On steady-state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation
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On steady‑state joint distribution of an infinite buffer batch service Poisson queue with single and multiple vacation G. K. Tamrakar1 · A. Banerjee1 Accepted: 7 May 2020 © Operational Research Society of India 2020
Abstract This article considers a single server, infinite buffer, bulk service Poisson queue with single and multiple vacation. The customers are served in batches following ‘general bulk service’ (GBS) rule. The customers are arriving according to the Poisson process, and the service time of the batches follows an exponential distribution. Using bivariate probability generating function (PGF) method the steady-state joint distributions of the queue content and server content (when server is busy), and joint distribution of the queue content and type of the vacation taken by the server (when server is in vacation) have been obtained. Here by the ‘type of the vacation’ we mean the queue length at vacation initiation epoch. The information about these joint distributions may help in increasing the system performance. Finally, several numerical examples are carried out using MAPLE software to verify the analytical results. Keywords Bulk-service · Bivariate probability generating function · Infinite buffer queue · Single vacation · Multiple vacation
1 Introduction In the service facility center, it is often observed that the server becomes unavailable for a certain period of its job duration and during that period, also called the vacation period, either it takes rest or performs some supplementary jobs. Queueing model, along with the server’s vacation is known as vacation queuing model, and is proposed by Levy and Yechiali [29]. In [29] authors introduced the concept of single vacation (SV) and multiple vacation (MV) in M/G/1 queue. After this significant addition to the literature of queueing theory, many researchers paid huge attention to vacation models due to its unambiguous applicability * A. Banerjee [email protected] G. K. Tamrakar [email protected] 1
Indian Institute of Technology (BHU) Varanasi, Varanasi, India
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in transportation, production manufacturing, telecommunication system, etc. The excellent survey on vacation models has been done by Doshi [10] and Ke et al. [23]. Good literature on vacation models is found in the books by Takagi [46] and Tian and Zhang [48]. It has been observed that several vacation rules are famous amongst researchers, e.g., single vacation (SV), multiple vacations (MV), working vacation (WV), Bernoulli Vacation (BV), etc. After a busy period (by busy period we mean the total time when server is continuously busy in serving customers) the server takes the vacation of random length and at the end of the vacation, either it serves the waiting customers with different service policies (viz., (exhaustive){gated}[limited] service policy) or 1. stays in dormant state and wait for required minimum number of customers to initiate the service according to the service rule under consideration. 2. goes for anoth
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