of the decay parameter for most transient Markovian queueing models is still an open problem. In particular, we still know little about decay properties of the queueing model discussed in Chen et al [8]. Note that, however, if the bulk-service is not presented, the decay parameter and the associated properties have been examined in Li and Chen [27,28].…”

Ergodicity and Perturbation Bounds for Inhomogeneous Birth and Death Processes with Additional Transitions from and to the Origin

“…Since the publication of Chen and Renshaw (1997), some papers have appeared devoted to the study of birth and death processes with "mass exodus" and "mass arrivals when empty" (see, e.g., Chen et al, 2010;Chen and Renshaw, 2004;Li and Chen, 2013;Gaidamaka et al, 2014), and what they all have in common is that transition intensities are assumed to be constants independent of time. We tried to look further and see if the time-dependent analysis of such birth and death processes can be extended to the case when all possible transition intensities are non-random functions of time.…”