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.

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.

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.

The Performance Analysis Of A M/M/2/2＋ 1 Retrial Queue With Unreliable Server

The M/M/r/r＋d retrial queuing system with unreliable server is considered. The customers arrive according to a Poisson process and the service time distribution is negative exponential. The life time of the server and repair times are also negative exponential. If the system is full at the time of arrival of a customer, the customer enters into an orbit. From the orbit the customer tries his luck. The time between two successive retrial follows negative exponential distribution. The model is analyzed using Matrix Geometric Method. The joint distribution of system size and orbit size in steady state is studied. Some system performance measures are obtained. We also provide numerical examples by taking particular values to the parameters.

A Spectrum Occupancy Model for Primary Users in Cognitive Radio Systems

There are abundant research results related to cognitive radio systems (CR systems), but using queueing models to portray CR systems is a new research trend. In this paper, a single-server retrial cognitive radio system with a linear retrial rate has been considered. The system has two types of users: primary users and secondary users. Secondary users have no effect on primary users because primary users have preemptive precedence. As a result, our purpose is to examine some performance indicators such as the expected queue length for primary users, the probability of the system being idle or occupied by a secondary user, and the probability of the system being busy. This paper begins by deriving the expressions for the generating functions based on the balance equations, so that we can calculate our goal conveniently.

A New Congestion Prevention Policy for the Router

The performance of one router will degrade and the network will have congestion because many packets arrive in the router at the same time. A new congestion prevention policy for router, which is based on the M/G/1 retrial queue with Bernoulli vacation and generic retrial, vacation, setup and service time, is proposed. In this policy, the packets from other routers find the router busy and the packets will be queued in the orbit in accordance with a first-in-first-out (FIFO) discipline which the router starts to process. Only the packet at the head of the queue is allowed access to the router for processing. The ergodical condition of this model is analyzed and the probability generating functions of content of system are obtained. At last, some performance measures and numerical results are given.

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.

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.

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.

Optimal Design for a Retrial Queueing System with State-Dependent Service Rate

This paper considers a single server retrial queue in which a state-dependent service policy is adopted to control the service rate. Customers arrive in the system according to a Poisson process and the service times and inter-retrial times are all exponentially distributed. If the number of customers in orbit is equal to or less than a certain threshold, the service rate is set in a low value and it also can be switched to a high value once this number exceeds the threshold. The stationary distribution and two performance measures are obtained through the partial generating functions. It is shown that this state-dependent service policy degenerates into a classic retrial queueing system without control policy under some conditions. In order to achieve the social optimal strategies, a new reward-cost function is established and the global numerical solutions, obtained by Canonical Particle Swarm Optimization algorithm, demonstrate that the managers can get more benefits if applying this state-dependent service policy compared with the classic model.

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.

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 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.

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 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.

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.

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.

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.

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.