Analysis of cyclic queueing networks with parallelism and vacation

The aim of this paper is to improve the machine interference model with vacation to deal with more recent problems of the communication area. To this scope the model is extended to include parallelism in the vacation station. The underlying Markov process is analyzed and a state arrangement is found that yields an efficient matrix-analytic technique that substantially lowers down the time- and space-complexity of standard methods. A numerical example of the method effectiveness is presented, and an example of resource allocation is introduced that finds applications in the QoS management of wireless networks. The author is thankful to the anonymous referees for the improvements their comments have earned to the quality of the presentation and to the completeness of the paper. The author is thankful to Giuseppe Iazeolla, whose careful reading of the original draft of this paper led to significant improvements in its overall quality. This work was partially supported by funds from the FIRB project “Performance Evaluation of Complex Systems: Techniques Methodologies and Tools” and by the University of Roma TorVergata project on High Performance ICT Platforms.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

Comments on “multi-server system with single working vacation”

Lin and Ke consider the M/M/R queue with working vacation [Chuen-Horng Lin, Jau-Chuan Ke, Multi-server system with single working vacation, Appl. Math. Modell. (2008), doi:10.1016/j.apm.2008.10.006], and derive a computable explicit form for rate matrix R$\mathfrak{R}$ of the geometric approach and the stationary probabilities of the queue. However, it contains some errors concerning the terminology, notations and the final form of rate matrix R$\mathfrak{R}$. This note shows that the classical Quasi-birth-death (QBD) formulation of the M/M/R queue with working vacation naturally leads to the infinitesimal generator matrix of the QBD process and the probability interpretation of matrices involving in the infinitesimal generator matrix.     

Numerical investigation of finite-source multiserver systems with different vacation policies

Systems with vacations are usually modeled and analyzed by queueing theory, and almost all works assume that the customer source is infinite and the arrival process is Poisson. This paper aims to present an approach for modeling and analyzing finite-source multiserver systems with single and multiple vacations of servers or all stations, using the Generalized Stochastic Petri nets model. We show how this high level formalism, allows a simple construction of detailed and compact models for such systems and to obtain easily the underlying Markov chains. However, for real vacation systems, the models may have a huge state space. To overcome this problem, we give the algorithms for automatically computing the infinitesimal generator, for the different vacation policies. In addition, we develop the formulas of the main exact stationary performance indices. Through numerical examples, we discuss the effect of server number, vacation rate and vacation policy on the system’s performances.     

Performance analysis of MAP/G/1 queue with working vacations and vacation interruption

We consider the MAP/G/1 queue with working vacations and vacation interruption. We obtain the queue length distribution with the method of supplementary variable, combined with the matrix-analytic method and censoring technique. We also obtain the system size distribution at pre-arrival epoch and the Laplace–Stieltjes transform (LST) of waiting time.     

Performance analysis of GI/M/1 queue with working vacations and vacation interruption

In this paper, we analyze a single-server vacation queue with a general arrival process. Two policies, working vacation and vacation interruption, are connected to model some practical problems. The GI/M/1 queue with such two policies is described and by the matrix analysis method, we obtain various performance measures such as mean queue length and waiting time. Finally, using some numerical examples, we present the parameter effect on the performance measures and establish the cost and profit functions to analyze the optimal service rate η during the vacation period.     

An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule

This paper treats an M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Whenever the system becomes empty at a service completion instant, the server goes for a single working vacation. In the working vacation, a customer is served at a lower speed, and if there are customers in the queue at the instant of a service completion, the server is resumed to a regular busy period with probability p   (i.e., the vacation is interrupted) or continues the vacation with probability 1-p$1-p$. Using the matrix analytic method, we obtain the distribution for the stationary queue length at departure epochs. The joint distribution for the stationary queue length and service status at the arbitrary epoch is also obtained by using supplementary variable technique. We also develop a variety of stationary performance measures for this system and give a conditional stochastic decomposition result. Finally, several numerical examples are presented.     

Analysis of MAP/PH1,PH2/1 queue with vacations and optional secondary services

In this paper we study a queueing model in which the customers arrive according to a Markovian arrival process (MAP). There is a single server who offers services on a first-come-first-served basis. With a certain probability a customer may require an optional secondary service. The secondary service is provided by the same server either immediately (if no one is waiting to receive service in the first stage) or waits until the number waiting for such services hits a pre-determined threshold. The model is studied as a QBD-process using matrix-analytic methods and some illustrative examples are discussed.     

一类新颖的冷贮备可修系统的维修模型(英文)

本文研究由两不同型部件组成的冷贮备可修系统的最优化更换策略问题.在该系统中,修理工可多重休假且修理设备具有优先权.通过引入一类推广的泊松过程来模拟系统的运作状态,并结合新定义的更换周期,从理论上找出该系统经长期运行单位时间的期望损失的具体表达式.最后,通过事例,从直观上验证文中所涉及到的一些理论结果.     

BMAP/G/1/N queue with vacations and limited service discipline

We consider a finite buffer single server queue with batch Markovian arrival process (BMAP), where server serves a limited number of customer before going for vacation(s). Single as well as multiple vacation policies are analyzed along with two possible rejection strategies: partial batch rejection and total batch rejection. We obtain queue length distributions at various epochs and some important performance measures. The Laplace–Stieltjes transforms of the actual waiting time of the first customer and an arbitrary customer in an accepted batch have also been obtained.