Jung Woo Baek

BMAP/G/1 Queue Under D-Policy: Queue Length Analysis

ABSTRACT We study the queue length distribution of a queueing system with BMAP arrivals under D-policy. The idle server begins to serve the customers only when the sum of the service times of all waiting customers exceeds some fixed threshold D. We derive the vector generating functions of the queue lengths both at a departure and at an arbitrary point of time. Mean queue lengths are derived and a numerical example is presented.

A Markov‐modulated fluid flow queueing model under D‐policy

We consider a Markov-modulated fluid flow queueing model under D-policy. As soon as the fluid level reaches zero, the server becomes idle. During the idle period, fluid arrives from outside according to an underlying continuous time Markov chain (UMC) and the idle server does not process the fluid. We consider two increase patterns of fluid during the idle period: vertical increase (Type-V) and linear increase (Type-L). The idle server is reactivated only when the cumulative fluid level in the system exceeds a predetermined threshold value D. We derive the distributions of fluid level and mean performance measures for both types. We also present cost optimization model to minimize average operating cost per unit time. Copyright

Analysis of the Mx/G/1 queue under D-policy

Abstract We study the queue length of the M X /G/1 queue under D-policy. We derive the queue length PGF at an arbitrary point of time. Then, we derive the mean queue length. As special cases, M/G/1, M X /M/1, and M/M/1 queue under D-policy are investigated. Finally, the effects of employing D-policy are discussed.

A factorization property for BMAP/G/1 vacation queues under variable service speed

Abstract This paper proposes a simple factorization principle that can be used efficiently and effectively to derive the vector generating function of the queue length at an arbitrary time of the BMAP/G/1/ queueing systems under variable service speed. We first prove the factorization property. Then we provide moments formula. Lastly we present some applications of the factorization principle.

Analysis of discrete-time MAP/G/1 queue under workload control

In this paper, we analyze the discrete-time MAP/G/1 queue under the D-policy in which the idle server resumes its service only when the accumulated workload exceeds the predetermined threshold D. We first derive the probability generating functions of the queue length, workload, waiting time, and sojourn time distributions. Then we derive the mean performance measures. Lastly we present our numerical experience. Through the numerical examples we show the effects of the threshold and the variability of the service times on the mean performance measures, and the effect of the correlation between arrivals.

A workload factorization for BMAP/G/1 vacation queues under variable service speed

Abstract This paper proposes a simple factorization property for the workload distribution of the BMAP/G/1/ vacation queues under variable service speed. The server provides service at different service speeds depending on the phases of the underlying Markov chain. Using the factorization principle, the workload distribution at any arbitrary time point can be easily derived only by obtaining the distribution during the idle period. We prove the factorization property and the moments formula. Lastly, we provide some applications of our factorization principle.

A MAP-modulated fluid flow model with multiple vacations

We consider a MAP-modulated fluid flow queueing model with multiple vacations. As soon as the fluid level reaches zero, the server leaves for repeated vacations of random length V until the server finds any fluid in the system. During the vacation period, fluid arrives from outside according to the MAP (Markovian Arrival Process) and the fluid level increases vertically at the arrival instance. We first derive the vector Laplace–Stieltjes transform (LST) of the fluid level at an arbitrary point of time in steady-state and show that the vector LST is decomposed into two parts, one of which the vector LST of the fluid level at an arbitrary point of time during the idle period. Then we present a recursive moments formula and numerical examples.

Analysis of discrete-time Geo/G/1 queue under the D -policy

In this paper, we present a theoretical and analytical framework to analyze the discrete-time queue in which the on-off of the server is controlled by workload threshold. For this purpose, we consider the discrete-time Geo/G/1 queue under the D-policy. We derive the queue length, waiting time, sojourn time distributions and the mean performance measures. We then perform numerical computations to see the effect of the threshold and the variability of the service time distributions on the system performance. We will also see if our discrete-time results can approximate the continuous-time M/G/1/D-policy queueing system. The approach of this paper can be used as a basis for the analysis of more complex queueing systems in which the server is controlled by workload.

Performance of the MAP/G/1 Queue Under the Dyadic Control of Workload and Server Idleness

This paper studies the steady-state queue length process of the MAP/G/1 queue under the dyadic control of the D-policy and multiple server vacations. We derive the probability generating function of the queue length and the mean queue length. We then present computational experiences and compare the MAP queue with the Poisson queue.

Threshold Workload Control in the BMAP/G/1 Queue

We analyze the distribution of the workload in the queueing system with batch Markovian arrival process (BMAP) controlled according to the D-policy. The server begins to serve the customers only when the total workload exceeds some predetermined threshold value D. We first explore the stochastic nature of the idle period. Then we derive the Laplace-Stieltjes transform of the workload and the mean values at an arbitrary time. Finally we provide computational experiences and discuss the effects of the BMAP arrivals. Our study shows that the naive Poisson assumption heavily underestimates the workload in a queueing system. We also discuss the effect of the workload threshold control