A Complete and Simple Solution to a Discrete-Time Finite-Capacity BMAP/D/c Queue

We consider a discrete-time multi-server finite-capacity queueing system with correlated batch arrivals and deterministic service times (of single slot), which has a variety of potential applications in slotted digital telecommunication systems and other related areas. For this queueing system, we present, based on Markov chain analysis, not only the steady-state distributions but also the transient distributions of the system length and of the system waiting time in a simple and unified manner. From these distributions, important performance measures of practical interest can be easily obtained. Numerical examples concerning the superposition of certain video traffics are presented at the end.

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.

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.

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.

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 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.

Stationary Analysis of Geo/Geo/1 Queue with Two-Speed Service and the Optimal Switching Threshold for the Service Rate

This paper considers a Geo/Geo/1 queueing system with infinite capacity, in which the service rate changes depending on the workload. Initially, when the number of customers in the system is less than a certain threshold L, low service rate is provided for cost saving. On the other hand, the high service rate is activated as soon as L customers accumulate in the system and such service rate is preserved until the system becomes completely empty even if the number of customers falls below L. The steady-state probability distribution and the expected number of customers in the system are derived. Through the first-step argument, a recursive algorithm for computing the first moment of the conditional sojourn time is obtained. Furthermore, employing the results of regeneration cycle analysis, the direct search method is also implemented to determine the optimal value of L for minimizing the long-run average cost rate function.

具有两类平行顾客的M/M/1重试排队系统均衡策略研究

本文研究具有两类平行顾客且服务台可靠的M/M/1重试排队系统的均衡策略.在该排队系统中,两类顾客平行到达,并服从不同参数的负指数分布.当顾客进入系统时,若观察到服务台为空,将立刻开始服务;若观察到服务台处于忙期,则进入重试空间等待重试.在完全可见和几乎可见两种情形下,基于“收益-成本”理论提出合理的效用函数并对两类平行顾客进行均衡分析.此外,建立单位时间的社会收益函数,给出最优社会效益分析.最后运用数值分析直观地表示出随着系统参数的改变,顾客行为策略的变化情况.

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.

RELIABILITY INDICES OF DISCRETE-TIME Geo^X/G/1 QUEUEING SYSTEM WITH UNRELIABLE SERVICE STATION AND MULTIPLE ADAPTIVE DELAYED VACATIONS

This paper considers the discrete-time GeoX/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, hi, 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.

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.

Exact Tail Asymptotics for a Discrete-time Preemptive Priority Queue

In this paper, we consider a discrete-time preemptive priority queue with different service com- pletion probabilities for two classes of customers, one with high-priority and the other with low-priority. This model corresponds to the classical preemptive priority queueing system with two classes of independent Poisson customers and a single exponential server. Due to the possibility of customers＇ arriving and departing at the same time in a discrete-time queue, the model considered in this paper is more complicated than the continuous- time model. In this model, we focus on the characterization of the exact tail asymptotics for the joint stationary distribution of the queue length of the two types of customers, for the two boundary distributions and for the two marginal distributions, respectively. By using generating functions and the kernel method, we get the exact tail asymptotic properties along the direction of the low-priority queue, as well as along the direction of the high-priority queue.

Performance Analysis of a Discrete-Time Queue with Working Breakdowns and Searching for the Optimum Service Rate in Working Breakdown Period

This paper deals with a discrete-time Geo/Geo/1 queueing system with working breakdowns in which customers arrive at the system in variable input rates according to the states of the server. The server may be subject to breakdowns at random when it is in operation. As soon as the server fails, a repair process immediately begins. During the repair period, the defective server still provides service for the waiting customers at a lower service rate rather than completely stopping service.We analyze the stability condition for the considered system. Using the probability generating function technique, we obtain the probability generating function of the steady-state queue size distribution.Also, various important performance measures are derived explicitly. Furthermore, some numerical results are provided to carry out the sensitivity analysis so as to illustrate the effect of different parameters on the system performance measures. Finally, an operating cost function is formulated to model a computer system and the parabolic method is employed to numerically find the optimum service rate in working breakdown period.

带有两类故障和延迟维修的M/G/1重试排队

研究带两类故障和延迟维修的M/G/1重试排队模型。针对系统平稳性及其平稳分布的问题,首先采用嵌入马尔科夫链和补充变量法得到稳态条件和平稳概率分布,然后用概率母函数研究系统队长分布,最后通过数值算例分析参数对系统性能指标的影响。

ANALYTICAL MODELS BASED DISCRETE-TIME QUEUEING FOR THE CONGESTED NETWORK

Congestion is one of the well-studied problems in computer networks,which occurs when the request for network resources exceeds the buffer capacity.Many active queue management techniques such as BLUE and RED have been proposed in the literature to control congestions in early stages.In this paper,we propose two discrete-time queueing network analytical models to drop the arrival packets in preliminary stages when the network becomes congested.The first model is based on Lambda Decreasing and it drops packets from a probability value to another higher value according to the buffer length.Whereas the second proposed model drops packets linearly based on the current queue length.We compare the performance of both our models with the original BLUE in order to decide which of these methods offers better quality of service.The comparison is done in terms of packet dropping probability,average queue length,throughput ratio,average queueing delay,and packet loss rate.

Analysis of a Discrete-Time Geo/G/1 Queue in a Multi-Phase Service Environment with Disasters

This paper considers a discrete-time Geo/G/1 queue in a multi-phase service environment,where the system is subject to disastrous breakdowns, causing all present customers to leave the system simultaneously. At a failure epoch, the server abandons the service and the system undergoes a repair period. After the system is repaired, it jumps to operative phase i with probability qi, i = 1, 2 ···, n.Using the supplementary variable technique, we obtain the distribution for the stationary queue length at the arbitrary epoch, which are then used for the computation of other performance measures. In addition, we derive the expected length of a cycle time, the generating function of the sojourn time of an arbitrary customer, and the generating function of the server’s working time in a cycle. We also give the relationship between the discrete-time queueing system to its continuous-time counterpart. Finally,some examples and numerical results are presented.

有可选到达、服务台可修的M/G/1重试排队系统

考虑了一个具有重试,可选择到达,反馈,服务台可修的M/G/1排队系统.研究了顾客到达后具有两种选择:或以概率q直接进入重试组,在重试组中要求接受服务;或者以概率1-q接触服务台,如果服务台处于闲期,则立刻接受服务,否则进入重试组,顾客一旦服务完毕后,可以以概率1-p离开系统或者以概率p返回重试组再次要求服务的情况.求得系统稳态时一些排队指标和可靠性指标.

基于重试、不耐烦M/M/s/k+M排队的呼叫中心性能分析

为对呼叫中心整体性能和ACD(自动话务分配)的统计数据进行科学的分析,针对呼叫中心中顾客到达有遇忙音而重试(retrial)和在ACD中排队时会因不耐烦(impatience)而放弃等待的特点,讨论了重试和不耐烦M/M/s/k+M排队模型.尝试一种新的求解方法对模型求解,给出解析解和有关指标的计算公式,并给出数值计算示例和在单个服务台有同等服务强度的情况下,大系统更能使顾客感到满意等结论.