On tandem blocking queues with a common retrial queue

We consider systems of tandem blocking queues having a common retrial queue. The model represents dynamics of short TCP transfers in the Internet. Analytical results are available only for a specific example with two queues in tandem. We propose approximation procedures involving simple analytic expressions, based on mean value analysis (MVA) and on fixed point approach (FPA). The mean sojourn time of a job in the system and the mean number of visits to the orbit queue are estimated by the MVA which needs as an input the fractions of blocked jobs in the primary queues. The fractions of blocked jobs are estimated by FPA. Using a benchmark example of the system with two primary queues, we conclude that the approximation works well in the light traffic regime. We note that our approach becomes exact if the blocking probabilities are fixed. Finally, we consider two optimization problems regarding minimizing mean total sojourn time of a job in the system: (i) finding the best order of queues and (ii) allocating a given capacity among the primary queues.     

Decomposing the queue length distribution of processor-sharing models into queue lengths of permanent customer queues

We obtain a decomposition result for the steady state queue length distribution in egalitarian processor-sharing (PS) models. In particular, for multi-class egalitarian PS queues, we show that the marginal queue length distribution for each class equals the queue length distribution of an equivalent single class PS model with a random number of permanent customers. Similarly, the mean sojourn time (conditioned on the initial service requirement) for each class can be obtained by conditioning on the number of permanent customers. The decomposition result implies linear relations between the marginal queue length probabilities, which also hold for other PS models such as the egalitarian PS models with state-dependent system capacity that only depends on the total number of customers in the system. Based on the exact decomposition result for egalitarian PS queues, we propose a similar decomposition for discriminatory processor-sharing (DPS) models, and numerically show that the approximation is accurate for moderate differences in service weights.     

The distribution of network states during residence times in product form queueing networks

Product form queueing networks are so named because their equilibrium state probabilities have a simple product form. This simple form has lead to computationally efficient techniques for obtaining solutions of these networks. Recent work on the probability of network states at those instants when a customer completes service at a service center (departure instants) or arrives at a service center (arrival instants) has revealed a similar product form. The arrival instant result provides a simple, intuitive explanation of the Mean Value Analysis solution technique for product form networks with First-Come-First-Served service centers.In this paper we derive the distribution of network states seen by a particular customer while resident at a particular service center. This distribution too has a relatively simple product form. We use this information to explain in an intuitive way the MVA solution technique for a more general class of network, those containing load independent Processor Sharing and last-Come-First-Served-Preemptive-Resume service centers. It is hoped that, just as the intuitive explanation of the response time formula for FCFS centers has led to approximation techniques for non-separable FCFS centers, this new information may provide approximation techniques for non-separable centers with other scheduling disciplines such as preemptive and non-preemptive priority.     

Violation Probability in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We focus here upon performance requirements in the form of an upper bound on the probability of failing to achieve a specified minimum transmission rate or a specified minimum average rate. For an M/G/l processor-sharing queue, we give a closed-form expression for this violation probability. We derive closed-form expressions for the marginal service rate with respect to the violation probability and to the minimum transmission rate, and characterize when each is binding. We then consider the effect of using connection access control by modeling an M/G/l/K processor-sharing queue, and discuss the relationship between queue service rate, queue limit, violation probability, and blocking probability. Finally, we consider a two-class discriminatory processor-sharing queue, and discuss what combinations of class weighting and service rate can be used to achieve specified minimum rate violation probabilities for both classes.     

CoMoM: Efficient Class-Oriented Evaluation of Multiclass Performance Models

We introduce the class-oriented method of moments (CoMoM), a new exact algorithm to compute performance indexes in closed multiclass queuing networks. Closed models are important for performance evaluation of multitier applications, but when the number of service classes is large, they become too expensive to solve with exact methods such as mean value analysis (MVA). CoMoM addresses this limitation by a new recursion that scales efficiently with the number of classes. Compared to the MVA algorithm, which recursively computes mean queue lengths, CoMoM also carries on in the recursion information on higher-order moments of queue lengths. We show that this additional information greatly reduces the number of operations needed to solve the model and makes CoMoM the best-available algorithm for networks with several classes. We conclude the paper by generalizing CoMoM to the efficient computation of marginal queue-length probabilities, which finds application in the evaluation of state-dependent attributes such as quality-of-service metrics.     

Entropy maximised queueing networks with blocking and multiple job classes

The method of entropy maximisation (MEM) is applied in a state space partitioning mode for the approximation of the joint stationary queue length distribution of an M/M/1/N queue with finite capacity, N( > 1), multiple and distinct classes of jobs, R( > 1), under a complete buffer sharing scheme and mixed service disciplines drawn from the first-come-first-served (FCFS), last-come-first-served with (LCFS-PR) or without (LCFS-NPR) preemption and processor sharing (PS) rules. The marginal and aggregate maximum entropy (ME) queue length distributions and the associated blocking probabilities per class are also determined. These ME results in conjunction with the first moments of the effective flows are used, as building blocks, in order to establish a new product-form approximation for arbitrary exponential open queueing networks with multiple classes of jobs under repetitive-service (RS) blocking with random destination (RD). It is verified that the ME approximation reduces to the exact truncated solution of open multi-class reversible queueing networks. Numerical experiments demonstrate a good accuracy level of ME statistics in relation to simulation. Moreover, recent extentions of MEM for arbitrary GE-type queueing networks with RS-RD blocking and multiple classes of jobs are presented.     

Throughput in Processor-Sharing Queues

Processor-sharing queues are often used to model file transmission in networks. While sojourn time is a common performance metric in the queueing literature, average transmission rate is the more commonly discussed metric in the networking literature. Whereas much is known about sojourn times, there is little known about the average service rate experienced by jobs in processor-sharing queues. We first define the average rate as observed by users and by the queue. In an M/M/1 processor-sharing queue, we give closed-form expressions for these average rates, and prove a strict ordering amongst them. We prove that the queue service rate (in bps) is an increasing function of the minimum required average transmission rate, and give a closed-form expression for the marginal cost associated with such a performance requirement. We then consider the effect of using connection access control by modeling an M/M/1/K processor-sharing queue. We give closed-form expressions for average transmission rates, and discuss the relationship between the queue service rate (in bps), the queue limit, the average rate, and the blocking probability     

Analysis of a two-layered network by means of the power-series algorithm

We consider an extension of the classical machine-repair model, also known as the computer-terminal model or time-sharing model. As opposed to the classical model, we assume that the machines, apart from receiving service from the repairman, supply service themselves to queues of products. The extended model can be viewed as a two-layered queueing network, of which the first layer consists of two separate queues of products. Each of these queues is served by its own machine. The marginal and joint queue length distributions of the first-layer queues are hard to analyse in an exact fashion. Therefore, we apply the power-series algorithm to this model to obtain the light-traffic behaviour of the queue lengths symbolically. This leads to two accurate approximations for the marginal mean queue length. The first approximation, based on the light-traffic behaviour, is in closed form. The second approximation is based on an interpolation between the light-traffic behaviour and heavy-traffic results for the mean queue length. The obtained approximations are shown to work well for arbitrary loaded systems. The proposed numerical algorithm and approximations may prove to be very useful for system design and optimisation purposes in application areas such as manufacturing, computer systems and telecommunications.     

Joining the right queue: a state-dependent decision rule

The problem of assigning customers to one of several parallel queues so as to minimize the average time spent in the system (sojourn time) is studied as a Markov decision process. It is shown how the approach developed by K.R. Krishman and T.J. Ott (Proc. 25th IEEE Conf. Decision Contr. Dec. 1986, p.2124-8) to investigate state-dependent routing of voice traffic for blocking minimization can also be used for sojourn minimization for data traffic. For queues in parallel, this approach produces a rule, called the `separable' rule, which is a generalization of the `join the shortest queue' rule to the case of dissimilar queues, reducing to the shortest queue rule when the queues are all alike. Numerical results show that in cases where the queues are dissimilar in both the service rates and numbers of their servers, the separable rule is strikingly superior to the shortest queue rule; if the dissimilarities are limited to differences in the service rates, the separable rule practically always is better than the shortest queue rule; if the dissimilarities consist only of the numbers of servers being different, then the shortest queue rule does better than the separable rule in most instances     

A generalized method of moments for closed queueing networks

We introduce a new solution technique for closed product-form queueing networks that generalizes the Method of Moments (MoM), a recently proposed exact algorithm that is several orders of magnitude faster and memory efficient than the established Mean Value Analysis (MVA) algorithm. Compared to MVA, MoM recursively computes higher-order moments of queue lengths instead of mean values, an approach that remarkably reduces the computational costs of exact solutions, especially on models with large numbers of jobs.In this paper, we show that the MoM recursion can be generalized to include multiple recursive branches that evaluate models with different numbers of queues, a solution approach inspired by the Convolution algorithm. Combining the approaches of MoM and Convolution simplifies the evaluation of normalizing constants and leads to large computational savings with respect to the recursive structure originally proposed for MoM.     

Analysis of a finite buffer queue with different scheduling and push-out schemes

A queueing system M₁, M₂/G₁, G₂/1/N with different scheduling and push-out scheme is analyzed in this paper. This work is motivated by the study of the performance of an output link of ATM switches with traffic of two classes with different priorities. However, the queueing model developed in this paper is more general than that of the output link of ATM switches with two-class priority traffic. General service time distributions are allowed for classes 1 and 2 and a general service discipline function, ₁(i, j), is introduced where ₁(i, j) is the probability that a class 1 packet will be served, given that there are i class 1 and j class 2 packets waiting for service. An exact solution is obtained for the loss probabilities for classes 1 and 2, the queue length distribution and the mean waiting time for class 1. The queue length distribution and the mean waiting time for class 2 are calculated approximately. It is shown that the approximation is an upper bound and the error due to the approximation is very small when the loss probability of class 2 is small (e.g., less than 0.01).     

Exact analysis of performance models by the Method of Moments

Performance evaluation of distributed systems and service-oriented architectures is often based on stochastic models, such as closed queueing networks which are commonly solved by the Mean Value Analysis (MVA) algorithm. However, the MVA is unable to solve models with hundreds or thousands of users accessing services of multiple classes, a configuration that is often useful to predict the performance of real-world applications. This paper introduces the Method of Moments (MoM), the first exact algorithm for solving closed queueing networks with large population sizes.Compared to the MVA algorithm, which is based on a recursive evaluation of mean queue-lengths, MoM defines a recursion on higher-order moments of queue-lengths that is solved at each step by a linear system of equations. This approach dramatically decreases the costs of an exact analysis compared to the MVA approach. We prove that MoM requires log-quadratic time and log-linear space in the total population size, whereas MVA complexity expressions grow combinatorially as the product of class populations. This extends the feasibility of exact methods to a much larger family of multiclass performance models than those that can be solved by the MVA algorithm.     

Lead time control in multi-class multi-stage assembly systems with finite capacity

In this paper, we model multi-class multi-stage assembly systems with finite capacity as queueing networks. It is assumed that different classes (types) of products are produced by the production system and products’ orders for different classes are received according to independent Poisson processes. Each service station of the queueing network specifies a manufacturing or assembly operation, in that processing times for different types of products are independent and exponentially distributed random variables with service rates, which are controllable, and the queueing discipline is First Come First Served (FCFS). Different types of products may be different in their routing sequences of manufacturing and assembly operations. For modeling multi-class multi-stage assembly systems, we first consider every class separately and convert the queueing network of each class into an appropriate stochastic network. Then, by using the concept of continuous-time Markov processes, a system of differential equations is created to obtain the distribution function of manufacturing lead time for any type of product, which is actually the time between receiving the order and the delivery of finished product. Furthermore, we develop a multi-objective model with three conflicting objectives to optimally control the service rates, and use goal attainment method to solve a discrete-time approximation of the original multi-objective continuous-time problem.     

TinyOS中多优先级任务队列调度策略研究

针对TinyOS先来先服务调度策略中重要任务不能及时响应的不足,提出一种基于多优先级任务队列的调度策略。该调度策略将原来一个任务队列增加为三个优先级队列并引入抢占机制,最高优先级队列中的任务在满足抢占原则时才可以抢占其他队列正在执行的任务,任务只能在不同队列之间发生抢占,这样既减少了上下文切换,又保证了重要任务的优先执行。实验结果表明,该调度策略在不影响原有系统性能的情况下,提高了TinyOS对重要任务的响应性能。     

A new blocking problem from Java-based schedulers

We consider a form of blocking, which is typical in client–server systems including those implemented under the Enterprise JavaBean (EJB) specification. The novel feature is that tasks must wait for one of a number of parallel queues to clear its outstanding work. Thus, blocking time is the minimum of sojourn times at the parallel queues. Under certain simplifying assumptions, we solve this model for the probability distribution of blocking time and obtain a simple formula for its mean value. We then use this result in an aggregate server model of a larger queueing network in which further non-standard techniques are included to represent this form of blocking. We compare our approximate results against simulation data, obtaining good agreement for both system throughput and queue length probability distributions at equilibrium.     

Performance analysis of networks of queues under active queue management scheme

Analysis of networks of queues under repetitive service blocking mechanism has been presented in this paper. Nodes are connected according to an arbitrary configuration and each node in the networks employs an active queue management (AQM) based queueing policy to guarantee certain quality of service for multiple class external traffic. This buffer management scheme has been implemented using queue thresholds. The use of queue thresholds is a well known technique for network traffic congestion control. The analysis is based on a queue-by-queue decomposition technique where each queue is modelled as a GE/GE/1/N queue with single server, R (R ⩾ 2) distinct traffic classes and {N = N₁, N₂, … , N_R} buffer threshold values per class under first-come-first-serve (FCFS) service rule. The external traffic is modelled using the generalised exponential (GE) distribution which can capture the bursty property of network traffic. The analytical solution is obtained using the maximum entropy (ME) principle. The forms of the state and blocking probabilities are analytically established at equilibrium via appropriate mean value constraints. The initial numerical results demonstrate the credibility of the proposed analytical solution.     

Modelling manufacturing systems: A birth-death approach

An approximation method for modelling a manufacturing system is introduced. The system is considered as a queueing network, where each queue is limited in size, and interarrival and processing times are exponentially distributed. The birth-death approach is considered and an approximation method to reduce the dimension of the model is developed. The results are the marginal probability distribution of the number of units in each queue; other performance indices, such as mean queue lengths, utilizations of the working stations, and throughput can be easily obtained. The general procedure is applied to model, for example, queues in tandem, a split node, and a more complex network of queues. Simulation and, when possible, comparison with the exact solution show an acceptable error level of the proposed method.     

A maximum entropy priority approximation for a stableG/G/1 queue

Summary The principle of maximum entropy is used under two different sets of mean value constraints to analyse a stableG/G/1 queue withR priority classes under preemptive-resume (PR) and non-preemptive head-of-line (HOL) scheduling disciplines. New one-step recursions for the maximum entropy state probabilities are established and closed form approximations for the marginal queue length distribution per priority class are derived. To expedite the utility of the maximum entropy solutions exact analysis, based on the generalised exponential (GE) distribution, is used to approximate the marginal mean queue length and idle state probability class constraints for both the PR and HOLG/G/1 priority queues. Moreover, these results are used as building blocks in order to provide new approximate formulae for the mean and coefficient of variation of the effective priority service-time and suggest a maximum entropy algorithm for general open queueing networks with priorities in the context of the reduced occupancy approximation (ROA) method. Numerical examples illustrate the accuracy of the proposed maximum entropy approximations in relation to simulations involving different interarrival-time and service-time distributions per class. Comments on the extension of the work to more complex types of queueing systems are included.This work is sponsored in part by the Science and Engineering Research Council (SERC), UK, under grant GR/D/12422 and in part by the Ministry of Higher Education of the Algerian Government     

A probabilistic approach to estimate the mean waiting times in the earliest deadline first polling

In queueing system, the mean waiting times of messages are important measures to characterize the quality of service (QoS) under various requirements. In a time-critical system, message transactions which cannot meet deadline constraints might lead to catastrophic consequences. Currently, the waiting time estimations using the first-come-first-served (FCFS) and priority (PRI) strategies are already well developed. However, in the case of multi-queue dynamic environments, these quantities are more difficult to analyze due to multiple classes of messages are considered. In this paper, we aim to consider a polling system consisting of a number of parallel infinite-capacity single-server queues. We propose a probabilistic approach to derive the waiting times for different classes of messages by using non-preemptive earliest deadline first (EDF) polling policy. The resulting formula can also lead to the FCFS polling and PRI polling by altering the relative deadlines. Moreover, the bounds of waiting times are discussed. The accuracy of the proposed algorithm is established by comparisons with simulation results. The runtime results are in very good convergence with the theoretical predictions made by our formulas, in terms of prediction accuracies of waiting times and untimely service ratios of messages under various scenarios and timing constraints.     

Higher-order distributional properties in closed queueing networks

In many real-life computer and networking applications, the distributions of service times, or times between arrivals of requests, or both, can deviate significantly from the memoryless negative exponential distribution that underpins the product-form solution for queueing networks. Frequently, the coefficient of variation of the distributions encountered is well in excess of one, which would be its value for the exponential. For closed queueing networks with non-exponential servers there is no known general exact solution, and most, if not all, approximation methods attempt to account for the general service time distributions through their first two moments.We consider two simple closed queueing networks which we solve exactly using semi-numerical methods. These networks depart from the structure leading to a product-form solution only to the extent that the service time at a single node is non-exponential. We show that not only the coefficients of variation but also higher-order distributional properties can have an important effect on such customary steady-state performance measures as the mean number of customers at a resource or the resource utilization level in a closed network.Additionally, we examine the state that a request finds upon its arrival at a server, which is directly tied to the resulting quality of service. Although the well-known Arrival Theorem holds exactly only for product-form networks of queues, some approximation methods assume that it can be applied to a reasonable degree also in other closed queueing networks. We investigate the validity of this assumption in the two closed queueing models considered. Our results show that, even in the case when there is a single non-exponential server in the network, the state found upon arrival may be highly sensitive to higher-order properties of the service time distribution, beyond its mean and coefficient of variation.This dependence of mean numbers of customers at a server on higher-order distributional properties is in stark contrast with the situation in the familiar open M/G/1 queue. Thus, our results put into question virtually all traditional approximate solutions, which concentrate on the first two moments of service time distributions.