Dynamic scheduling of a single-server two-class queue with constant retrial policy

We analyze the non-preemptive assignment of a single server to two infinite-capacity retrial queues. Customers arrive at both queues according to Poisson processes. They are served on first-come-first-served basis following a cost-optimal routing policy. The customer at the head of a queue generates a Poisson stream of repeated requests for service, that is, we have a constant retrial policy. All service times are exponential, with rates depending on the queues. The costs to be minimized consist of costs per unit time that a customer spends in the system. In case of a scheduling problem that arise when no new customers arrive an explicit condition for server allocation to the first or the second queue is given. The condition presented covers all possible parameter choices. For scheduling that also considers new arrivals, we present the conditions under which server assignment to either queue 1 or queue 2 is cost-optimal.     

An approximate analysis of a cyclic server queue with limited service and reservations

In this paper, we examine a queueing problem motivated by the pipeline polling protocol in satellite communications. The model is an extension of the cyclic queueing system withM-limited service. In this service mechanism, each queue, after receiving service on cyclej, makes a reservation for its service requirement in cyclej + 1. The main contribution to queueing theory is that we propose an approximation for the queue length and sojourn-time distributions for this discipline. Most approximate studies on cyclic queues, which have been considered before, examine the means only. Our method is an iterative one, which we prove to be convergent by using stochastic dominance arguments. We examine the performance of our algorithm by comparing it to simulations and show that the results are very good.     

Transient analysis of a single server queue with catastrophes, failures and repairs

A transient solution is obtained analytically using continued fractions for the system size in an M/M/1 queueing system with catastrophes, server failures and non-zero repair time. The steady state probability of the system size is present. Some key performance measures, namely, throughput, loss probability and response time for the system under consideration are investigated. Further, reliability and availability of the system are analysed. Finally, numerical illustrations are used to discuss the system performance measures.      

A discrete-time priority queue with two-class customers and bulk services

This paper studies the behavior of a discrete queueing system which accepts synchronized arrivals and provides synchronized services. The number of arrivals occurring at an arriving point may follow any arbitrary discrete distribution possessing finite first moment and convergent probability generating function in ¦ z ¦ 1 + with > 0. The system is equipped with an infinite buffer and one or more servers operating in synchronous mode. Service discipline may or may not be prioritized. Results such as the probability generating function of queue occupancy, average queue length, system throughput, and delay are derived in this paper. The validity of the results is also verified by computer simulations.The work reported in this paper was supported by the National Science Council of the Republic of China under Grant NSC1981-0404-E002-04.     

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。     

Large deviation analysis of the single server queue

We establish the large deviation principle (LDP) for the virtual waiting time and queue length processes in the GI/GI/1 queue. The rate functions are found explicitly. As an application, we obtain the logarithmic asymptotics of the probabilities that the virtual waiting time and queue length exceed high levels at large times. Additional new results deal with the LDP for renewal processes and with the derivation of unconditional LDPs for conditional ones. Our approach applies in large deviations ideas and methods of weak convergence theory.This work was supported in part by AT&T Bell Labs.     

Delay analysis of discrete-time priority queue with structured inputs

This paper investigates a discrete-time priority queue with multi-class customers. Applying a delay-cycle analysis, we explicitly derive the probability generating function of the waiting time for an individual class in a geometric batch input queue under preemptive-resume and head-of-the-line priority rules. The conservation law and waiting time characterization for a general class of discrete-time queues are also presented. The results in this paper cover several previous results as special cases.     

Strategic behaviour in a tandem queue with alternating server

Queueing Systems - This paper considers an unobservable two-site tandem queueing system attended by an alternating server. We study the strategic customer behaviour under two threshold-based...     

Two-server queue with one server idle below a threshold

A birth-death queueing system with two identical servers, first-come first-served discipline, and Poisson arrivals is considered. Only one of the servers is active when the number of customers in the system does not exceed a prescribed threshold, whereas both are active above the threshold. The problem of determining the equilibrium density of the waiting time is formulated. A generating function is given for the Laplace transform of the density of the waiting time, and it is pointed out that it leads to an explicit expression for this quantity. Explicit expressions are obtained for the first and second moments of the waiting and sojourn times, and they are compared with the corresponding quantities for a single-server system with the same state-dependent mean service rates.     

Optimal control of a removable server in an M/G/1 queue with finite capacity

We consider an M/G/1 system with finite capacity L in which the removable server applies a (ν, N) policy; a classical cost structure is imposed and the total expected cost per unit time in the steady state is considered. For the M/M/1 situation, Hersh and Brosh [3] analysed the policies with 0⩾ν<N⩽L and established that the best of them is characterized either by ν = 0 or by N = L. By a different and quite easy way and for a general service time distribution, we prove that an optimal policy has the form (ν = 0, 0 ⩽ N ⩽ L + 1), where the (0, 0) and (0, L + 1) policies consist in never closing or never opening the station respectively. Moreover, we describe a precise technique to analyse the policy space and to determine easily the optimal policy.     

Analysis of a single server queue with semi-Markovian service interruption

In this paper we analyze a discrete-time single server queue where the service time equals one slot. The numbers of arrivals in each slot are assumed to be independent and identically distributed random variables. The service process is interrupted by a semi-Markov process, namely in certain states the server is available for service while the server is not available in other states. We analyze both the transient and steady-state models. We study the generating function of the joint probability of queue length, the state and the residual sojourn time of the semi-Markov process. We derive a system of Hilbert boundary value problems for the generating functions. The system of Hilbert boundary value problems is converted to a system of Fredholm integral equations. We show that the system of Fredholm integral equations has a unique solution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Numerical analysis of a discrete-time finite capacity queue with a burst arrival

We study a discrete-time, multi-server, finite capacity queue with a burst arrival. Once the first job of a burst arrives at the queue, the successive jobs will arrive every time slot until the last job of the burst arrives. The number of jobs and the inter-arrival time of bursts are assumed to be generally distributed, and the service time is assumed to be equal to one slot. We propose an efficient numerical method to exactly obtain the job loss probability, the waiting time distribution and the mean queue length using an embedded Markov chain at the arrival instants of bursts.     

A single server queue with service interruptions

In this paper we characterize the queue-length distribution as well as the waiting time distribution of a single-server queue which is subject to service interruptions. Such queues arise naturally in computer and communication problems in which customers belong to different classes and share a common server under some complicated service discipline. In such queues, the viewpoint of a given class of customers is that the server is not available for providing service some of the time, because it is busy serving customers from a different class. A natural special case of these queues is the class of preemptive priority queues. In this paper, we consider arrivals according the Markovian Arrival Process (MAP) and the server is not available for service at certain times. The service times are assumed to have a general distribution. We provide numerical examples to show that our methods are computationally feasible. This revised version was published online in June 2006 with corrections to the Cover Date.     

Optimal threshold policies in a two-class preemptive priority queue with admission and termination control

We consider a two-class 1 preemptive priority queue in which there are two essential, on-line decisions that have to be taken. The first is the decision to either accept or reject new type-1 or type-2 jobs. The second is the decision to abort jobs, i.e., to remove any type-1 or type-2 jobs from the system. We show that there exist optimal threshold policies for these two types of, decisions.     

Stability analysis of parallel server systems under longest queue first

We consider the stability of parallel server systems under the longest queue first (LQF) rule. We show that when the underlying graph of a parallel server system is a tree, the standard nominal traffic condition is sufficient for the stability of that system under LQF when interarrival and service times have general distributions. Then we consider a special parallel server system, which is known as the X-model, whose underlying graph is not a tree. We provide additional “drift” conditions for the stability and transience of these queueing systems with exponential interarrival and service times. Drift conditions depend in general on the stationary distribution of an induced Markov chain that is derived from the underlying queueing system. We illustrate our results with examples and simulation experiments. We also demonstrate that the stability of the LQF depends on the tie-breaking rule used and that it can be unstable even under arbitrary low loads.     

A retrial queue with redundancy and unreliable server

Retrial queues are useful in the stochastic modelling of computer and telecommunication systems amongst others. In this paper we study a version of the retrial queue with variable service. Such a point of view gives another look at the unreliable retrial queueing problem which includes the redundancy model.By using the theory of piecewise Markovian processes, we obtain the analogue of the Pollaczek-Khintchine formula for such retrial queues, which is useful for operations researchers to obtain performance measures of interest.     

A single server queue with gated processor-sharing discipline

In this paper we consider a single server queue in which arrivals occur according to a Poisson process and each customer's service time is exponentially distributed. The server works according to the gated process-sharing discipline. In this discipline, the server provides service to a batch of at mostm customers at a time. Once a batch of customers begins service, no other waiting customer can receive service until all members of the batch have completed their service. For this queue, we derive performance characteristics, such as waiting time distribution, queue length distribution etc. For this queue, it is possible to obtain the mean conditional response time for a customer whose service time is known. This conditional response time is a nonlinear function (as opposed to the linear case for the ordinary processor-sharing queue). A special case of the queue (wherem=) has an interesting and unusual solution. For this special case, the size of the batch for service is a Markov chain whose steady state distribution can be explicitly written down. Apart from the contribution to the theory of Markov chains and queues, the model may be applicable to scheduling of computer and communication systems.