Separable Queueing Networks Research Articles

Solution properties and convergence of an approximate mean value analysis algorithm

We present the solution properties and convergence results of an approximate Mean Value Analysis (MVA) algorithm, the Queue Line (QL) algorithm, for solving separable queueing networks. We formally prove that the QL algorithm is always more accurate than, and yet has the same computational complexity as the Bard-Schweitzer Proportional Estimation algorithm, the most popular approximate MVA algorithm for solving this type of queueing networks.

Experiments with improved approximate mean value analysis algorithms

Approximate mean value analysis (MVA) is a popular technique for analyzing queueing networks because of the efficiency and accuracy that it affords. In this paper, we present a new software package, called the improved approximate mean value analysis library (IAMVAL), which can be easily integrated into existing commercial and research queueing network analysis packages. The IAMVAL packages include two new approximate MVA algorithms, the queue line (QL) algorithm and the fraction line (FL) algorithm, for analyzing multiple class separable queueing networks. The QL algorithm is always more accurate than, and yet has approximately the same computational efficiency as, the Bard–Schweitzer proportional estimation (PE) algorithm, which is currently the most widely used approximate MVA algorithm. The FL algorithm has the same computational cost and, in noncongested separable queueing networks where queue lengths are quite small, yields more accurate solutions than both the QL and PE algorithms.

A decomposition approximation method for multiclass BCMP queueing networks with multiple-server stations

Closed multiclass separable queueing networks can in principle be analyzed using exact computational algorithms. This, however, may not be feasible in the case of large networks. As a result, much work has been devoted to developing approximation techniques, most of which is based on heuristic extensions of the mean value analysis (MVA) algorithm. In this paper, we propose an alternative approximation method to analyze large separable networks. This method is based on an approximation method for non-separable networks recently proposed by Baynat and Dallery. We show how this method can be efficiently used to analyze large separable networks. It is especially of interest when dealing with networks having multiple-server stations. Numerical results show that this method has good accuracy.

MRE hierarchical decomposition of general queueing network models

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.

Analysis of a class of distributed queues with applications

Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various methods by which the processors in the LAN can simulate the availability of centralized packet transport facility, but whose service incorporates a particular type of changeover time known as ‘moving server’ overhead. First we describe the operation of moving server systems in general, for both First-Come-First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis where we transform the moving server system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a major component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors.

PAM—A noniterative approximate solution method for closed multichain queueing networks

Approximate MVA algorithms for separable queueing networks are based upon an iterative solution of a set of modified MVA formulas. Although each iteration has a computational time requirement of O( MK 2) or less, many iterations are typically needed for convergence to a solution. ( M denotes the number of queues and K the number of closed chains or customer classes.) We present some faster approximate solution algorithms that are noniterative. They are suitable for the analysis and design of communication networks which may require tens to hundreds, perhaps thousands, of closed chains to model flow-controlled virtual channels. The basis of our method is the distribution of a chain's population proportional to loads to get initial estimates of mean queue lengths. This is the same basis used in the derivation of proportional upper bounds for single-chain networks; for a multichain network, such a proportional distribution leads to approximations rather than upper bounds of chain throughputs. Nevertheless, these approximate solutions provide chain throughputs, mean end-to-end delays, and server utilizations that are sufficiently accurate for the analysis and design of communication networks and possibly other distributed systems with a large number of customer classes. Three PAM algorithms of increasing accuracy are presented. Two of them have time and space requirements of O( MK). The third algorithm has a time requirement of O( MK 2) and a space requirement of O( MK).

An approximate product form for a class of degradable queueing networks

An approximate product form for a class of degradable queueing networks (DQNs) is presented. DQNs emerge from a synthesis of the well-known separable queueing networks (BCMP networks) and the classical machine-repair-man model. Under the assumption that failure and repair events which change the level of performance are much less frequent compared with the service completions of tasks moving through the network, the presented product from approximates the exact probability distribution very closely. Because DQNs are nearcompletely decomposable, the error of approximation depends on the degree of coupling between the different modes of degraded performance. In accordance with the theory of nearcomplete decomposability, a numerical example shows that the approximation error is proportional to the degree of coupling.

PAM-a noniterative approximate solution method for closed multichain queueing networks

Approximate MVA algorithms for separable queueing networks are based upon an iterative solution of a set of modified MVA formulas. Although each iteration has a computational time requirement of O(MK 2 ) or less, many iterations are typically needed for convergence to a solution. ( M denotes the number of queues and K the number of closed chains or customer classes.) We present some faster approximate solution algorithms that are noniterative . They are suitable for the analysis and design of communication networks which may require tens to hundreds, perhaps thousands, of closed chains to model flow-controlled virtual channels. Three PAM algorithms of increasing accuracy are presented. Two of them have time and space requirements of O(MK) . The third algorithm has a time requirement of O(MK 2 ) and a space requirement of O(MK) .

Two classes of performance bounds for closed queueing networks

Two classes of performance bounds for separable queueing networks are presented, one for single-chain networks and one for multichain networks. Unlike most bounds for single-chain networks, our bounds are not based upon the consideration of balanced networks. Instead, they are obtained by assuming mean queue lengths to be proportional to server loads; hence, they are called proportional bounds. Proportional bounds are tighter than balanced bounds because individual server loads are retained as parameters in a bound's formula. For the same reason, they require more computational effort than balanced bounds. We also show how proportional bounds are related to balanced bounds. Next we present generalized bounds that are calculated iteratively over sequences of population sizes; our method extends that of Eager and Sevcik (1983). These generalized bounds are shown to have a nested property. Furthermore, we present optimal population sequences, over all sequences of the same length, for getting the tightest upper and lower bounds. The other emphasis of this paper is on performance bounds for networks with many closed chains and many service centers. Bounding techniques are especially important for multichain networks since the computation time and space requirements are often so large that an exact solution is not feasible. Models of communication networks typically have many routing chains which are characterized by a sparseness property. In the computation of our performance bounds for multichain networks, we improve their accuracy by making use of routing information and exploiting the sparseness property.

An analysis of an approximation algorithm for queueing networks

Queueing network models have proved to be useful aids in computer system performance prediction. Numerous approximation algorithms have been devised to allow queueing networks to be analyzed with a reduced computational expense. In particular, for separable queueing networks several popular iterative approximation algorithms are available. One disadvantage of these algorithms has been the lack of results concerning their behavior, such as whether or not iteration convergence is guaranteed. This paper presents an analysis of an approximate MVA algorithm proposed by Schweitzer (1979). It is proved that the equations defining the algorithm have a unique solution when there is only a single customer class, and iteration initializations that yield monotonic convergence to this solution are exhibited. It is also proved that the solution is pessimistic relative to the exact queueing network solution.

Performance bound hierarchies for queueing networks

In applications of queueing network models to computer system performance prediction, the computational effort required to obtain an exact equilibrium solution of a model may not be justified by the accuracy actually required. In these cases, there is a need for approximation or bounding techniques that can provide the necessary information at reduced cost. This paper presents Performance Bound Hierarchies (PBHs) for single class separable queueing networks consisting of fixed rate and delay service centers. A PBH consists of a hierarchy of upper (pessimistic) or lower (optimistic) bounds on mean system residence time. (The bounds can also be expressed as bounds on system throughput or center utilizations.) Each successive member requires more computational effort, and in the limit, the bounds converge to the exact solution.

A decomposition approach to modelling high service time variability

The predictive ability of queueing network models can be greatly enhanced if these models include the effects of system characteristics such as high service time variability and simultaneous resource possession, which violate the assumptions required for their efficient exact solution. In this paper we present a new approximate solution technique for queueing networks that include Coxian servers to represent resources at which customers have high service time variability. Our approach is unique in several respects: it is based directly on the theory of near-complete decomposability, it is non-iterative (performance measures for the queueing network of interest are expressed as linear combinations of the performance measures of a set of separable queueing networks), and it is conceptually and computationally simple.

A decomposition approach to modelling high service time variability

Abstract The predictive ability of queueing network models can be greatly enhanced if these models include the effects of system characteristics such as high service time variability and simultaneous resource possession, which violate the assumptions required for their efficient exact solution. In this paper we present a new approximate solution technique for queueing networks that include Coxian servers to represent resources at which customers have high service time variability. Our approach is unique in several respects: it is based directly on the theory of near-complete decomposability, it is non-iterative (performance measures for the queueing network of interest are expressed as linear combinations of the performance measures of a set of separable queueing networks), and it is conceptually and computationally simple.

Multiple class memory constrained queueing networks

Most computer systems have a memory constraint: a limit on the number of requests that can actively compete for processing resources, imposed by finite memory resources. This characteristic violates the conditions required for queueing network performance models to be <i>separable,</i> i.e., amenable to efficient analysis by standard algorithms. Useful algorithms for analyzing models of memory constrained systems have been devised only for models with a single customer class. In this paper we consider the multiple class case. We introduce and evaluate an algorithm for analyzing multiple class queueing networks in which the classes have independent memory constraints. We extend this algorithm to situations in which several classes share a memory constraint. We sketch a generalization to situations in which a subsystem within an overall system model has a population constraint. Our algorithm is compatible with the extremely time- and space-efficient iterative approximate solution techniques for separable queueing networks. This level of efficiency is mandatory for modelling large systems.

Performance bound hierarchies for queueing networks

In applications of queueing network models to computer system performance prediction, the computational effort required to obtain an exact equilibrium solution of a model may not be justified by the accuracy actually required. In these cases, there is a need for approximation or bounding techniques that can provide the necessary information at reduced cost. This paper presents Performance Bound Hierarchies (PBHs) for single class separable queueing networks consisting of fixed rate and delay service centers. A PBH consists of a hierarchy of upper (pessimistic) or lower (optimistic) bounds on mean system residence time. (The bounds can also be expressed as bounds on system throughput or center utilizations.) Each successive member requires more computational effort, and in the limit, the bounds converge to the exact solution.

Balanced job bound analysis of queueing networks

Applications of queueing network models to computer system performance prediction typically involve the computation of their equilibrium solution. When numerous alternative systems are to be examined and the numbers of devices and customers are large, however, the expense of computing the exact solutions may not be warranted by the accuracy required. In such situations, it is desirable to be able to obtain bounds on the system solution with very little computation. Asymptotic bound analysis (ABA) is one technique for obtaining such bounds. In this paper, we introduce another bounding technique, called balanced job bounds (BJB), which is based on the analysis of systems in which all devices are equally utilized. These bounds are tighter than ABA bounds in many cases, but they are based on more restrictive assumptions (namely, those that lead to separable queueing network models).

The solution of separable queueing network models using mean value analysis

Because it is more intuitively understandable than the previously existing convolution algorithms, Mean Value Analysis (MVA) has gained great popularity as an exact solution technique for separable queueing networks. However, the derivations of MVA presented to date apply only to closed queueing network models. Additionally, the problem of the storage requirement of MVA has not been dealt with satisfactorily. In this paper we address both these problems, presenting MVA solutions for open and mixed load independent networks, and a storage maintenance technique that we postulate is the minimum possible of any “reasonable” MVA technique.

Balanced job bound analysis of queueing networks

Applications of queueing network models to computer system performance prediction typically involve the computation of their equilibrium solution. When numerous alternative systems are to be examined and the numbers of devices and customers are large, however, the expense of computing the exact solutions may not be warranted by the accuracy required. In such situations, it is desirable to be able to obtain bounds on the system solution with very little computation. Asymptotic bound analysis (ABA) is one technique for obtaining such bounds. In this paper, we introduce another bounding technique, called balanced job bounds (BJB), which is based on the analysis of systems in which all devices are equally utilized. These bounds are tighter than ABA bounds in many cases, but they are based on more restrictive assumptions (namely, those that lead to separable queueing network models).

An exact solution method for the general class of closed separable queueing networks

In this paper we present a convolution algorithm for the full class of closed, separable queueing networks. In particular, the algorithm represents an alternative method to those already known for the solution of networks with class changes, and is the first efficient algorithm to deal with Lam-type networks [11]. As an application of the algorithm, we study a simple queueing network with disk I/O devices connected to a single CPU through a single channel. The algorithm is then used to develop a simple, accurate approximation for the blocking of disk devices that takes place when a customer using a disk is waiting for or in service at the channel.

An exact solution method for the general class of closed separable queueing networks

In this paper we present a convolution algorithm for the full class of closed, separable queueing networks. In particular, the algorithm represents an alternative method to those already known for the solution of networks with class changes, and is the first efficient algorithm to deal with Lam-type networks [11]. As an application of the algorithm, we study a simple queueing network with disk I/O devices connected to a single CPU through a single channel. The algorithm is then used to develop a simple, accurate approximation for the blocking of disk devices that takes place when a customer using a disk is waiting for or in service at the channel.