Optimal Control Policy of M/G/1 Queueing System with Delayed Randomized Multiple Vacations Under the Modified Min(N,D)-Policy Control

Based on the number of customers and the server’s workload,this paper proposes a modified Min(N,D)-policy and discusses an M/G/1 queueing model with delayed randomized multiple vacations under such a policy.Applying the well-known stochastic decomposition property of the steady-state queue size,the probability generating function of the steady-state queue length distribution is obtained.Moreover,the explicit expressions of the expected queue length and the additional queue length distribution are derived by some algebraic manipulations.Finally,employing the renewal reward theorem,the explicit expression of the long-run expected cost per unit time is given.Furthermore,we analyze the optimal policy for economizing the expected cost and compare the optimal Min(N,D)-policy with the optimal N-policy and the optimal D-policy by using numerical examples.

Transient Analysis of a Two-Heterogeneous Severs Queue with Impatient Behaviour and Multiple Vacations

We consider an M/M/2 queueing system with two-heterogeneous servers and multiple vacations. Customers arrive according to a Poisson process. However, customers become impatient when the system is on vacation. We obtain explicit expressions for the time dependent probabilities,mean and variance of the system size at time t by employing probability generating functions, continued fractions and properties of the modified Bessel functions. Finally, two special cases are provided.

On a BMAP/G/1 G-queue with Setup Times and Multiple Vacations

In this paper, we consider a BMAP/G/1 G-queue with setup times and multiple vacations. Arrivals of positive customers and negative customers follow a batch Markovian arrival process （BMAP） and Markovian arrival process （MAP） respectively. The arrival of a negative customer removes all the customers in the system when the server is working. The server leaves for a vacation as soon as the system empties and is allowed to take repeated （multiple） vacations. By using the supplementary variables method and the censoring technique, we obtain the queue length distributions. We also obtain the mean of the busy period based on the renewal theory.

Product Form Solution of a Queuing-Inventory System with Lost Sales and Server Vacation

In this study,the authors consider an M/M/1 queuing system with attached inventory under an(s,S)control policy.The server takes multiple vacations whenever the inventory is depleted.It is assumed that the lead time and the vacation time follow exponential distributions.The authors formulate the model as a quasi-birth-and-dearth(QBD)process and derive the stability condition of the system.Then,the stationary distribution in product form for the joint process of the queue length,the inventory level,and the server’s status is obtained.Furthermore,the conditional distributions of the inventory level when the server is on and operational,and when it is off due to a vacation,are derived.Using the stationary distribution,the authors obtain some performance measures of the system.The authors investigate analytically the effect of the server’s vacation on the performance measures.Finally,several numerical examples are presented to investigate the effects of some parameters on the performance measures,the optimal policy,and the optimal cost.

PERFORMANCE ANALYSIS OF POWER SAVING CLASS OF TYPE Ⅲ IN IEEE 802.16E WITH USER INITIATED TRAFFIC

A novel method for analysing the performance of power saving class of Type Ⅲ in IEEE 802.16e is proposed,which is applicable to design,maintenance and management for mobile wireless metropolitan area network.Considering the memoryless nature of user initiated packet arrival,a Geom/G/1 queue model with multiple vacations and setup period is built to capture the principle for the power saving class of Type Ⅲ.By using an embedded Markov chain method and the boundary state variable theory,we obtain the queueing measures such as queueing length,waiting time and busy cycle in steady state.Correspondingly,we derive explicitly the performance measures for the power saving class of Type Ⅲ in terms of handover ratio,energy saving ratio,and average packet response time.Based on numerical results,we develop a cost function to determine numerically the optimal length of sleep window and the minimal cost with different offered loads.

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.

Spectrum of the Operator Corresponding to the M/M/1 Queueing Model with Vacations and Multiple Phases of Operation and Application

We describe the point spectrum of the operator which corresponds to the M/M/1 queueing model with vacations and multiple phases of operation.Then by using this result we prove that the essential growth bound of the C0-semigroup generated by the operator is 0,the C0-semigroup is not compact,not eventually compact,even not quasi-compact.Moreover,we verify that it is impossible that the time-dependent solution of the M/M/1 queueing model with vacations and multiple phases of operation exponentially converges to its steady-state solution.In addition,we obtain the spectral radius and essential spectral radius of the C0-semigroup.Lastly,we discuss other spectrum of the operator and obtain a set which belongs to the union of its continuous spectrum and residual spectrum.

THE BULK INPUT M~[X]/M/1 QUEUE WITH WORKING VACATIONS

In this paper, we analyze a bulk input M^[X]/M/1 queue with multiple working vacations. A quasi upper triangle transition probability matrix of two-dimensional Markov chain in this model is obtained, and with the matrix analysis method, highly complicated probability generating function（PGF） of the stationary queue length is firstly derived, from which we got the stochastic decomposition result for the stationary queue length which indicates the evident relationship with that of the classical M^[X]/M/1 queue without vacation. It is important that we find the upper and the lower bounds of the stationary waiting time in the Laplace transform order using the properties of the conditional Erlang distribution. Furthermore, we gain the mean queue length and the upper and the lower bounds of the mean waiting time.

Transient Behavior of a Single-Server Markovian Queue with Balking and Working Vacation Interruptions

This paper studies the time-dependent analysis of an M/M/1 queueing model with single,multiple working vacation,balking and vacation interruptions.Whenever the system becomes empty,the server commences working vacation.During the working vacation period,if the queue length reaches a positive threshold value‘k’,the working vacation of the server is interrupted and it immediately starts the service in an exhaustive manner.During working vacations,the customers become discouraged due to the slow service and possess balking behavior.The transient system size probabilities of the proposed model are derived explicitly using the method of generating function and continued fraction.The performance indices such as average and variance of system size are also obtained.Further,numerical simulations are presented to analyze the impact of system parameters.