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 considers a discrete-time Geo/G/1 queue under the Min（N, D）-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of ...This paper considers a discrete-time Geo/G/1 queue under the Min（N, D）-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first （Min（N, D）-policy）. By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of tile transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n＋. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution, l^trthermore, the important relations between the steady state queue length distributions at different time epochs n , n and n＋ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.展开更多
This paper considers the discrete-time GeoX/G/1 queueing model with unreliable service station and multiple adaptive delayed vacations from the perspective of reliability research. Following problems will be discussed...This paper considers the discrete-time GeoX/G/1 queueing model with unreliable service station and multiple adaptive delayed vacations from the perspective of reliability research. Following problems will be discussed： 1） The probability that the server is in a ＂generalized busy period＂ at time n; 2） The probability that the service station is in failure at time n, i.e., the transient unavailability of the service station, and the steady state unavailability of the service station; 3） The expected number of service station failures during the time interval （0, hi, and the steady state failure frequency of the service station; 4） The expected number of service station breakdowns in a server＇s ＂generalized busy period＂. Finally, the authors demonstrate that some common discrete-time queueing models with unreliable service station are special cases of the model discussed in this paper.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 under Grant Nos.71171138,71301111,71571127the Scientific Research Innovation&Application Foundation of Headmaster of Hexi University under Grant Nos.XZ2013-06,XZ2013-09
文摘This paper considers a discrete-time Geo/G/1 queue under the Min（N, D）-policy in which the idle server resumes its service if either N customers accumulate in the system or the total backlog of the service times of the waiting customers exceeds D, whichever occurs first （Min（N, D）-policy）. By using renewal process theory and total probability decomposition technique, the authors study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursive expression of the z-transformation of tile transient queue length distribution and the recursive formula for calculating the steady state queue length at arbitrary time epoch n＋. Meanwhile, the authors obtain the explicit expressions of the additional queue length distribution, l^trthermore, the important relations between the steady state queue length distributions at different time epochs n , n and n＋ are also reported. Finally, the authors give numerical examples to illustrate the effect of system parameters on the steady state queue length distribution, and also show from numerical results that the expressions of the steady state queue length distribution is important in the system capacity design.
基金supported in part by the National Natural Science Foundation of China under Grant Nos. 71171138,70871084the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No.200806360001
文摘This paper considers the discrete-time GeoX/G/1 queueing model with unreliable service station and multiple adaptive delayed vacations from the perspective of reliability research. Following problems will be discussed： 1） The probability that the server is in a ＂generalized busy period＂ at time n; 2） The probability that the service station is in failure at time n, i.e., the transient unavailability of the service station, and the steady state unavailability of the service station; 3） The expected number of service station failures during the time interval （0, hi, and the steady state failure frequency of the service station; 4） The expected number of service station breakdowns in a server＇s ＂generalized busy period＂. Finally, the authors demonstrate that some common discrete-time queueing models with unreliable service station are special cases of the model discussed in this paper.
