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1.

Polling system models are extensively used to model a large variety of computer and communication networks as well as production and service systems in which multiple customer classes or a number of distinct items compete for the capacity of a common server or production facility. In this paper we describe an efficient approximation method for the steady state distributions of the queue sizes and waiting times. This method is highly accurate as demonstrated by an extensive numerical study. In addition, it is highly adaptable to a variety of arrival patterns and switching protocols, including exhaustive and gated regimes, simple cyclical systems as well as general polling tables. For a system with

*N*stations, one finds the first*K*probability density function values of the steady state queue size in any given station in*O*(max(*N, K*^{2}) time only. When executed on an IBM system RS/6000, we have observed an average CPU time of less than 1 second for systems with as many as 50 stations over a large variety of parameter settings. 相似文献2.

W. Stadje 《Queueing Systems》1989,4(1):85-92

For a M/M/1 queueing system with group arrivals of random size the transition probabilities of the queue size process and the distribution of the maximal queue size during a time interval [0,

*t*) are calculated. Simple formulae for the corresponding Laplace transforms are given. 相似文献3.

We consider an M/M/m retrial queue and investigate the tail asymptotics for the joint distribution of the queue size and the number of busy servers in the steady state. The stationary queue size distribution with the number of busy servers being fixed is asymptotically given by a geometric function multiplied by a power function. The decay rate of the geometric function is the offered load and independent of the number of busy servers, whereas the exponent of the power function depends on the number of busy servers. Numerical examples are presented to illustrate the result. 相似文献

4.

This paper deals with a single server working vacation queueing model with multiple types of server breakdowns. In a working vacations queueing model, the server works at a different rate instead of being completely idle during the vacation period; the arrival rate varies according to the server’s status. It is assumed that the server is subject to interruption due to multiple types of breakdowns and is sent immediately for repair. Each type of breakdown requires a finite random number of stages of repair. The life time of the server and the repair time of each phase are assumed to be exponentially distributed. We propose a matrix–geometric approach for computing the stationary queue length distribution. Various performance indices namely the expected length of busy period, the expected length of working vacation period, the mean waiting time and average delay, etc. are established. In order to validate the analytical approach, by taking illustration, we compute numerical results. The sensitivity analysis is also performed to explore the effect of different parameters. 相似文献

5.

We consider a discrete time single server queueing system where the arrival process is governed by a discrete autoregressive
process of order

*p*(DAR(*p*)), and the service time of a customer is one slot. For this queueing system, we give an expression for the mean queue size, which yields upper and lower bounds for the mean queue size. Further we propose two approximation methods for the mean queue size. One is based on the matrix analytic method and the other is based on simulation. We show, by illustrations, that the proposed approximations are very accurate and computationally efficient. 相似文献6.

K. J. E. Carpio 《Queueing Systems》2007,55(2):123-130

The stationary processes of waiting times {If this assumption does not hold but the sequence of serial correlation coefficients {ρ

相似文献*W*_{ n }}_{ n = 1,2,…}in a*GI*/*G*/1 queue and queue sizes at successive departure epochs {*Q*_{n}}_{n = 1,2,…}in an*M*/*G*/1 queue are long-range dependent when 3 < κ_{ S }< 4, where κ_{ S }is the moment index of the independent identically distributed (i.i.d.) sequence of service times. When the tail of the service time is regularly varying at infinity the stationary long-range dependent process {*W*_{ n }} has Hurst index ½(5−κ_{ S }), i.e.${\rm sup} \left\{h : {\rm lim sup}_{n\to\infty}\, \frac{{\rm var}(W_1+\cdots+W_n)}{n^{2h}} = \infty \right\} = \frac{5-\kappa_S} {2}\,.$

_{ n }} of the stationary process {*W*_{ n }} behaves asymptotically as*cn*^{−α}for some finite positive*c*and α ∊ (0,1), where α = κ_{ S }− 3, then {*W*_{ n }} has Hurst index ½(5−κ_{ S }). If this condition also holds for the sequence of serial correlation coefficients {*r*_{ n }} of the stationary process {*Q*_{ n }} then it also has Hurst index ½(5κ_{ S })7.

《随机分析与应用》2013,31(3):739-753

**Abstract**We consider an

*M*

^{ x }/

*G*/1 queueing system with a random setup time, where the service of the first unit at the commencement of each busy period is preceded by a random setup time, on completion of which service starts. For this model, the queue size distributions at a random point of time as well as at a departure epoch and some important performance measures are known [see Choudhury, G. An

*M*

^{ x }/

*G*/1 queueing system with setup period and a vacation period. Queueing Sys.

**2000**,

*36*, 23–38]. In this paper, we derive the busy period distribution and the distribution of unfinished work at a random point of time. Further, we obtain the queue size distribution at a departure epoch as a simple alternative approach to Choudhury4. Finally, we present a transform free method to obtain the mean waiting time of this model. 相似文献

8.

9.

We study a single server queue with batch arrivals and general (arbitrary) service time distribution. The server provides service to customers, one by one, on a first come, first served basis. Just after completion of his service, a customer may leave the system or may opt to repeat his service, in which case this customer rejoins the queue. Further, just after completion of a customer's service the server may take a vacation of random length or may opt to continue staying in the system to serve the next customer. We obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and some known results have been derived. A numerical illustration is provided. 相似文献

10.

In this paper, we study (

*N*,*L*) switch-over policy for machine repair model with warm standbys and two repairmen. The repairman (*R*_{1}) turns on for repair only when*N*-failed units are accumulated and starts repair after a set up time which is assumed to be exponentially distributed. As soon as the system becomes empty, the repairman (*R*_{1}) leaves for a vacation and returns back when he finds the number of failed units in the system greater than or equal to a threshold value*N*. Second repairman (*R*_{2}) turns on when there are*L*(>*N*) failed units in the system and goes for a vacation if there are less than*L*failed units. The life time and repair time of failed units are assumed to be exponentially distributed. The steady state queue size distribution is obtained by using recursive method. Expressions for the average number of failed units in the queue and the average waiting time are established. 相似文献