Revisiting the Model of Servicing Machines with Repairable Service Facility-A New Analyzing Idea and Some New Results

In this paper the model of servicing machines with repairable facility is further studied.By standard conditioning decomposition argument,two reliability indices-the probability that the service facility fails at time t and the expected number of failure occurring during（0,t] are discussed.Some important relations of them are given.Furthermore,some new reliability problems are presented and discussed as follows：1） The numbers of the service facility failures during the generalized service time and the generalized busy period;2） The asymptotic expansion of the expected failure number of the service facility during（0,t].A series of new reliability results of the service facility are obtained.

RELIABILITY INDICES OF DISCRETE-TIME Geo^X/G/1 QUEUEING SYSTEM WITH UNRELIABLE SERVICE STATION AND MULTIPLE ADAPTIVE DELAYED VACATIONS

这篇论文从可靠性研究的观点与不可靠的加油站和多重适应推迟的假期认为排队的分离时间的 GeoX/G/1 是模型。后面的问题将被讨论：1 ) 服务者在在时间 n 的一个概括忙时期的概率；2 ) 加油站在在时间 n 的失败的概率，即，加油站的短暂 unavailability，和加油站的稳定的州的 unavailability；3 ) 加油站失败的期望的数字在时间间隔期间(0， n ] ，并且加油站的稳定的州的失败频率；4 ) 加油站故障在的期望的数字一概括的服务者忙碌时期。最后，作者证明有不可靠的加油站的一些普通分离时间的排队模型是在这篇论文讨论的模型的特殊情况。

The Structure of Departure Process and Optimal Control Strategy N^＊ for Geo/G/1 Discrete-Time Queue with Multiple Server Vacations and Min(N, V)-Policy

This paper considers the departure process and the optimal control strategy for a discretetime Geo/G/1 queueing model in which the system operates under the control of multiple server vacations and Min(N, V)-policy. Using the law of total probability decomposition, the renewal theory and the probability generating function technique, the transient and the steady-state probabilities that the server is busy at any epoch n^+ are derived. The authors also obtain the explicit expression of the probability generating function for the expected number of departures occurring in the time interval (0^+, n^+] from any initial state. Meanwhile, the relationship among departure process, server's state process and service renewal process in server busy period is found, which shows the special structure of departure process. Especially, some corresponding results of departure process for special discrete-time queues are directly gained by our results. Furthermore, the approximate expansion for calculating the expected number of departures is presented. In addition, some other important performance measures,including the expected length of server busy period, server's actual vacation period and busy cycle period etc., are analyzed. Finally, some numerical results are provided to determine the optimum value N*for minimizing the system cost under a given cost structure.

Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period

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.

Queue Size Distribution of Geo/G/1 Queue Under the Min(N,D)-Policy

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 the 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.Furthermore,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.

THE RECURSIVE SOLUTION OF QUEUE LENGTH FOR Geo/G/1 QUEUE WITH N-POLICY

This paper considers a discrete-time queue with N-policy and LAS-DA(late arrival system with delayed access) discipline.By using renewal process theory and probability decomposition techniques,the authors derive the recursive expressions of the queue-length distributions at epochs n^-,n^+,and n.Furthermore,the authors obtain the stochastic decomposition of the queue length and the relations between the equilibrium distributions of the queue length at different epochs(n^-,n^+,n and departure epoch D_n).

The Recursive Solution for Geom/G/1(E,SV) Queue with Feedback and Single Server Vacation

Using recursive method, this paper studies the queue size properties at any epoch n＋ in Geom/G/ I（E, SV） queueing model with feedback under LASDA （late arrival system with delayed access） setup. Some new results about the recursive expressions of queue size distribution at different epoch （n＋, n, n-） are obtained. Furthermore the important relations between stationary queue size distribution at different epochs are discovered. The results are different from the relations given in M/G/1 queueing system. The model discussed in this paper can be widely applied in many kinds of communications and computer network.

Spatio-Temporal Deformation of Kink-Breather to the (2+1)-Dimensional Potential Boiti–Leon–Manna–Pempinelli Equation

In the paper, the rational breather soliton and kink solitary wave solution of the (2+1)-dimensional PBLMP equation are obtained by adopting Hirota bilinear method and selecting different test functions. Furthermore, it has been found that the fusion and degeneration of the kink solitary wave occur when interaction between the rational breather soliton and the kink solitary wave happens. These phenomena are very helpful in researching soliton dynamical complexity in the higher dimensional systems.

TRANSIENT QUEUE SIZE DISTRIBUTION SOLUTION OF GEOM/G/1 QUEUE WITH FEEDBACK-A RECURSIVE METHOD

This paper considers the Geom/G/1 queueing model with feedback according to a latearrival system with delayed access (LASDA).Using recursive method,this paper studies the transientproperty of the queue size from the initial state N(0^+)=i.Some new results about the recursiveexpression of the transient queue size distribution at any epoch n^+ and the recursive formulae of theequilibrium distribution are obtained.Furthermore,the recursive formulae of the equilibrium queuesize distribution at epoch n,and n are obtained,too.The important relations between stationaryqueue size distributions at different epochs are discovered (being different from the relations given inM/G/1 queueing system).The model discussed in this paper can be widely applied in all kinds ofcommunications and computer network.

A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS

In this paper,an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods.In order to efficiently solve the generated linear large-scale system,the generalized minimal residual(GMRES)algorithm is applied.For accelerating the convergence rate of the it erative,the St rang-type,Chantype and P-type preconditioners are introduced.The suggested met hod can reach higher order accuracy both in space and in time than the existing met hods.When the used boundary value method is Ak1,K2-stable,it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1.It implies that the iterative solution is convergent rapidly.Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.