Queueing systems on a circle

Consider a ring on which customers arrive according to a Poisson process. Arriving customers drop somewhere on the circle and wait there for a server who travels on the ring. Whenever this server encounters a customer, he stops and serves the customer according to an arbitrary service time distribution. After the service is completed, the server removes the client from the circle and resumes his journey.We are interested in the number and the locations of customers that are waiting for service. These locations are modeled as random counting measures on the circle. Two different types of servers are considered: The polling server and the Brownian (or drunken) server. It is shown that under both server motions the system is stable if the traffic intensity is less than 1. Furthermore, several earlier results on the configuration of waiting customers are extended, by combining results from random measure theory, stochastic integration and renewal theory.     

Single line queue with recurrent repeated demands

We analyze a model queueing system in which customers cannot be in continuous contact with the server, but must call in to request service. If the server is free, the customer enters service immediately, but if the server is occupied, the unsatisfied customer must break contact and reapply for service later. There are two types of customer present who may reapply. First transit customers who arrive from outside according to a Poisson process and if they find the server busy they join a source of unsatisfied customers, called the orbit, who according to an exponential distribution reapply for service till they find the server free and leave the system on completion of service. Secondly there are a number of recurrent customers present who reapply for service according to a different exponential distribution and immediately go back in to the orbit after each completion of service. We assume a general service time distribution and calculate several characterstic quantities of the system for both the constant rate of reapplying for service and for the case when customers are discouraged and reduce their rate of demand as more customers join the orbit.     

A single-server discrete-time retrial <Emphasis Type="Italic">G</Emphasis>-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.     

Polling and greedy servers on a line

A single server moves with speed on a line interval (or a circle) of length (circumference)L. Customers, requiring service of constant durationb, arrive on the interval (or circle) at random at mean rate customers per unit length per unit time. A customer's mean wait for service depends partly on the rules governing the server's motion. We compare two different servers: thepolling server and thegreedy server. Without knowing the locations of waiting customers, a polling server scans endlessly back and forth across the interval (or clockwise around the circle), stopping only where it encounters a waiting customer. Knowing where customers are waiting, a greedy server always travels toward the current nearest one. Except for certain extreme values of ,L, b, and, the polling and greedy servers are roughly equally effective. Indeed, the simpler polling server is often the better. Theoretical results show, in most cases, that the polling server has a high probability of moving toward the nearest customer, i.e. moving as a greedy server would. The greedy server is difficult to analyze, but was simulated on a computer.     

Stability of a spatial polling system with greedy myopic service

This paper studies a spatial queueing system on a circle, polled at random locations by a myopic server that can only observe customers in a bounded neighborhood. The server operates according to a greedy policy, always serving the nearest customer in its neighborhood, and leaving the system unchanged at polling instants where the neighborhood is empty. This system is modeled as a measure-valued random process, which is shown to be positive recurrent under a natural stability condition that does not depend on the server??s scan radius. When the interpolling times are light-tailed, the stable system is shown to be geometrically ergodic. The steady-state behavior of the system is briefly discussed using numerical simulations and a heuristic light-traffic approximation.     

Exponential Queues with Server Vacations

In this paper, we analyse a queueing system where the server may take a vacation. The customers arrive at the service facility according to a Poisson process, and are served if the server is available (not on vacation). We consider two models: when the server vacation cycle is independent of and dependent on the number of customers in the system. The infinitesimal generators of the underlying Markov processes have a block tri-diagonal structure, and we provide a matrix geometric solution. When the vacation cycle is independent of the customer queue length, we present a simple load-dependent approximation that is fairly accurate.     

Questionnaire design improvement and missing item scores estimation for rapid and efficient decision making

In this paper we consider a single-server, cyclic polling system with switch-over times and Poisson arrivals. The service disciplines that are discussed, are exhaustive and gated service. The novel contribution of the present paper is that we consider the reneging of customers at polling instants. In more detail, whenever the server starts or ends a visit to a queue, some of the customers waiting in each queue leave the system before having received service. The probability that a certain customer leaves the queue, depends on the queue in which the customer is waiting, and on the location of the server. We show that this system can be analysed by introducing customer subtypes, depending on their arrival periods, and keeping track of the moment when they abandon the system. In order to determine waiting time distributions, we regard the system as a polling model with varying arrival rates, and apply a generalised version of the distributional form of Little??s law. The marginal queue length distribution can be found by conditioning on the state of the system (position of the server, and whether it is serving or switching).     

A single-server discrete-time queue with correlated positive and negative customer arrivals

An MMBP/Geo/1 queue with correlated positive and negative customer arrivals is studied. In the infinite-capacity queueing system, positive customers and negative customers are generated by a Bernoulli bursty source with two correlated geometrically distributed periods. I.e., positive and negative customers arrive to the system according to two different geometrical arrival processes. Under the late arrival scheme (LAS), two removal disciplines caused by negative customers are investigated in the paper. In individual removal scheme, a negative customer removes a positive customer in service if any, while in disaster model, a negative customer removes all positive customers in the system if any. The negative customer arrival has no effect on the system if it finds the system empty. We analyze the Markov chains underlying the queueing systems and evaluate the performance of two systems based on generating functions technique. Some explicit solutions of the system, such as the average buffer content and the stationary probabilities are obtained. Finally, the effect of several parameters on the system performance is shown numerically.     

On optimal polling policies

In a single-server polling system, the server visits the queues according to a routing policy and while at a queue, serves some or all of the customers there according to a service policy. A polling (or scheduling) policy is a sequence of decisions on whether to serve a customer, idle the server, or switch the server to another queue. The goal of this paper is to find polling policies that stochastically minimize the unfinished work and the number of customers in the system at all times. This optimization problem is decomposed into three subproblems: determine the optimal action (i.e., serve, switch, idle) when the server is at a nonempty queue; determine the optimal action (i.e., switch, idle) when the server empties a queue; determine the optimal routing (i.e., choice of the queue) when the server decides to switch. Under fairly general assumptions, we show for the first subproblem that optimal policies are greedy and exhaustive, i.e., the server should neither idle nor switch when it is at a nonempty queue. For the second subproblem, we prove that in symmetric polling systems patient policies are optimal, i.e., the server should stay idling at the last visited queue whenever the system is empty. When the system is slotted, we further prove that non-idling and impatient policies are optimal. For the third subproblem, we establish that in symmetric polling systems optimal policies belong to the class of Stochastically Largest Queue (SLQ) policies. An SLQ policy is one that never routes the server to a queue known to have a queue length that is stochastically smaller than that of another queue. This result implies, in particular, that the policy that routes the server to the queue with the largest queue length is optimal when all queue lengths are known and that the cyclic routing policy is optimal in the case that the only information available is the previous decisions.This work was supported in part by NSF under Contract ASC-8802764.     

带止步与中途退出策略的M~x/M/1/N单重工作休假排队系统

研究了带有止步和中途退出的M~x/M/1/N单重工作休假排队系统.顾客成批到达,到达后每批中的顾客,或者以概率b决定进入队列等待服务,或者以概率1-b止步(不进入系统).顾客进入系统后可能因为等待的不耐烦而在没有接受服务的情况下离开系统(中途退出).系统中一旦没有顾客,服务员立即进入单重工作休假.首先,利用马尔科夫过程理论建立了系统稳态概率满足的方程组.其次利用矩阵解法求出了稳态概率的矩阵解并得到了系统的平均队长、平均等待队长以及顾客的平均消失概率等性能指标.最后通过数值例子分析了工作休假时的低服务率η和休假率θ这两个参数对系统平均队长的影响.     

Stability and performance of greedy server systems

Consider a queueing system in which arriving customers are placed on a circle and wait for service. A traveling server moves at constant speed on the circle, stopping at the location of the customers until service completion. The server is greedy: always moving in the direction of the nearest customer. Coffman and Gilbert conjectured that this system is stable if the traffic intensity is smaller than 1; however, a proof or counterexample remains unknown. In this review, we present a picture of the current state of this conjecture and suggest new related open problems.     

Single line queue with repeated demands

We analyze a model of a queueing system in which customers can only call in to request service: if the server is free, the customer enters service immediately, but if the service system is occupied, the unsatisfied customer must break contact and reinitiate his request later. Such a customer is said to be in “orbit”. In this paper we consider three models characterized by the discipline governing the order of re-request of service from orbit. First, all customers in orbit can reapply, but are discouraged and reduce their rate of demand as more customers join the orbit. Secondly, the FCFS discipline operates for the unsatisfied customers in orbit. Finally, the LCFS discipline governs the customers in orbit and the server takes an exponentially distributed vacation after each service is completed. We calculate several characteristics quantities of such systems, assuming a general service-time distribution and different exponential distributions for the times between arrivals of first and repeat requests.     

Stochastic bounds for a polling system

In this note we consider two queueing systems: a symmetric polling system with gated service at allN queues and with switchover times, and a single-server single-queue model with one arrival stream of ordinary customers andN additional permanently present customers. It is assumed that the combined arrival process at the queues of the polling system coincides with the arrival process of the ordinary customers in the single-queue model, and that the service time and switchover time distributions of the polling model coincide with the service time distributions of the ordinary and permanent customers, respectively, in the single-queue model. A complete equivalence between both models is accomplished by the following queue insertion of arriving customers. In the single-queue model, an arriving ordinary customer occupies with probabilityp _i a position at the end of the queue section behind theith permanent customer,i = l, ...,N. In the cyclic polling model, an arriving customer with probabilityp _i joins the end of theith queue to be visited by the server, measured from its present position.For the single-queue model we prove that, if two queue insertion distributions {p _{i, i} = l, ...,N} and {q _{i, i} = l, ...,N} are stochastically ordered, then also the workload and queue length distributions in the corresponding two single-queue versions are stochastically ordered. This immediately leads to equivalent stochastic orderings in polling models.Finally, the single-queue model with Poisson arrivals andp ₁ = 1 is studied in detail.Part of the research of the first author has been supported by the Esprit BRA project QMIPS.     

服务台不可靠的重试排队系统均衡分析

本文研究服务台不可靠的M/M/1常数率重试排队系统中顾客的均衡进队策略, 其中服务台在正常工作和空闲状态下以不同的速率发生故障。在该系统中, 服务台前没有等待空间, 如果到达的顾客发现服务台处于空闲状态, 该顾客可占用服务台开始服务。否则, 如果服务台处于忙碌状态, 顾客可以选择留下信息, 使得服务台在空闲时可以按顺序在重试空间中寻找之前留下信息的顾客进行服务。当服务台发生故障时, 正在被服务的顾客会发生丢失, 且系统拒绝新的顾客进入系统。根据系统提供给顾客的不同程度的信息, 研究队长可见和不可见两种信息情形下系统的稳态指标, 以及顾客基于收入-支出函数的均衡进队策略, 并建立单位时间内服务商的收益和社会福利函数。比较发现, 披露队长信息不一定能提高服务商收益和社会福利。     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

Ergodicity of a polling network

The polling network considered here consists of a finite collection of stations visited successively by a single server who is following a Markovian routing scheme. At every visit of a station a positive random number of the customers present at the start of the visit are served, whereupon the server takes a positive random time to walk to the station to be visited next. The network receives arrivals of customer groups at Poisson instants, and all customers wait until served, whereupon they depart from the network. Necessary and sufficient conditions are derived for the server to be able to cope with the traffic. For the proof a multidimensional imbedded Markov chain is studied.     

Strategic joining in M/M/1 retrial queues

The equilibrium and socially optimal balking strategies are investigated for unobservable and observable single-server classical retrial queues. There is no waiting space in front of the server. If an arriving customer finds the server idle, he occupies the server immediately and leaves the system after service. Otherwise, if the server is found busy, the customer decides whether or not to enter a retrial pool with infinite capacity and becomes a repeated customer, based on observation of the system and the reward–cost structure imposed on the system. Accordingly, two cases with respect to different levels of information are studied and the corresponding Nash equilibrium and social optimization balking strategies for all customers are derived. Finally, we compare the equilibrium and optimal behavior regarding these two information levels through numerical examples.     

Multiserver queue with addressed retrials

This paper presents a multiserver retrial queueing system with servers kept apart, thereby rendering it impossible for one to know the status (idle/busy) of the others. Customers proceeding to one channel will have to go to orbit if the server in it is busy and retry after some time to some channel, not necessarily the one already tried. Each orbital customer, independently of others, chooses the server randomly according to some specified probability distribution. Further this distribution is identical for all customers. We assume that the same ‘orbit’ is used by all retrial customers, between repeated attempts, to access the servers. We derive the system state probability distribution under Poisson arrival process of external customers, exponentially distributed service times and linear retrial rates to access the servers. Several system state characteristics are obtained and numerical illustrations provided. AMS subject classification: Primary 60K25 60K20     

Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines

The stability of a cyclic polling system, with a single server and two infinite-buffer queues, is considered. Customers arrive at the two queues according to independent batch Markovian arrival processes. The first queue is served according to the gated service discipline, and the second queue is served according to a state-dependent time-limited service discipline with the preemptive repeat-different property. The state dependence is that, during each cycle, the predetermined limited time of the server’s visit to the second queue depends on the queue length of the first queue at the instant when the server last departed from the first queue. The mean of the predetermined limited time for the second queue either decreases or remains the same as the queue length of the first queue increases. Due to the two service disciplines, the customers in the first queue have higher service priority than the ones in the second queue, and the service fairness of the customers with different service priority levels is also considered. In addition, the switchover times for the server traveling between the two queues are considered, and their means are both positive as well as finite. First, based on two embedded Markov chains at the cycle beginning instants, the sufficient and necessary condition for the stability of the cyclic polling system is obtained. Then, the calculation methods for the variables related to the stability condition are given. Finally, the influence of some parameters on the stability condition of the cyclic polling system is analyzed. The results are useful for engineers not only checking whether the given cyclic polling system is stable, but also adjusting some parameters to make the system satisfy some requirements under the condition that the system is stable.     

Analysis of M/M/S queues with 1-limited service

In this paper, a multiple server queue, in which each server takes a vacation after serving one customer is studied. The arrival process is Poisson, service times are exponentially distributed and the duration of a vacation follows a phase distribution of order 2. Servers returning from vacation immediately take another vacation if no customers are waiting. A matrix geometric method is used to find the steady state joint probability of number of customers in the system and busy servers, and the mean and the second moment of number of customers and mean waiting time for this model. This queuing model can be used for the analysis of different kinds of communication networks, such as multi-slotted networks, multiple token rings, multiple server polling systems and mobile communication systems.