Cycle time distribution of cyclic networks with blocking

An algorithm is provided to compute the cycle time distribution for cyclic closed exponential queueing networks with N finite capacity nodes and blocking. The blocking phenomenon is due to the nodes with finite capacity queues which can be used to represent systems with limited resources. Moreover, a recursive algorithm is provided to evaluate the cycle time moments, and a closed form solution, both in terms of distribution and moments of the cycle time, is provided for the model under special constraints.     

An approximate MVA algorithm for exponential, class-dependent multiple servers

Mean value analysis (MVA) is an efficient algorithm for determining the mean sojourn time, the mean queue length, and the throughput in a closed multiclass queueing network. It provides exact results for the class of product-form networks. Often different classes have different service requirements in FCFS queues, but such networks are not of product form. There are several possibilities to compute performance measure for such nodes and networks. In this paper we present an approximation formula for multiple-server FCFS queues with class-dependent service times as a Norton flow equivalent product node, where the departure rate of any class depends on the number of customers of all classes in the queue. We will use this approximation in the sojourn time formula of some exact and approximate MVA algorithms.     

Markov-chain model and algorithmic procedure for the performance analysis of closed cyclic queues

This paper presents a model and an algorithmic procedure to analyze closed cyclic queues that are subject to blocking. We consider the first two moments of the processing time and present the fitting of phase-type distributions such that the number of phases and transitions is minimal. Using phase-type distributions, we enable the analysis of queueing systems with processing times with any coefficient of variation. We model the closed cyclic queues subject to blocking as continuous-time Markov chains. The implementation procedure covers the state-space generation and the determination of the infinitesimal generator matrix. Apart from rounding errors, we obtain exact results for the queueing model this way. The results are useful as reference values for the output of approximate approaches. Further, the algorithmic procedure enables a repeated analysis of different configuration alternatives as needed in optimization procedures. Though the method is very fast for small cyclic queues, it takes a long computation time for larger systems. Furthermore, the size of the queueing model to be analyzed is restricted due to the limited working memory with its present-day capacity. In a numerical study, the computation times for different configurations are investigated, limits in the size of the applicable queueing model are given, and numerical results of the performance measures are provided.     

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.     

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.     

MRE hierarchical decomposition of general queueing network models

Summary The principle of Minimum Relative Entropy (MRE), given fully decomposable subset and aggregate mean queue length, utilisation and flow-balance constraints, is used in conjunction with asymptotic connections to infinite capacity queues, to derive new analytic approximations for the conditional and marginal state probabilities of single class general closed queueing network models (QNMs) in the context of a multilevel variable aggregation scheme. The concept of subparallelism is applied to preserve the flow conservation and a universal MRE hierarchical decomposition algorithm is proposed for the approximate analysis of arbitrary closed queueing networks with single server queues and general service-times. Heuristic criteria towards an optimal coupling of network's units at each level of aggregation are suggested. As an illustration, the MRE algorithm is implemented iteratively by using the Generalised Exponential (GE) distributional model to approximate the service and asymptotic flow processes in the network. This algorithm captures the exact solution of separable queueing networks, while for general queueing networks it compares favourably against exact solutions and known approximations.This work is sponsored by the Science and Engineering Research Council (SERC), UK, under grant GR/F29271     

Perfect sampling for closed queueing networks

In this paper we investigate coupling from the past (CFTP) algorithms for closed queueing networks. The stationary distribution has a product form only in a very limited number of particular cases when queue capacity is finite, and numerical algorithms are intractable due to the cardinality of the state space. Moreover, closed networks do not exhibit any monotonic property enabling efficient CFTP. We derive a bounding chain for the CFTP algorithm for closed queueing networks. This bounding chain is based on a compact representation of sets of states that enables exact sampling from the stationary distribution without considering all initial conditions in the CFTP. The coupling time of the bounding chain is almost surely finite, and numerical experiments show that it is close to the coupling time of the exact chain.     

A recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks

This paper is concerned with reliable multistation series queueing networks. Items arrive at the first station according to a Poisson distribution and an operation is performed on each item by a server at each station. Every station is allowed to have more than one server with the same characteristics. The processing times at each station are exponentially distributed. Buffers of nonidentical finite capacities are allowed between successive stations. The structure of the transition matrices of these specific type of queueing networks is examined and a recursive algorithm is developed for generating them. The transition matrices are block-structured and very sparse. By applying the proposed algorithm the transition matrix of a K-station network can be created for any K. This process allows one to obtain the exact solution of the large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures can be calculated.Scope and purposeThe exact analysis of queueing networks with multiple servers at each workstation and finite capacities of the intermediate queues is extremely difficult as for even the case of exponential operation (service or processing) times the Markovian chain that models the system consists of a huge number of states which grows exponentially with the number of stations, the number of servers at each station and the queue capacity of each intermediate queue of the resulting system. The scope and purpose of the present paper is to analyze and provide a recursive algorithm for generating the transition matrices of multistation multiserver exponential reliable queueing networks. By applying the proposed algorithm one may create the transition matrix of a K-station queueing network for any K. This process allows one to obtain the exact solution of the resulting large sparse linear system by the use of the Gauss–Seidel method. From the solution of the linear system the throughput and other performance measures of the system can be obtained.     

On the nonconcavity of throughput in certain closed queueing networks

Analytic queueing network models are being used to analyze various optimization problems such as server allocation, design and capacity issues, optimal routing, and workload allocation. The mathematical properties of the relevant performance measures, such as throughput, are important for optimization purposes and for insight into system performance.We show that for closed queueing networks of m arbitrarily connected single server queues with n customers, throughput, as a function of a scaled, constrained workload, is not concave. In fact, the function appears to be strictly quasiconcave. There is a constraint on the total workload that must be allocated among the servers in the network. However, for closed networks of two single server queues, we prove that our scaled throughput is concave when there are two customers in the network and strictly quasi-concave when there are more than two customers. The mathematical properties of both the scaled throughput and reciprocal throughput are demonstrated graphically for closed networks of two and three single server queues.     

非乘积解随机Petri网的乘积形式近似求解

讨论了非乘积解随机Petri网的近似求解问题。将Marie方法引入到随机Petri网的近似分析中,利用随机Petri网中已有的结论将该方法中的分解原则推广到更一般的情形,使其应用范围更广,利用运算分析法对这些分解原则作了形式化描述,在此基础上,给出了有关结论的数学证明。最后,对这种近似方法作了误差分析,找出了产生误差的原因。实验数据表明本文所给的近似方法应用广且有效。     

Performance Analysis of Retrial Queueing Systems Using Generalized Stochastic Petri Nets

We consider retrial queueing systems, in which an arriving customer finding the server busy, may repeat his call after a random duration. The consideration of repeated calls introduces great analytical difficulties. In fact, detailed analytical results exist for some special retrial queueing systems, while for many others, the performance evaluation is limited to numerical algorithms, approximation methods and simulation. Retrial queues have been widely used to model many problems in telephone switching systems, telecommunication and computer networks. In this paper, we show a method of modelling and analysing retrial queueing systems, using the Generalized Stochastic Petri nets (GSPNs).     

A hybrid method for performance analysis of G/G/m queueing networks

Open queueing networks are useful for the performance analysis of numerous real systems. Since exact results exist only for a limited class of networks, decomposition methods have been extensively used for approximate analysis of general networks. This procedure is based on several approximation steps. Successive approximations made in this approach can lead to a considerable error in the output. In particular, there are no general accurate formulas for computing the mean waiting time and the inter-departure variance in general multiple-server queues. This causes the results from decomposition methods when applied to G/G/m queueing networks to be very approximative and to significantly deviate from actual performance values. We suggest substituting some approximate formulae by low-cost simulation estimates in order to obtain more accurate results when benefiting from the speed of an analytical method. Numerical experiments are presented to show that the proposed approach provides improved performance.     

Fluid limits and diffusion approximations for networks of multi-server queues in heavy traffic

A Functional Strong-Law-of-Large-Numbers (known as fluid limit or fluid approximation) and a Functional Central Limit Theorem (known as diffusion approximation) are proved for both open and closed networks of multi-server queues in heavy traffic. The fluid limit is a reflected piecewise linear deterministic process, and the diffusion limit is a reflected Brownian motion; both limiting processes are on the nonnegative orthant for open networks and on the nonnegative unit simplex for closed networks. The results generalize the existing ones for networks of single-server queues.     

Optimal buffer allocation in finite closed networks with multiple servers

In this study, we present an optimal buffer allocation procedure for closed queueing networks with finite buffers. The performance measures are evaluated using the expanded mean value analysis, and the solution procedure is incorporated into a nonlinear optimization scheme to arrive at the sub-optimal buffer space vector. The effectiveness of the method is demonstrated through several numerical experiments. Discussions on convergence and computational complexity are also included.     

Equivalencies between open and closed queueing networks with finite buffers

Equivalencies between open and closed models of queueing networks with finite buffers are investigated. Interarrival times in open networks and service times are assumed to be exponentially distributed. Using these equivalencies, bounds on the throughput of open networks are established.     

A review on queueing network models with finite capacity queues for software architectures performance prediction

A review is carried out on how queueing network models with blocking have been applied so far into the performance evaluation and prediction of Software Architectures (SA). Queueing network models with finite capacity queues and blocking have recently been introduced and applied as more realistic models of systems with finite capacity resources and population constraints. Queueing network models have been often adopted as models for the evaluation of software performance. Starting from our own experience, we observe the need of a more accurate definition of the performance models of SA to capture some features of the communication systems. We consider queueing networks with finite capacity and blocking after service (BAS) to represent some synchronization constraints that cannot be easily modeled with queueing network models with infinite capacity queues. We investigate the use of queueing networks with blocking as performance models of SA with concurrent components and synchronous communication. Queueing theoretic analysis is used to solve the queueing network model and study the synchronous communication and performance of concurrent software components. Our experience is supported by other approaches that also propose the use of queueing networks with blocking. Directions for future research work in the field are included.     

Mean-delay approximation for cyclic-service queueing systems

This paper investigates mean delays in a single-server queueing model with cyclic service. An approximate analysis is performed for an arbitrary number of queues, constant switch-over time, exhaustive service discipline, Poisson arrival processes, and general service-time distributions. Neither inter-arrival nor service-time distributions have to be identical for the different queues. A major value of our approximation lies in the simplicity of its numerical evaluation for an arbitrary number of queues and any traffic pattern. Extensive comparisons with simulation results show the high accuracy of the approximation over a wide range of parameters.     

QBD approximations of a call center queueing model with general patience distribution

This paper investigates a computationally practical way for analyzing a call center queueing model, i.e., a finite-capacity, multi-server queueing model, where each server goes on a single vacation. Poisson arrival process and exponential service and vacation times are assumed. We also assume that each customer may leave the queue due to impatience. Customers’ patience times are i.i.d. random variables with a general distribution. Level-dependent finite QBD (quasi-birth–death) processes are employed to approximate such a queueing model. Two approaches are considered. The first one uses the phase-type (PH) distribution to approximate the general patience distribution, whereas the second one is based on the idea of replacing the eventual reneging of customers with balking. We find that the first approach is almost impossible to compute numerically due to the exponential growth of the size of the block matrices in a level-dependent finite QBD. We examine the validity and applicability of the approximation based on the second approach and show that it gives us a practical way to obtain performance measures of call center systems in practical scale with sufficiently reasonable accuracy.     

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     

Buffer allocation in general single-server queueing networks

The optimal buffer allocation in queueing network systems is a difficult stochastic, non-linear, integer mathematical programming problem. Moreover, the objective function, the constraints or both are usually not available in closed form, making the problem even harder. A good approximation for the performance measures is thus essential for a successful buffer allocation algorithm. A recently published two-moment approximation formula to obtain the optimal buffer allocation in general service time single queues is examined in detail, based on which a new algorithm is proposed for the buffer allocation in single-server general service time queueing networks. Computational results and simulation results are shown to evaluate the efficacy of the approach in generating optimal buffer allocation patterns.