Discrete-time Geo/G/1 retrial queues with general retrial time and Bernoulli vacation

This paper considers a discrete-time Geo/G/1 retrial queue where the retrial time has a general distribution and the server is subject to Bernoulli vacation policy.It is assumed that the server, after each service completion,begins a process of search in order to find the following customer to be served with a certain probability,or begins a single vacation process with complementary probability. This paper analyzes the Markov chain underlying the queueing system and obtain its ergodicity condition.The generating functions of the number of customers in the orbit and in the system are also obtained along with the marginal distributions of the orbit size when the server is idle,busy or on vacation.Finally,the author gives two stochastic decomposition laws,and as an application the author gives bounds for the proximity between the system size distributions of the model and the corresponding model without retrials.     

A single server retrial queue with general retrial times and two-phase service

An M / G / 1 retrial queue with a first-come-first-served (FCFS) orbit, general retrial time, two-phase service and server breakdown is investigated in this paper. Customers are allowed to balk and renege at particular times. Assume that the customers who find the server busy are queued in the orbit in accordance with an FCFS discipline. All customers demand the first “essential” service, whereas only some of them demand the second “optional” service, and the second service is multioptional. During the service, the server is subject to breakdown and repair. Assume that the retrial time, the service time, and the repair time of the server are all arbitrarily distributed. By using the supplementary variables method, the authors obtain the steady-state solutions for both queueing and reliability measures of interest. This research is supported by the National Natural Science Foundation of China under Grant No. 10871020.     

Analysis of a discrete-time GI/Geo/1/N queue with multiple working vacations

This paper analyzes a finite-buffer renewal input single server discrete-time queueing system with multiple working vacations. The server works at a different rate rather than completely stopping working during the multiple working vacations. The service times during a service period, service time during a vacation period and vacation times are geometrically distributed. The queue is analyzed using the supplementary variable and the imbedded Markov-chain techniques. We obtain steady-state system length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The analysis of actual waiting-time distribution and some performance measures are carried out. We present some numerical results and discuss special cases of the model.     

Optimal control of an M/G/1 retrial queue with vacations

In this note, we consider an M/G/1 retrial queue with server vacations, when retrial times, service times and vacation times are arbitrary distributed. The distribution of the number of customers in the system in stationary regime is obtained in terms of generating function. Next, we give heavy traffic approximation of such distribution. We show that the system size can be decomposed into two random variables, one of which corresponds to the system size of the ordinary M/G/1 FIFO queue without vacation. Such a stochastic decomposition property is useful for the computation of performance measures of interest. Finally, we solve simple problems of optimal control of vacation and retrial policies.     

第二类故障期间以概率p进入的M/G/1可修排队

假设服务台在忙期和闲期内都可能发生故障,且具有不同的故障率,并且在闲期的故障状态期间到达顾客以概率p进入系统.使用全概率分解技术和利用拉普拉斯变换工具,研究了服务台的瞬态不可用度、稳态不可用度、(0,t]时间内的平均故障次数和稳态故障频度,获得服务台一些重要的可靠性结果,并且分别讨论了当p取值为0和取值为1时的特殊情况.     

延迟多重休假MX/G/1排队系统的队长分布

考虑延迟多重休假的M^x／G／1排队，在假定延迟时间、休假时间和服务时间都是一般概率分布函数下，研究了队长的瞬态和稳态性质、通过引进“服务员忙期”，导出了在任意时刻t瞬态队长分布的L变换的递推表达式和稳态队长分布的递推表达式，以及平稳队长的随机分解．     

混合延迟消失制Geo1←+Geo2/Geo1,Geo2/s/s+K排队系统

研究了具有两类顾客的带优先权的Geo1←+Geo2/Geo1,Geo2/s/s+K排队系统,第一类顾客具有延迟损失性,第二类顾客具有损失性,并且第二类顾客对第一类顾客具有抢占优先权.使用矩阵几何解的方法,得到了两类顾客的平均队长、损失率和系统利用率等性能指标,最后通过Matlab软件给出了一些具体数值例子.     

A batch arrival retrial queue with starting failures,feedback and admission control

This paper is concerned with the analysis of a feedback M^[X]/G/1 retrial queue with starting failures and general retrial times. In a batch, each individual customer is subject to a control admission policy upon arrival. If the server is idle, one of the customers admitted to the system may start its service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn-on the server, which takes negligible time. If the server is started successfully (with a certain probability), the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. The Markov chain underlying the considered queueing system is studied and the necessary and sufficient condition for the system to be stable is presented. Explicit formulae for the stationary distribution and some performance measures of the system in steady-state are obtained. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.     

重试,反馈M/M/s/k排队的呼叫中心性能分析

由于CTI(计算机电话集成)技术的发展,使呼叫中心得到广泛的应用.与呼叫中心实现技术的发展相比,对呼叫中心管理的研究显得有些滞后,而针对呼叫中心排队模型的研究,更是如此.针对呼叫中心服务系统中的重试和反馈问题,考虑一种带重试和反馈的M/M/s/k排队模型.将等待位置和服务台数推广到有限个.在模型求解过程中,尝试采用矩阵迭代的新方法,使求解过程简单明了.然后,采用逼近的方法给出模型的数值解,并得出反馈对系统的影响随系统负荷的增大而快速增大等结论.     

On the single server retrial queue with priority subscribers and server breakdowns

The author concerned the reliability evaluation as well as queueing analysis of M1, M2/G1, G2/1 retrial queues with two different types of primary customers arriving according to independent Poisson flows. In the case of blocking, the first type of customers can be queued whereas the second type of customers must leave the service area but return after some random period of time to try their luck again. The author assumes that the server is unreliable and it has a service-type dependent, exponentially distributed life time as well as a service-type dependent, generally distributed repair time. The necessary and sufficient condition for the system to be stable is investigated. Using a supplementary variable method, the author obtains a steady-state solution for queueing measures, and the transient as well as the steady-state solutions for reliability measures of interest.     

Queue size distribution of Geo/G/1 queue under the Min(<Emphasis Type="Italic">N,D</Emphasis>)-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.     

Reliability indices of discrete-time Geox/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations

This paper considers the discrete-time Geo~x/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,n],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.     

Recursive Solution of Queue Length Distribution for Geo/G/1 Queue with Delayed Min(N,D)-Policy

In this paper we consider a discrete-time Geo/G/1 queue with delayed Min(N, D)-policy.Using renewal process theory, total probability decomposition technique and z-transform, we study the transient and equilibrium properties of the queue length from an arbitrary initial state, and obtain both the recursive expressions of the transient state queue length distribution and the steady state queue length distribution at arbitrary time epoch n~+. Furthermore, we derive the important relations between equilibrium queue length distributions at different time epochs n~-, n and n~+. Finally, we give some numerical examples about capacity decision in queueing systems to demonstrate the application of the analytical results reported in this paper.     

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 M/PH/1 queue with working vacations and vacation interruption

We study an M/PH/1 queue with phase type working vacation and vacation interruption where the vacation time follows a phase type distribution. The server serves the customers at a lower rate in a vacation period. The server comes back to the regular busy period at a service completion without completing the vacation. Such policy is called vacation interruption. In terms of quasi birth and death process and matrix-geometric solution method, we obtain the stationary queue length distribution. Moreover we obtain the conditional stochastic decomposition structures of queue length and waiting time when the service time distribution in the regular busy period is exponential.     

The M/M/1 queue with working vacations and vacation interruptions

In this paper, we study the M/M/1 queue with working vacations and vacation interruptions. The working vacation is introduced recently, during which the server can still provide service on the original ongoing work at a lower rate. Meanwhile, we introduce a new policy：, the server can come back from the vacation to the normal working level once some indices of the system, such as the number of customers, achieve a certain value in the vacation period. The server may come back from the vacation without completing the vacation. Such policy is called vacation interruption. We connect the above mentioned two policies and assume that if there are customers in the system after a service completion during the vacation period, the server will come back to the normal working level. In terms of the quasi birth and death process and matrix-geometric solution method, we obtain the distributions and the stochastic decomposition structures for the number of customers and the waiting time and provide some indices of systems.     

Analysis of a continuous time SM[K]/PH[K]/1/FCFS queue: Age process, sojourn times, and queue lengths

This paper studies a continuous time queueing system with multiple types of customers and a first-come-first-served service discipline.Customers arrive according to a semi-Markov arrival process and the service times of individual types of customers have PH-distributions.A GI/M/1 type Markov process for a generalized age process of batches of customers is constructed.The stationary distribution of the GI/M/1 type Markov process is found explicitly and,consequently,the distributions of the age of the batch in service,the total workload in the system,waiting times,and sojourn times of different batches and different types of customers are obtained.The paper gives the matrix representations of the PH-distributions of waiting times and sojourn times.Some results are obtained for the distributions of queue lengths at departure epochs and at an arbitrary time.These results can be used to analyze not only the queue length,but also the composition of the queue.Computational methods are developed for calculating steady state distributions related to the queue lengths,sojourn times,and waiting times.     

The recursive solution of queue length for <Emphasis Type=Italic>Geo/G</Emphasis>/1 queue with <Emphasis Type=Italic>N</Emphasis>-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).     

Analysis of batch arrival queue with randomized vacation policy and an un-reliable server

This paper examines an M^[x]G/1 queueing system with an unreliable server and a delayed repair,in which the server operates a randomized vacation policy with multiple vacations.Whenever the system is empty,the server immediately takes a vacation.If there is at least one customer found waiting in the queue upon returning from a vacation,the server will be immediately activated for service.Otherwise,if no customers are waiting for service at the end of a vacation,the server either remains idle with probability p or leaves for another vacation with probability 1-p.Whenever one or more customers arrive when the server is idle,the server immediately starts providing service for the arrivals.The server may also meet an unpredictable breakdown and the repair may be delayed.For such a system the authors derive the distributions of some important system characteristics,such as the system size distribution at a random epoch and at a departure epoch,the system size distribution at the busy period initiation epoch,and the distribution of the idle period and the busy period.The authors perform a numerical analysis for changes in the system characteristics,along with changes in specific values of the system parameters.A cost effectiveness maximization model is constructed to explain the benefits of such a queueing system.     

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.