On impatience in M/M/1/N/DWV queue with vacation interruption
Croatian Operational Research Review
In this paper, we establish a cost optimization analysis for an M/M/1/N queuing system with differentiated working vacations, Bernoulli schedule vacation interruption, balking and reneging. Once the system is empty, the server waits a random amount of time before he goes on working vacation during which service is provided with a lower rate. At the instant of the service achievement in the vacation period, if there are customers present in the system, the vacation is interrupted and the server
... ted and the server returns to the regular busy period with probability β or continues the working vacation with probability 1 − β . Whenever the working vacation is ended, the server comes back to the normal busy period. If the system is empty, the server can take another working vacation of shorter duration. In addition, it is supposed that during both busy and working vacation periods, arriving customers may become impatient with individual timers exponentially distributed. Explicit expressions for the steady-state system size probabilities are derived using recursive technique. Further, interesting performance measures are explicitly obtained. Then, we construct a cost model in order to determine the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Finally, numerical illustrations are added to validate the theoretical results. may come back from the vacation to the regular working level once the number of customers attain a certain value in the vacation period. Then, Li et. al  considered a GI/M/1 queuing model with working vacation and vacation interruption. Baba  dealt with a M/P H/1 queuing model under working vacations and vacation interruption. An M/G/1 queuing system with single working vacation and vacation interruption under Bernoulli schedule has been examined in Gao and Liu . Lee and Kim  presented the sojourn time distribution of an M/G/1 queue with a single working vacation and vacation interruption. Later, the transient analysis of an infinite capacity sigle server Markovian queue with differentiated vacations has been done in Vijayashree and Janani . Majid et. al  treated a M/M/1 queuing model with working vacation and vacation interruption under Bernoulli schedule. Recently, significant results on the subject have been presented, see for instance, Ameur et. al , Majid and Manoharan , and Rajadurai  and the reference therein. Various queuing situations occur where customers tend to be discouraged by a long queue. As a consequence, customers decide either not to join the queue (balk) or leave after logging into the queue without being served because of their impatience (renege). Numerous researchers have been attracted by the analysis of the impatience behavior in queuing models with vacation and working vacation. Yue et. al  dealt with impatience behavior in an M/M/1/ queue under multiple working vacation policy. A finite buffer renewal input queuing model with balking and multiple working vacations has been carried out by Vijaya Laxmi and Jyothsna . Then, Vijaya Laxmi and Jyothsna  analyzed the impatience behaviour in a queuing model with Bernoulli schedule vacation interruption. Later, Bouchentouf and Yahiaoui  investigated a M/M/1 queuing model with feedback, reneging and retention of reneged customers, multiple working vacations and Bernoulli schedule vacation interruption. Majid and Manoharan  provided the analysis of an infinite-space Markovian multi-server queuing model with single and multiple synchronous working vacations. Afroun et. al  used a Q-matrix method for the study of an unreliable M/M/1/N queuing model with customer's impatience. Recently, Bouchentouf et. al , Bouchentouf and Guendouzi [5, 6], Bouchentouf and Medjahri , and Sampath and Liu  gave some important papers in this area. In this investigation, we consider a finite-buffer Markovian single server queuing system with waiting server, balking and reneging, under differentiated working vacations and Bernoulli schedule vacation interruption at which the server is subject to two types of working vacation, namely: working vacation after the busy period and working vacation taken immediately after the server has just returned from previous working vacation to find that there are no customers in the queue. During working vacation period, the service is supposed to be interrupted under the Bernoulli schedule. The rest of the paper is organized as follows. Section 2 presents the description of the queuing model. In Section 3 we derive the stationary distribution of the system. In Section 4 we deduce important characteristics of the system. In Section 5 we construct a cost model in order to determine the optimal values of service rates, simultaneously, to minimize the total expected cost per unit time by using a quadratic fit search method (QFSM). Numerical demonstrations are given in Section 6. Finally, Section 7 concludes the paper. Description of the model Consider a finite-buffer single server queuing system subject to differentiated working vacations, Bernoulli schedule vacation interruption, waiting server and customers' impatience. Figure 1 shows the state transition diagram of the considered model.