非强占优先权下的M/M/1排队

分析带有两个优先权的非强占M/M/1系统的性能,用补充变量法构造向量马尔可夫过程对此排队系统的状态转移方程进行分析,得到两类顾客在非强占优先权的队长联合分布的母函数,进一步讨论,得出了服务台被两类顾客占有和闲置的概率以及两类信元各自的平均队长.     

Analysis of nonpreemptive priority queues with multiple servers and two priority classes

In this paper, we model a priority multiserver queueing system with two priority classes. A high priority customer has nonpreemptive priority over low priority customers. The approaches for solving the problem are the state-reduction based variant of Kao, the modified boundary algorithm of Latouche, the logarithmic reduction algorithm of Latouche and Ramaswami, and the power-series method of Blanc. The objectives of this paper are to present a power-series implementation for the priority queue and to evaluate the relative efficiencies of alternative procedures to compute various performance characteristics. In the paper, we find that at times the logarithmic reduction algorithm may not perform as well as expected and the power-series approach can occasionally pose numerical difficulties.     

Waiting times of a finite-capacity multi-server model with non-preemptive priorities

We consider a non-preemptive head of the line multi-server priority model with finite capacity. The arrival processes of the different priority classes are independent Poisson processes. The service times are exponentially distributed and identical for the different priority classes. The model is described by a homogeneous continuous-time Markov chain. For the two-class model we derive an explicit representation of its steady-state distribution. Applying matrix-analytic methods we calculate the Laplace-Stieltjes Transform of the actual waiting time of each priority class of a p-class system.     

Optimal server scheduling in nonpreemptive finite-population queueing systems

We consider a finite-population queueing system with heterogeneous classes of customers and a single server. For the case of nonpreemptive service, we fully characterize the structure of the server's optimal service policy that minimizes the total average customer waiting costs. We show that the optimal service policy may never serve some classes of customers. For those classes that are served, we show that the optimal service policy is a simple static priority policy. We also derive sufficient conditions that determine the optimal priority sequence.     

Stability analysis of N-model systems under a static priority rule

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.     

An algorithmic approach for a special class of Markov chains

In this paper, we develop an algorithmic method for the evaluation of the steady state probability vector of a special class of finite state Markov chains. For the class of Markov chains considered here, it is assumed that the matrix associated with the set of linear equations for the steady state probabilities possess a special structure, such that it can be rearranged and decomposed as a sum of two matrices, one lower triangular with nonzero diagonal elements, and the other an upper triangular matrix with only very few nonzero columns. Almost all Markov chain models of queueing systems with finite source and/or finite capacity and first-come-first-served or head of the line nonpreemptive priority service discipline belongs to this special class.     

Effect of global FCFS and relative load distribution in two-class queues with dedicated servers

In this paper, we investigate multi-class multi-server queueing systems with global FCFS policy, i.e., where customers requiring different types of service—provided by distinct servers—are accommodated in one common FCFS queue. In such scenarios, customers of one class (i.e., requiring a given type of service) may be hindered by customers of other classes. The purpose of this paper is twofold: to gain (qualitative and quantitative) insight into the impact of (i) the global FCFS policy and (ii) the relative distribution of the load amongst the customer classes, on the system performance. We therefore develop and analyze an appropriate discrete-time queueing model with general independent arrivals, two (independent) customer classes and two class-specific servers. We study the stability of the system and derive the system-content distribution at random slot boundaries; we also obtain mean values of the system content and the customer delay, both globally and for each class individually. We then extensively compare these results with those obtained for an analogous system without global FCFS policy (i.e., with individual queues for the two servers). We demonstrate that global FCFS, as well as the relative distribution of the load over the two customer classes, may have a major impact on the system performance.     

Time-dependent analysis of a queue with batch arrivals andn levels of nonpreemptive priority

A queueing system with batch arrivals andn classes of customers with nonpreemptive priorities between them is considered. Each batch arrives according to the Poisson distribution and contains customers of all classes while the service times follow arbitrary distributions with different probability density functions for each class. For such a model the system states probabilities both in the transient and in the steady state are analysed and also expressions for the Laplace transforms of the busy period densities for each class and for the general busy period are obtained.     

Modeling traffic flow interrupted by incidents

A steady-state M/M/c queueing system under batch service interruptions is introduced to model the traffic flow on a roadway link subject to incidents. When a traffic incident happens, either all lanes or part of a lane is closed to the traffic. As such, we model these interruptions either as complete service disruptions where none of the servers work or partial failures where servers work at a reduced service rate. We analyze this system in steady-state and present a scheme to obtain the stationary number of vehicles on a link. For those links with large c values, the closed-form solution of M/M/∞ queues under batch service interruptions can be used as an approximation. We present simulation results that show the validity of the queueing models in the computation of average travel times.     

An Eigenvalue Approach to Analyzing a Finite Source Priority Queueing Model

In this paper, we present a novel approach to determining the steady-state distribution for the number of jobs present in a 2-class, single server preemptive priority queueing model where the low priority source population is finite. Arrivals are assumed to be Poisson with exponential service times. The system investigated is a quasi birth and death process, and the joint distribution is derived via the method of generalized eigenvalues. Using this approach, we are able to obtain all eigenvalues and corresponding eigenvectors explicitly. Furthermore, we link this method to the matrix analytic approach by obtaining an explicit solution for the rate matrix R. Two numerical examples are given to illustrate the procedure and highlight some important computational features.     

基于排队的两类顾客易逝品库存控制策略研究

本文采用排队理论在面向两类顾客需求的环境下讨论易逝品库存排队系统控制策略问题。首先刻画面向两类顾客服务且具有马尔科夫结构的易逝品库存系统排队模型,获得库存水平状态的稳态概率分布以及作为库存成本控制的系统稳态性能指标。接下来,构建库存控制成本函数及考虑服务水平约束的库存控制优化模型,设计了改进的遗传算法。最后,数值算例揭示出系统参数的敏感性和相应的管理启示。     

Multi-server queueing systems with cooperation of the servers

In this paper, we consider an N-server queueing model with homogeneous servers in which customers arrive according to a stationary Poisson arrival process. The service times are exponentially distributed. Two new customer’s service disciplines assuming simultaneous service of arriving customer by all currently idle servers are discussed. The steady state analysis of the queue length and sojourn time distribution is performed by means of the matrix analytic methods. Numerical examples, which illustrate advantage of introduced disciplines comparing to the classical one, are presented.     

A two-class discrete-time queueing model with two dedicated servers and global FCFS service discipline

This paper considers a simple discrete-time queueing model with two types (classes) of customers (types 1 and 2) each having their own dedicated server (servers A and B resp.). New customers enter the system according to a general independent arrival process, i.e., the total numbers of arrivals during consecutive time slots are i.i.d. random variables with arbitrary distribution. Service times are deterministically equal to 1 slot each. The system uses a “global FCFS” service discipline, i.e., all arriving customers are accommodated in one single FCFS queue, regardless of their types. As a consequence of the “global FCFS” rule, customers of one type may be blocked by customers of the other type, in that they may be unable to reach their dedicated server even at times when this server is idle, i.e., the system is basically non-workconserving. One major aim of the paper is to estimate the negative impact of this phenomenon on the queueing performance of the system, in terms of the achievable throughput, the system occupancy, the idle probability of each server and the delay. As it is clear that customers of different types hinder each other more as they tend to arrive in the system more clustered according to class, the degree of “class clustering” in the arrival process is explicitly modeled in the paper and its very direct impact on the performance measures is revealed. The motivation of our work are systems where this kind of blocking is encountered, such as input-queueing network switches or road splits.     

Computation of the steady state distribution for multi-server retrial queues with phase type service process

We consider a multi-server retrial queueing system with the Batch Markovian Arrival Process and phase type service time distribution. Such a general queueing system suits for modeling and decision making in many real life objects including modern wireless communication networks. Behavior of such a system is described by the level dependent multi-dimensional Markov chain. Blocks of the generator of this chain, which is the block structured matrix of infinite size, can have large size if the number of servers is large and distribution of service time is not exponential. Due to this fact, the existing in literature algorithms allow to compute key performance measures of such a system only for a small number of servers. Here we describe the algorithm that allows to compute the stationary distribution of the system for larger number of servers and numerically illustrate its advantage. Importance of taking into account correlation in the arrival process is numerically demonstrated.     

Calculation of delay characteristics for multiserver queues with constant service times

We consider a discrete-time infinite-capacity queueing system with a general uncorrelated arrival process, constant-length service times of multiple slots, multiple servers and a first-come-first-served queueing discipline. Under the assumption that the queueing system can reach a steady state, we first establish a relationship between the steady-state probability distributions of the system content and the customer delay. Next, by means of this relationship, an explicit expression for the probability generating function of the customer delay is obtained from the known generating function of the system content, derived in previous work. In addition, several characteristics of the customer delay, namely the mean value, the variance and the tail distribution of the delay, are derived through some mathematical manipulations. The analysis is illustrated by means of some numerical examples.     

Open queueing networks in discrete time-some limit theorems

A finite number of nodes, each with a single server and infinite buffers, is considered in discrete time. The service may be FIFO and the service times are constant. The external arrivals and the routing decision variables form a general stationary sequence. Stability of the system is proved under these assumptions. Extension to multiple servers at a node and general stationary distributions holds. If the external input is i.i.d. and the routing is Markovian then stochastic ordering, continuity of stationary distributions, rates of convergence, a functional CLT and a functional LIL and various other limit theorems for the queue length process are also proved. Generalizations to multiple servers at nodes, customers with priority, multiple customer classes, general service length and Markov modulated external arrival cases are discussed.     

Containment of socially optimal policies in multiple-facility Markovian queueing systems

We consider a Markovian queueing system with N heterogeneous service facilities, each of which has multiple servers available, linear holding costs, a fixed value of service and a first-come-first-serve queue discipline. Customers arriving in the system can be either rejected or sent to one of the N facilities. Two different types of control policies are considered, which we refer to as ‘selfishly optimal’ and ‘socially optimal’. We prove the equivalence of two different Markov Decision Process formulations, and then show that classical M/M/1 queue results from the early literature on behavioural queueing theory can be generalized to multiple dimensions in an elegant way. In particular, the state space of the continuous-time Markov process induced by a socially optimal policy is contained within that of the selfishly optimal policy. We also show that this result holds when customers are divided into an arbitrary number of heterogeneous classes, provided that the service rates remain non-discriminatory.     

A queueing system where customers require a random number of servers simultaneously

A closed single node queueing system with multiple classes is analyzed numerically. The node consists of M identical servers fed by a single queue. Each customer of class r, 1 ⩽ r ⩽ M, acquires r servers simultaneously at the beginning of its service. All r servers are released at the same time upon completion of its service. The service time of a class r customer is exponentially distributed with a mean depending on r. This queueing model is analyzed with a view to obtaining performance measures such as throughput, distribution of busy servers, and queue-length distribution.     

Workload in queues having priorities assigned according to service time

System designers often implement priority queueing disciplines in order to improve overall system performance; however, improvement is often gained at the expense of lower priority cystomers. Shortest Processing Time is an example of a priority discipline wherein lower priority customers may suffer very long waiting times when compared to their waiting times under a democratic service discipline. In what follows, we shall investigate a queueing system where customers are divided into a finitie number of priority classes according to their service times.We develop the multivariate generating function characterizing the joint workload among the priority classes. First moments obtained from the generating function yield traffic intensities for each priority class. Second moments address expected workloads, in particular, we obtain simple Pollaczek-Khinchine type formulae for the classes. Higher moments address variance and covariance among the workloads of the priority classes.This work was supported in part by National Science Foundation Grant DDM-8913658.     

Instability of LAS multiclass queueing networks

We provide an example of a strictly subcritical multiclass queueing network which is unstable under the least attained service (LAS) service protocol. It is a reentrant line with two servers and six customer classes. The customer interarrival times in our system are bounded below and have finite exponential moments, while the corresponding service times are deterministic. As a special case, we obtain a deterministic, strictly subcritical unstable LAS network.