• Communications for Statistical Applications and Methods
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• v.25 no.4
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• pp.443-451
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• 2018

$P^M_{\lambda}$-service policy is a workload dependent hysteretic policy. The policy has two service states comprised of the ordinary stage and the fast stage. An ordinary service stage is initiated by the arrival of a customer in an idle state. When the workload of the server surpasses threshold ${\lambda}$, the ordinary service stage changes to the fast service state, and it continues until the system is empty. These service stages alternate in this manner. When the cost of changing service stages is high, the hysteretic policy is more efficient than the threshold policy, where a service stage changes immediately into the other service stage at either case of the workload's surpassing or crossing down a threshold. $P^M_{\lambda}$-service policy is a modification of $P^M_{\lambda}$-policy proposed to control finite dams, and also an extension of the well-known D-policy. The distributions of the stationary workload of $P^M_{\lambda}$-service policy and its variants are studied well. However, there is no known result on the sojourn time distribution. We prove that there is a relation between the sojourn time of a customer and the first up-crossing time of the workload process over the threshold ${\lambda}$ after the arrival of the customer. Using the relation and the duality of M/G/1 and G/M/1 queues, we obtain conditional sojourn time distributions in M/G/1 and G/M/1 queues under the policy.

• Journal of the Korean Statistical Society
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• v.36 no.2
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• pp.285-297
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• 2007

We consider the $P_{\lambda}^M-service$ policy for an M/G/1 queueing system in which the workload is monitored randomly at discrete points in time. If the level of the workload exceeds a threshold ${\lambda}$ when it is monitored, then the service rate is increased from 1 to M instantaneously and is kept as M until the workload reaches zero. By using level-crossing arguments, we obtain explicit expressions for the stationary distribution of the workload in the system.

• East Asian mathematical journal
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• v.24 no.1
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• pp.45-55
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• 2008

We consider an M/G/1 queue with $P^M_{\lambda}$-service policy, which is a two-stage service policy. The server starts to serve with rate 1 if a job arrives to the sever in idle state. If the workload of the system upcrosses $\lambda$, then the service rate is changed to M and this rate continues until the system is empty. It costs to change the service rate to M and maintaining the rate. When the expectation of the stationary workload is supposed to be less than a given value, we derive the optimal value of M.

• 한국데이터정보과학회:학술대회논문집
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• 2005.04a
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• pp.25-29
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• 2005

We analyze an M/G/1 queueing system under $P_{\lambda,\tau}^M$ service policy. By using the level crossing theory and solving the corresponding integral equations, we obtain the stationary distribution of the workload in the system explicitly.

• Proceedings of the Korean Statistical Society Conference
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• 2003.10a
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• pp.189-194
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• 2003

We consider an M/G/1 queueing system under P$_{\lambda}^{M}$-service policy. As soon as the workload exceeds threshold ${\lambda}$ > 0, the service rate is increased from 1 to M ${\geq}$ 1 and is kept until the system becomes empty. After assigning several costs, we show that there exists a unique M minimizing the long-run average cost per unit time.

• Journal of the Korean Data and Information Science Society
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• v.24 no.4
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• pp.941-948
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• 2013

We introduce the $P^M_{{\lambda},{\tau}}$ service policy, as a generalized two-stage service policy of the $P^M_{\lambda}$ policy of Bae et al. (2002) for an M/G/1 queueing system. By using the level crossing theory and solving the corresponding integral equations, we obtain the explicit expression for the stationary distribution of the workload in the system.

• Proceedings of the Korean Statistical Society Conference
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• 2004.11a
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• pp.301-306
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• 2004

We consider the $P_\lambda\;^M$ service policy for an M/G/1 queue in which the service rate is increased from 1 to M at the exponential setup time after the level of workload exceeds $\lambda$. The stationary distribution of the workload is explicitly obtained through the level crossing argument.