Two-Level Control Policy of an Unreliable Queueing System with Queue Size-Dependent Vacation and Vacation Disruption
01 Jan 2018-pp 373-382
TL;DR: In this paper, the authors analyse two-level control policy of an MX∕G(a, b)∕1 queueing system with fast and slow vacation rates and vacation disruption.
Abstract: The objective of the paper is to analyse two-level control policy of an MX∕G(a, b)∕1 queueing system with fast and slow vacation rates and vacation disruption. In the service completion epoch, if the queue length is less than ‘a’, then the server leaves for a vacation. In this model depending upon the queue length, the server is allowed to take two types of vacation called fast vacation and slow vacation. Addressing this in the service completion epoch, if the queue length ψ(say) is less than β where β ζ, where a − 1 ≥ ζ > β during service completion, then the server leaves for fast vacation. During slow vacation if the queue length reaches the value ζ, then the server breaks the slow vacation and switches over to fast vacation. Also if the queue length attains the threshold value ‘a’ during fast vacation, then the server breaks the fast vacation too and moves to tune-up process to start the service. After tune-up process service will be initiated only if ψ ≥ N(N > b). For the designed queueing system probability, generating function of the queue size at an arbitrary time epoch is obtained by using supplementary variable technique. Various performance characteristics will also be derived with suitable numerical illustrations. Cost-effective analysis is also carried out in the paper.
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TL;DR: An overview and literature survey on the performance modeling and analysis of single server, general service queueing system with service interruption using supplementary variable technique and factors causing service interruption such as unreliable server and server vacation are presented.
Abstract: In most of the queueing models, service is considered to be complete without any interruption. But in reality, queueing systems are subject to interruptions due to failure of server or any other cause. In the present article, we present an overview and literature survey on the performance modeling and analysis of single server, general service queueing system with service interruption using supplementary variable technique. The factors causing service interruption such as unreliable server and server vacation are elaborated. The brief of supplementary variable technique to establish the queue size distribution is explained for single server non-Markovian queueing models by incorporating the features of service interruption. The basic concepts and review of literature on the queues with server breakdown and/or vacationing server are described. The research works done during last 10 years (2010–2019) on queues with service interruption involving many other key concepts namely Bernoulli vacation, multiple vacation, bulk arrival, discouragement, etc. and queueing scenarios of service interruption are reported. Some specific applications are also highlighted.
16 citations
References
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TL;DR: In this paper, the authors assume that customers arrive at a counter according to a homogeneous Poisson process and are served in groups, according to the following policy: if there are less than L customers waiting at the time of a departure, the server must wait until there are L customers present, whereupon he serves them together.
Abstract: We assume that customers arrive at a counter according to a homogeneous Poisson process and are served in groups, according to the following policy: If there are less than L customers waiting at the time of a departure, the server must wait until there are L customers present, whereupon he serves them together. If there are L or more, but less than K(K ? L) customers waiting, all are served together. If there are K or more customers waiting, a group of K customers are served and the others must wait. The service times of successive groups are assumed to be conditionally independent given the bulk sizes, but may depend on their magnitude. We obtain 1. a description of the output process, 2. the queue length in discrete time, 3. the distribution of the busy period, 4. the queue length in continuous time and 5. some limit theorems for the number of customers served over a long period of time. The order of service is irrelevant in this paper. The method used throughout is that of the imbedded semi-Markov process.
318 citations
TL;DR: This paper studies the vacation policy of an M/G/1 queueing system with an unreliable server and startup and calls the policy modified T vacation policy, which is assumed that the server breaks down according to a Poisson process and his repair time has a general distribution.
54 citations
TL;DR: In this paper, the authors deal with batch service queues with vacations in which customers arrive according to a Poisson process and derive the queue length distributions both for single and multiple vacation cases.
Abstract: The paper deals with batch service queues with vacations in which customers arrive according to a Poisson process. Decomposition method is used to derive the queue length distributions both for single and multiple vacation cases. The authors look at other decomposition techniques and discuss some related open problems.
32 citations
TL;DR: This paper considers an M/G/1 repairable queueing system with N-policy and single vacation, in which the service station is subject to random breakdowns, and numerically determined the optimal threshold N∗ for minimizing the cost function.
Abstract: This paper considers an M/G/1 repairable queueing system with N-policy and single vacation, in which the service station is subject to random breakdowns. Once the service station breaks down, it is repaired by a repair facility. Moreover, the repair facility may fail during the repair period which results in repair interruptions. Failed repair facility resumes repair after a random period of time. Applying the renewal process theory and the probability decomposition technique, the probability that the service station is broken, the rate of occurrence of breakdowns of the service station, the probability that the repair facility is being replaced, the rate of occurrence of failures of the repair facility and the stochastic decomposition property of the reliability measures are obtained. Following the construction of the long-run expected cost function per unit time, we numerically determined the optimal threshold N
∗ for minimizing the cost function.
5 citations
01 Jan 2016
TL;DR: The proposed model, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained and various performance measures are obtained.
Abstract: In this paper, the server breakdown with interrupted vacation in a ( ) / , /1 MX G a b queuing system is considered. After completing a batch of service, if the server is breakdown with probability π , then the renovation of service station will be considered. After completing the renovation of service station or there is no breakdown of the server with probability (1 ) π − , if the server finds at least ‘ a ’ customers waiting for service say ξ , then the server serves a batch of min ( , ) b ξ customers, where b a ≥ . On the other hand, if the queue length is less than ‘ a ’, the server leaves for a secondary job (vacation) of random length. During the secondary job period, the secondary job is interrupted abruptly and the server resumes for primary service, if the queue size reaches ‘ a ’. On completion of the secondary job, the server remains in the system (dormant period) until the queue length reaches ‘ a ’. For the proposed model, the probability generating function of the steady state queue size distribution at an arbitrary time is obtained. Various performance measures are obtained. Numerical illustration is also given to justify the proposed model. The cost model is also developed.
4 citations