AN M/G/1 RETRIAL QUEUE WITH SECOND MULTI-OPTIONAL SERVICE,FEEDBACK AND UNRELIABLE SERVER

An M/G/1 retrial queue with two-phase service and feedback is studied in this paper, where the server is subject to starting failures and breakdowns during service. Primary customers get in the system according to a Poisson process, and they will receive service immediately if the server is available upon arrival. Otherwise, they will enter a retrial orbit and are queued in the orbit in accordance with a first-come-first-served （FCFS） discipline. Customers are allowed to balk and renege at particular times. All customers demand the first “essential” service, whereas only some of them demand the second “multi-optional” service. It is assumed that the retrial time, service time and repair time of the server are all arbitrarily distributed. The necessary and sufficient condition for the system stability is derived. Using a supplementary variable method, the steady-state solutions for some queueing and reliability measures of the system are obtained.

Availability and cost-benefit evaluation for a repairable retrial system with warm standbys and priority

This paper investigates a warm standby repairable retrial system with two types of components and a single reparman,where type 1 components have priority over type 2 in use.Failure and repair times for each type of component are assumed to be exponential distributions.The retrial feature is considered and the retrial time of each failed component is exponentially distributed.By using Markov process theory and matrix analytic method,the system steady-state probabil-ities are derived,and the system steady-state availability and some steady-state performance indices are obtained.Using the Bayesian approach,the system parameters can be estimated.The cost-benefit ratio function of the system is constructed based on the failed components and repairman's states.Numerical experiments are given to evaluate the effect of each parameter on the system steady-state availability and optimize the system cost-benefit ratio with repair rate as a decision variable.

AN M/M/C RETRIAL QUEUEING SYSTEM WITH BERNOULLI VACATIONS

In this paper, a steady-state Markovian multi-server retrial queueing system with Bernoulli vacation scheduling service is studied. Using matrix-geometric approach, various interesting and important system performance measures are obtained. Further, the probability descriptors like ideal retrial and vain retrial are provided. Finally, extensive numerical illustrations are presented to indicate the quantifying nature of the approach to obtain solutions to this queueing system.

A Batch Arrival Retrial Queue with Two Phases of Service and Bernoulli Vacation Schedule

We consider an MX/G/1 queueing system with two phases of heterogeneous service and Bernoulli vacation schedule which operate under a linear retrial policy. In addition, each individual customer is subject to a control admission policy upon the arrival. This model generalizes both the classical M/G/1 retrial queue with arrivals in batches and a two phase batch arrival queue with a single vacation under Bernoulli vacation schedule. We will carry out an extensive stationary analysis of the system, including existence of the stationary regime, embedded Markov chain, steady state distribution of the server state and number of customer in the retrial group, stochastic decomposition and cMculation of the first moment.

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.

A RepairableGeoX/G/1 Retrial Queue with Bernoulli Feedback and Impatient Customers

Abstract This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures.Diferent from standard batch arrival retrial queues with starting failures,we assume that each customer after service either immediately returns to the orbit for another service with probabilityθor leaves the system forever with probability 1θ（0≤θ〈1）.On the other hand,if the server is started unsuccessfully by a customer（external or repeated）,the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 q（0≤q〈1）.Firstly,we introduce an embedded Markov chain and obtain the necessary and sufcient condition for ergodicity of this embedded Markov chain.Secondly,we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time.We also derive a stochastic decomposition law.In the special case of individual arrivals,we develop recursive formulae for calculating the steady-state distribution of the orbit size.Besides,we investigate the relation between our discrete-time system and its continuous counterpart.Finally,some numerical examples show the influence of the parameters on the mean orbit size.

A Single-server Discrete-time Retrial G-queue with Server Breakdowns and Repairs

This paper concerns a discrete-time Geo/Geo/1 retrial queue with both positive and negative customers where the server is subject to breakdowns and repairs due to negative arrivals. The arrival of a negative customer causes one positive customer to be killed if any is present, and simultaneously breaks the server down. The server is sent to repair immediately and after repair it is as good as new. The negative customer also causes the server breakdown if the server is found idle, but has no effect on the system if the server is under repair. We analyze the Markov chain underlying the queueing system and obtain its ergodicity condition. The generating function 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 down. Finally, we present some numerical examples to illustrate the influence of the parameters on several performance characteristics of the system.

The MAP/PH/N Retrial Queue in a Random Environment

We consider the MAP/PH/N retrial queue with a finite number of sources operating in a finite state Markovian random environment. Two different types of multi-dimensional Markov chains are investigated describing the behavior of the system based on state space arrangements. The special features of the two formulations are discussed. The algorithms for calculating the stationary state probabilities are elaborated, based on which the main performance measures are obtained, and numerical examples are presented as well.

An M^([X])/G/1 Retrial G-queue with Single Vacation Subject to the Server Breakdown and Repair

An M[X]/G/1 retrial G-queue with single vacation and unreliable server is investigated in this paper. Arrivals of positive customers form a compound Poisson process, and positive customers receive service immediately if the server is free upon their arrivals; Otherwise, they may enter a retrial orbit and try their luck after a random time interval. The arrivals of negative customers form a Poisson process. Negative customers not only remove the customer being in service, but also make the server under repair. The server leaves for a single vacation as soon as the system empties. In this paper, we analyze the ergodical condition of this model. By applying the supplementary variables method, we obtain the steady-state solutions for both queueing measures and reliability quantities.

Discrete-time GGI/GG/1 Retrial Queues with Time-controlled Vacation Policies

A discrete-time GI/G/1 retrial queue with Bernoulli retrials and time-controlled vacation policies is investigated in this paper. By representing the inter-arrival, service and vacation tlmes using a Markov-based approach, we are able to analyze this model as a level-dependent quasi-birth-and-death （LDQBD） process which makes the model algorithmically tractable. Several performance measures such as the stationary probability distribution and the expected number of customers in the orbit have been discussed with two different policies： deterministic time-controlled system and random time-controlled system. To give a comparison with the known vacation policy in the literature, we present the exhaustive vacation policy as a contrast between these policies under the early arrival system （EAS） and the late arrival system with delayed access （LAS-DA）. Significant difference between EAS and LAS-DA is illustrated by some numerical examples.

Performance of the(BMAP_1, BMAP_2 )/(PH_1, PH_2 )/N Retrial Queueing System with Finite Buffer

This paper consider the （BMAP1, BMAP2）/（PH1, PH2）/N retrial queue with finite-position buffer. The behavior of the system is described in terms of continuous time multi-dimensional Markov chain. Arriving type I calls find all servers busy and join the buffer, if the positions of the buffer are insufficient, they can go to orbit. Arriving type II calls find all servers busy and join the orbit directly. Each server can provide two types heterogeneous services with Phase-type （PH） time distribution to every arriving call （including types I and II calls）, arriving calls have an option to choose either type of services. The model is quite general enough to cover most of the systems in communication networks. We derive the ergodicity condition, the stationary distribution and the main performance characteristics of the system. The effects of various parameters on the system performance measures are illustrated numerically.

Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson

In this paper, we consider an MMPP/M/1/1 retrial queue where incoming fresh calls arrive at the server according to a Markov modulated Poisson process (MMPP). Upon arrival, an incoming call either occupies the server if it is idle or joins a virtual waiting room called orbit if the server is busy. From the orbit, incoming calls retry to occupy the server in an exponentially distributed time and behave the same as a fresh incoming call. After an exponentially distributed idle time, the server makes an outgoing call whose duration is also exponentially distributed but with a different parameter from that of incoming calls. Our contribution is to derive the first order (law of large numbers) and the second order (central lim让 theorem) asymptotics for the distribution of the number of calls in the orb计 under the condition that the retrial rate is extremely low. The asymptotic results are used to obtain the Gaussian approximation for the distribution of the number of calls in the orbit. Our result generalizes earlier results where Poisson input was assumed.

N-policy for M^(x)/G/1 Unreliable Retrial G-Queue with Preemptive Resume and Multi-services

Bulk arrival retrial G-queue with impatient customers and multi-servicessubject to server breakdowns has been analyzed. The system allows the arrival oftwo types of customers: positive customers and negative customers in the system.The negative customers make the server fail if they find the server in busy state,whereas positive customers are served if the server is idle otherwise they join thevirtual pool of customers called orbit. The customers from the retrial orbit try theirchance again for the service. The customers have the option of obtaining more than oneservice. Moreover, the customers are impatient and may renege from the system withprobability (1−r ). The server is sent for repair as soon as it breakdowns;after repair,the service process starts again. Also, the server has the provision to initiate the servicewhen there areN customers accumulated in the system. Using supplementary variablestechnique and generating functions, various performance measures like reliabilityindices and long run probabilities have been obtained.

An Explicit Solution for a Series and Parallel Queue with Retrial,Losses,and Bernoulli Schedule

This paper deals with the series and parallel queueing system in which there are two servers whose service time follow two exponential distributions.Each arriving customer either enters into the tandem service with probability or joins the service of the single server with complementary probability.We assume that the customers of arriving at the first server who find the first server is busy join an orbit and retry to enter the server after some time and of arriving at the second server who find the second server is busy are lost.For this model,we obtain the explicit expressions of the joint stationary distribution between the number of customers in the orbit and the states of the servers.

DISCRETE-TIME Geo/G/1 RETRIAL QUEUES WITH GENERAL RETRIAL TIME AND BERNOULLI VACATION

这份报纸认为再审时间有一般分布和服务者的一个分离时间的 Geo/G/1 再审队列服从于 Bernoulli 假期政策。在每服务结束以后，服务者开始搜索的一个过程以便发现下列顾客以某个概率被供给，这被假定，或与互补概率开始一个单个假期过程。这份报纸分析位于排队系统下面的 Markov 链并且获得它的各态历经条件。当服务器是闲散的时，在轨道并且在系统的顾客的数字的产生函数也与轨道尺寸的边缘的分布一起被获得，忙或上假期。最后，作者给二随机的分解法律，并且没有再审，作为应用，作者为在模型和相应模型的系统尺寸分布之间的最近给界限。

A DISCRETE-TIME Geo/G/1 RETRIAL QUEUE WITH J-VACATION POLICY AND GENERAL RETRIAL TIMES

The authors discuss a discrete-time Geo/G/1 retrial queue with J-vacation policy and general retrial times.As soon as the orbit is empty,the server takes a vacation.However,the server is allowed to take a maximum number J of vacations,if the system remains empty after the end of a vacation.If there is at least one customer in the orbit at the end of a vacation,the server begins to serve the new arrivals or the arriving customers from the orbit.For this model,the authors focus on the steady-state analysis for the considered queueing system.Firstly,the authors obtain the generating functions of the number of customers in the orbit and in the system.Then,the authors obtain the closed-form expressions of some performance measures of the system and also give a stochastic decomposition result for the system size.Besides,the relationship between this discrete-time model and the corresponding continuous-time model is also investigated.Finally,some numerical results are provided.

ON THE SINGLE SERVER RETRIAL QUEUE WITH PRIORITY SUBSCRIBERS AND SERVER BREAKDOWNS

作者与根据独立泊松流动到达的主要顾客的二种不同类型使可靠性评估以及再审队列的排队分析担心了。在堵住的情况中，而顾客的第二种类型必须离开服务区，但是在时间的某随机的时期以后回来再碰碰他们的运气，顾客的第一种类型能被排队。作者假设服务者是不可靠的，它有一个服务类型依赖者，指数地分布式的生活时间以及一个服务类型依赖者，通常分布式的修理时间。为稳定的系统的必要、足够的条件被调查。用一个增补可变方法，作者为排队措施获得一个不变的解决方案，并且象兴趣的可靠性措施的不变的解决方案一样短暂。

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 balkand renege at particular times.Assume that the customers who find the server busy are queued inthe 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 multi-optional.During the service,the server is subject to breakdown and repair.Assume that the retrialtime,the service time,and the repair time of the server are all arbitrarily distributed.By using thesupplementary variables method,the authors obtain the steady-state solutions for both queueing andreliability measures of interest.