Asymptotic properties of queuing networks

IEEE Transactions on Software Engineering, 2000

In this paper, we present an approximate solution for the asymptotic behavior of relatively general queueing networks. In the particular case of networks with general service time distributions (i.e., fixed routing matrix, one or many servers per station, FIFO discipline), the application of the method gives relatively accurate results in a very short time. The approximate stationary state probabilities are identified with the solution of a nonlinear system. The proposed method is applicable to a larger class of queueing networks (dependent routing matrix, stations with fimite capacity, etc.). In this case, the structure of the network studied must satisfy certain decomposability conditions.

Queueing networks with finite capacity queues and blocking are applied to model systems with finite resources and population constraints, such as computer and communication systems, as well as traffic, production and manufacturing systems. Various blocking types can be defined to represent different system behaviors. When a customer attempts to enter a full capacity queue blocking occurs. The analysis of queueing networks with finite capacity queues is often based on approximate methods and simulation, since exact analytical techniques cannot be applied because of the synchronization constraint, except for a few special cases. Various approximate analytical methods have been proposed in literature and provide a solution in terms of average performance indices such as throughput and mean response time. These methods have different characteristics including model assumptions and constraints, type of blocking, algorithm accuracy and efficiency. We present a comparison of some significant approximate methods to analyze closed queueing networks with finite capacity queues. By experimental results we identify the condition under which one can appropriately select a given solution method. Experimental comparisons have been performed by the Queueing Networks with Blocking Analyzer (QNBA), a software tool developed for modelling and analysis of queueing network models with finite capacity queues and blocking.

Journal of the ACM, 1995

A new algorithm is developed for calculating normalization constants (partition functions) and moments of product-form steady-state distributions of closed queueing networks and related models. The essential idea is to numerically invert the generating function of the normalization constant and related generating functions appearing in expressions for the moments. It is known that the generating function of the normalization constant often has a remarkably simple form, but numerical inversion evidently has not been considered before. For p-dimensional transforms, as occur with queueing networks having p closed chains, the algorithm recursively performs p onedimensional inversions. The required computation grows exponentially in the dimension, but the dimension can often be reduced by exploiting conditional decomposition based on special structure. For large populations, the inversion algorithm is made more efficient by computing large sums using Euler summation. The inversion algorithm also has a very low storage requirement. A key ingredient in the inversion algorithm is scaling. An effective static scaling is developed for multichain closed queueing networks with only single-server and (optionally) infinite-server queues. An important feature of the inversion algorithm is a self-contained accuracy check, which allows the results to be verified in the absence of alternative algorithms.

Acta Informatica, 1976

We develop a method based on diffusion approximations in order to compute, under general conditions, the queue length distribution of a single queue in a network of queues. Several applications of this approach to computer network performance analysis and to time-sharing systems are presented. The accuracy of model predictions are evaluated by comparison with known exact results in particular cases, with simulation experiments and with the approximation method of Kobayashi and Reiser.

Lecture Notes in Computer Science, 1993

1999

Asymptotic behavior of queues is studied for large closed multi-class queueing networks consisting of one in nite server station with K classes and M processor sharing (PS) stations. A simple numerical procedure is derived that allows us to identify all bottleneck PS stations. The bottleneck station is de ned asymptotically as the station where the number of customers grows proportionally to the total number of customers in the network, as the latter increases simultaneously with service rates at PS stations. For the case when K = M = 2, the set of network parameters is identi ed that corresponds to each of the three possible types of behavior in heavy tra c: both PS stations are bottlenecks, only one PS station is a bottleneck, and a group of two PS stations is a bottleneck while neither PS station forms a bottleneck by itself. In the last case both PS stations are equally loaded by each customer class and their individual queue lengths, normalized by the large parameter, converge to uniformly distributed random variables.

Performance Evaluation, 1988

Analytical lower and upper bounds for the throughput of closed queueing networks with single and delay (infinite) servers are studied in this paper. The numerical evaluation of these bounds requires a small number of significant operations which is independent of the population N. This is in contrast to the exact computation of the throughput which requires at least O(N) operations as N tends to infinity. The bounds are given by simple closed-form analytical expressions and may be more suitable for various performance studies than the algorithmical form of the exact solution. In this paper, the previously known balanced-job bounds are generalized to networks containing delay servers (terminals) and a hierarchy of bounds is obtained for single and multiple class networks. For the single class network, further new bounds are derived: lower and upper bounds that require the evaluation of one square root and an upper bound that requires a constant number of exponentiations. This upper bound does not employ the balancing of server loadings and is especially useful for asymptotic analysis in the case of a large number of customers N.

We introduce a new framework supporting the bottleneck analysis of closed, multiclass BCMP queueing networks in the limiting regime where the number of jobs proportionally grows to infinity while keeping fixed other input parameters. First, we provide a weak convergence result for the limiting behavior of closed queueing networks, which is exploited to derive a sufficient and necessary condition establishing the existence of a single bottleneck. Then, we derive the new framework proposing efficient algorithms for the identification of queueing networks bottlenecks by means of linear programming. Our analysis reduces the computational requirements of existing techniques and, under general assumptions, it is able to handle load-dependent stations. We also establish a primal-dual relationship between our approach and a recent technique. This connection lets us extend the dual to deal with load-dependent stations, which is non-intuitive, and provides a unified framework for the enumeration of bottlenecks. Theoretical and practical insights on the asymptotic behavior of multiclass networks are shown as application of the proposed framework.

Humanitarian & Natural Sciences Journal, 2024

Queuing is one of the most usable tools that help in analyzing the performance of complex telecommunication and system networks. Thus, this term paper presents the performance measurements of computer networks with queuing technique. The paper covers the detail introduction of queuing theory and its various applications widely used for complex network/system environment.

Computers & Operations Research, 2013

This paper presents a new approach to evaluate the performance of general multi-class closed queuing networks. The approach uses parametric characterization of the traffic processes to derive two-moment approximations for performance measures at individual nodes. Based on these approximations, linkage equations are derived to establish the relationships between the various nodes in the network. These relationships result in a system of non-linear equations that is solved using an iterative procedure. Numerical studies comparing the performance of the approach with detailed simulations suggest that the approach yields fairly accurate estimates of performance measures without significant computational complexity.