فایل ورد کامل حل گذرای صف M/M/c با امتناع و انصراف

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در این مقاله ، صف M/M/c با امتناع و انصراف در نظر گرفته شد. معادلات سیستم در مورد امتناع و انصراف فرمول نویسی گردید. مولفین احتمالات گذرای اندازه صف را نیز با استفاده از تکنیک تابع مولد و خصوصیات تابع بسل، مطرح کردند. 

عنوان انگلیسی:Transient solution of the M/M/c queue with balking and reneging~~en~~

Introduction In real life, many queueing situations arise in which there may be tendency of customers to be discouraged by a long queue. As a result, the customers either decide not join the queue (i.e. balk) or depart after joining the queue without getting served due to impatience (i.e. renege). The importance of this system appears in many real life problems such as the situations involving impatient telephone switchboard customers, the hospital emergency rooms handling critical patients, and the inventory systems that store perishable goods [1]. So queueing systems with balking, reneging, or both were studied by many researchers. Haight [2] first presented the M/M/1 queue with balking. Al-Seedy and Kotb [3] considered the transient solution of a single-server system with balking concept. The M/M/1 queue with customers reneging was also proposed in [4]. The combined effects of balking and reneging in the M/M/1/N queue were investigated in [5,6]. The M/M/1/N Queue with balking, reneging and server vacations was analyzed in [7]. On the other hand, multi-server Markovian queues have been widely studied due to a significant role in day-by-day queueing situation. Reynolds [8] gave the stationary solution of a multi-server model with discouragement. A multi-server queueing model with balking and reneging was proposed by using diffusion approximation method (see [9]). Haghighi et al. [10] derived the steady state probabilities for multi-channel M/M/c queue under consideration of balking and reneging concepts. Hillier and Lieberman discussed various aspects of balking and reneging in [11]. Abou EI-Ata and Hariri [12] analyzed the multiple servers’ queueing system M/M/c/N with balking and reneging. A finite capacity priority queue with discouragement was discussed in [13]. Jain [14] obtained steady state queue size distribution and some other characteristics for M/M/m queue with discouragement and additional servers. The study of M/M/c/N queue with balking, reneging and server breakdowns was presented in [15]. The M/M/c/N queue with balking, reneging and synchronous vacation of partial serves was analyzed in [16]. An additional repairman for machine repair problem with balking, and reneging and spares was incorporated in [17]. Recently, Jain and Singh [18] implemented additional servers for M/M/m queueing model with balking and reneging. In this investigation, the authors will present another technique to compute transient probabilities for the M/M/c queue under consideration of balking and reneging. This technique is a straightforward application of generating functions. The transient probabilities will be expressed in terms of Bessel functions. This is an extension of results obtained for transient probabilities for the multi-server queue given in [19]. This paper is organized as follows. Section 2, gives a description of the queueing model. In Section 3, the equation of system in case balking and reneging is formulated. A simple differential equation is derived and the transient probabilities are obtained by using the properties of Bessel functions in the solution of this differential equation. An equation to evaluate the transient probabilities Pc1(t) is derived in Section 4. Conclusions are given in Section 5.

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