This paper proposes a new discrete-time Geo/G/1 queueing model under the control of bi-level randomized(p,N1,N2)-policy.That is,the server is closed down immediately when the system is empty.If N1(≥1)customers are ac...This paper proposes a new discrete-time Geo/G/1 queueing model under the control of bi-level randomized(p,N1,N2)-policy.That is,the server is closed down immediately when the system is empty.If N1(≥1)customers are accumulated in the queue,the server is activated for service with probability p(0≤p≤1)or still left off with probability(1−p).When the number of customers in the system becomes N_(2)(≥N1),the server begins serving the waiting customers until the system becomes empty again.For the model,firstly,we obtain the transient solution of the queue size distribution and the explicit recursive formulas of the stationary queue length distribution by employing the total probability decomposition technique.Then,the expressions of its probability generating function of the steady-state queue size and the expected steady-state queue size are presented.Additionally,numerical examples are conducted to discuss the effect of the system parameters on some performance indices.Furthermore,the steady-state distribution of queue length at epochs n−,n and outside observer’s observation epoch are explored,respectively.Finally,we establish a cost function to investigate the cost optimization problem under the constraint of the average waiting time.And the presented model provides a less expected cost as compared to the traditional N-policy.展开更多
This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject ...This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service.We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution.Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.展开更多
基金Supported by the National Natural Science Foundation of China(71571127)the Opening Fund of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things(2023WZJ02)。
文摘This paper proposes a new discrete-time Geo/G/1 queueing model under the control of bi-level randomized(p,N1,N2)-policy.That is,the server is closed down immediately when the system is empty.If N1(≥1)customers are accumulated in the queue,the server is activated for service with probability p(0≤p≤1)or still left off with probability(1−p).When the number of customers in the system becomes N_(2)(≥N1),the server begins serving the waiting customers until the system becomes empty again.For the model,firstly,we obtain the transient solution of the queue size distribution and the explicit recursive formulas of the stationary queue length distribution by employing the total probability decomposition technique.Then,the expressions of its probability generating function of the steady-state queue size and the expected steady-state queue size are presented.Additionally,numerical examples are conducted to discuss the effect of the system parameters on some performance indices.Furthermore,the steady-state distribution of queue length at epochs n−,n and outside observer’s observation epoch are explored,respectively.Finally,we establish a cost function to investigate the cost optimization problem under the constraint of the average waiting time.And the presented model provides a less expected cost as compared to the traditional N-policy.
基金Supported by the National Natural Science Foundation of China(71571127)the Training Fund Program of Excellent Paper of Sichuan Normal University([2016]4-1)
文摘This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service.We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution.Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.
