Abstract
The transient behavior of the finite-buffer queueing model with batch arrivals and generally distributed repeated vacations is analyzed. Such a system has potential applications in modeling the functioning of production systems, computer and telecommunication networks with energy saving mechanism based on cyclic monitoring the queue state (Internet of Things, wireless sensors networks, etc.). Identifying renewal moments in the evolution of the system and applying continuous total probability law, a system of Volterra-type integral equations for the time-dependent queue-size distribution, conditioned by the initial buffer state, is derived. A compact-form solution for the corresponding system written for Laplace transforms is obtained using an algebraic approach based on Korolyuk’s potential method. An illustrative numerical example presenting the impact of the service rate, arrival rate, initial buffer state and single vacation duration on the queue-size distribution is attached as well.
Highlights
We consider the queueing model with multiple vacation periods launched after the completion of each busy period of the system. Queueing systems of this type can be used in the practical modeling of the functioning of, for example, computer and telecommunication network nodes, in which an energy saving mechanism is implemented based on the cyclical checking the state of queue of packets waiting for processing)
For the purposes of the analysis, the approach based on the idea of the embedded Markov chain, continuous formula of total probability, integral equations and linear algebra is proposed
The closed-form formula for the Laplace transform of the transient queue-size distribution conditioned by the initial buffer state is obtained
Summary
Mathematical models of queueing systems with limited capacity of buffers accumulating customers waiting for service are widely used in practice. We consider the queueing model with multiple vacation periods launched after the completion of each busy period of the system Queueing systems of this type can be used in the practical modeling of the functioning of, for example, computer and telecommunication network nodes (in particular wireless, e.g., based on Wi-Fi or LTE standards), in which an energy saving mechanism is implemented based on the cyclical checking the state of queue of packets waiting for processing). We study the time-dependent queue-size distribution in the M X /G/1/N-type queueing model with a multiple vacation policy and an exhaustive service. Numerical examples illustrate analytical results, namely the impact of the vacation duration, arrival intensity, service speed, buffer size and the initial system state on the behavior of the queue-size distribution.
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