Abstract
We study a modified Markovian bulk-arrival and bulk-service queue incorporating general state-dependent control. The stopped bulk-arrival and bulk-service queue is first investigated, and the relationship between this stopped queue and the full queueing model is examined and exploited. Using this relationship, the equilibrium behaviour for the full queueing process 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 times and busy period distributions are answered in detail, and the Laplace transforms of these distributions are presented. Further properties regarding the busy period distributions including expectation and conditional expectation of busy periods are also explored.
Highlights
Markovian queues occupy a significant niche in applied probability
Note that bulk arrivals and bulk service queues are commonplace in scenarios such as industrial assembly lines, road traffic flow, the movement of aircraft passengers, etc., and the related models have extensive and important applications
We provide a couple of fundamental lemmas which will be our stepping stones for further analysis
Summary
Markovian queues occupy a significant niche in applied probability. Markovian queues play a very important role both in the development of general queueing models and in the theory and applications of continuous-time Markov chains. Being a Markovian model, we have more powerful methods and techniques such as Kolmogorov backward and forward equations and Ito’s excursion law which enable us to get many more fruitful results than for the M/G/1-type Markov chains, say Another reason for us to use the current approach is that the results obtained in this paper open the door and paves the way to study another advanced and extremely important topic of quasi-limiting distributions, including determining the decay parameter and invariant measure, and functions which reveal deep properties regarding transient behaviour of our current queuing models. As far as quasi-limiting distributions are concerned, the behaviour of our current model is remarkably different to say, the M/G/1-type Markov chains We shall discuss this topic in a couple of subsequent papers.
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