• Journal of the Korean Statistical Society
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• 제36권2호
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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.

• Communications for Statistical Applications and Methods
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• 제25권4호
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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.

• 한국데이터정보과학회:학술대회논문집
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• 한국데이터정보과학회 2005년도 춘계학술대회
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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.

• East Asian mathematical journal
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• 제24권1호
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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.

• 한국통계학회:학술대회논문집
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• 한국통계학회 2003년도 추계 학술발표회 논문집
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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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• 제24권4호
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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.

• Journal of the Korean Statistical Society
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• 제26권1호
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• pp.147-154
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• 1997

An infinite dam with a compound Poisson input having exponential jumps is considered. As an output policy, we adopt the $P_{\lambda}$$^{M}$ Policy. After assigning costs to the dam we obtain the long-rum average cost per unit time of operating the dam and find the optimal values of .lambda. and M which minimize the long-run average cost.t.

• 한국통계학회:학술대회논문집
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• 한국통계학회 2004년도 학술발표논문집
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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.

• Journal of the Korean Data and Information Science Society
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• 제22권6호
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• pp.1089-1096
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• 2011

본 논문에서는 댐으로 물이 복합 포아송 과정을 따라 유입되고, 댐 수위가 적정수준 ${\lambda}$를 넘어서게 되면 방출률 M으로 물을 방출하는 무한 댐 모형이 고려된다. 한번 내리는 비의 양이 지수분포를 따르는 경우, 정상상태에서 댐 수위의 확률적 성질들을 구한다. 댐 운영과 관리에 관련된 여러 비용을 고려한 후, 장시간에 걸친 단위시간당 평균 비용을 구하고, 이를 최소화하는 방출률 M과 적정수준${\lambda}$가 유일하게 존재함을 보인다. MATLAB을 이용하여 수치적인 결과도 함께 보인다.

• Journal of the Korean Statistical Society
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• 제32권3호
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• pp.311-311
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• 2003

This paper contains the following errors. 1. "$I_{\{x>D\}}{\lambda}dt_{p0}(t)s(x)$" should be added to the right hand side of (2.3). 2. "$I_{\{x>D\}}{\lambda}_{p0}(t)s(x)$" should be added to the right hand side of (2.6). 3. "$I_{\{x>D\}}{\lambda}_{p0}s(x)$" should be added to the right hand side of (2.9). 4. In Theorem 2.3 and its proof, "${\lambda}{\int}_{0}^{D}f(y)s(x-y)dy$" appears three times (including one in (2.20)). To each of these, "${\lambda}_{po}s(x)$" should be added. 5. In Remark 2.5, "${\lambda}dt_{p0}/s(x)dx" should be added to "${\int}_{0}^{D}{\lambda}dt\;s(x-y)dxf(y)dy$". As a result of these corrections, a simpler proof of Theorem 2.3 becomes available. Substituting (2.18), (2.21), (2.22) into the left hand side of (2.20) and comparing the result with (2.10), we have the right hand side of (2.20).