On the Time to Buffer Overflow in a Queueing Model with a General Independent Input Stream and Power-Saving Mechanism Based on Working Vacations.

Martyna Kobielnik,Wojciech Kempa

doi:10.3390/s21165507

Martyna Kobielnik, Wojciech Kempa

Open Access

https://doi.org/10.3390/s21165507

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Abstract

A single server queue with a limited buffer and an energy-saving mechanism based on a single working vacation policy is analyzed. The general independent input stream and exponential service times are considered. When the queue is empty after a service completion epoch, the server lowers the service speed for a random amount of time following an exponential distribution. Packets that arrive while the buffer is saturated are rejected. The analysis is focused on the duration of the time period with no packet losses. A system of equations for the transient time to the first buffer overflow cumulative distribution functions conditioned by the initial state and working mode of the service unit is stated using the idea of an embedded Markov chain and the continuous version of the law of total probability. The explicit representation for the Laplace transform of considered characteristics is found using a linear algebra-based approach. The results are illustrated using numerical examples, and the impact of the key parameters of the model is investigated.

Highlights

The problem of reducing energy consumption is global
We study a finite-buffer GI/M/1/N queueing model with generaltype independent input flow of jobs, exponentially distributed service times, and a single working vacation policy
We investigated a finite-capacity queueing model with an independent general input flow, exponential service times, and a single working vacation policy

Summary

Introduction

The problem of reducing energy consumption is global. This results in large-scale research on algorithms supporting power-saving control and the accompanying technical solutions. An energy-saving mechanism based on a threshold-controlled multiple vacation policy was considered in [27] as a model for the operation a wireless sensor network node. We study a finite-buffer GI/M/1/N queueing model with generaltype independent input flow of jobs, exponentially distributed service times, and a single working vacation policy. Applying an analytic approach based on the idea of embedded Markov chain and linear algebra, we find the closed-form representation for the Laplace transform of the time to the first buffer overflow distribution, conditioned by the initial system state and working mode of the service unit.

Model Description and Auxiliary Result

Solution for the Normal Mode Let us denote

Solution for the WV Mode

Numerical Examples

Conclusions