Approximations For Queueing Networks Research Articles

The two-moment three-parameter decomposition approximation of queueing networks with exponential residual renewal processes

We propose a two-moment three-parameter decomposition approximation of general open queueing networks by which both autocorrelation and cross correlation are accounted for. Each arrival process is approximated as an exponential residual (ER) renewal process that is characterized by three parameters: intensity, residue, and decrement. While the ER renewal process is adopted for modeling autocorrelated processes, the innovations method is used for modeling the cross correlation between randomly split streams. As the interarrival times of an ER renewal process follow a two-stage mixed generalized Erlang distribution, viz., MGE(2), each station is analyzed as an MGE(2)/G/1 system for the approximate mean waiting time. Variability functions are also used in network equations for a more accurate modeling of the propagation of cross correlations in queueing networks. Since an ER renewal process is a special case of a Markovian arrival process (MAP), the value of the variability function is determined by a MAP/MAP/1 approximation of the departure process. Numerical results show that our proposed approach greatly improves the performance of the parametric decomposition approximation of open queueing networks.

Estimating manufacturing cycle time and throughput in flow shops with process drift and inspection

Process drift is a common occurrence in many manufacturing processes where machines become dirty (leading to more contamination) or processing parameters degrade, negatively affecting system performance. Statistical process control tracks process quality to determine when the process has gone out of control (has drifted beyond its specifications). This paper considers the case where parts examined at a downstream inspection station are used to determine when the upstream process is out of control. The manufacturing cycle time from the out of control process to the downstream inspection process influences the detection time that elapses until the out of control process is noticed and repaired. Because an out of control process produces more bad parts, the detection time affects the number of good parts produced and the throughput of the manufacturing system. This situation is common in many industries but no models of the phenomena exist. This paper presents a novel manufacturing system model based on queueing network approximations for estimating the manufacturing cycle time and throughput of such systems. These are important performance measures since they influence economic measures such as inventory costs and revenue. The model can be used for a variety of system design and analysis tasks. In particular, the model can be used to evaluate the placement of inspection stations in a process flow.

Taking Account of Correlations Between Streams in Queueing Network Approximations

In this paper we describe a new variation on QNA, the Queueing Network Analyzer (Whitt [38]). QNA calculates approximate congestion measures for a network of queues. The variant proposed here allows us to relax QNA's assumption that certain counts of events in the network are uncorrelated. Following QNA, the network is approximated in terms of three basic operations on streams, namely, queueing, splitting and superposition, and random quantities of interest are characterized approximately by two moments. The approach differs from QNA in that instead of expressing the network operations as transformations applied to two-moment characterizations of the inputs, it expresses these operations as regression relations between counts in the output streams and the input streams. In essence, we substitute approximate regression relationships between random variables for QNA's approximate moment relationships. This device allows us to enhance QNA's moment relationships with information about correlations between the streams. We compare the performance of the proposed approach with QNA on a simple feed-forward network to demonstrate the need for taking account of correlation. We also discuss some of the limitations of this approach, particularly its difficulty in handling feedback, and give some ideas to improve accuracy.

Optimal design of stochastic production lines: a dynamic programming approach

We consider the problem of choosing the number and type of machines for each station in a new production line where the sequence of processes (i.e., manufacturing recipe) has already been established. We formulate a model to minimize cost (investment plus operating) subject to constraints on throughput and cycle time. Using queueing network approximations within a dynamic programming framework, we develop a line design algorithm that works in station-wise fashion. For computational tractability, we must discretize a continuous state space. However, we are able to compute bounds on the error in the cost function as a guide to the appropriate choice of grid size. We conclude by applying our algorithm to an industrial problem that motivated this work.

Using an optimized queueing network model to support wafer fab design

We develop an Optimized Queueing Network (OQNet) capacity planning tool for supporting the design of new and reconfigured semiconductor fabrication facilities that makes use of queueing network approximations and an optimization routine. The basic problem addressed by this tool is to minimize the facility cost required to meet specified volume and cycle time targets. Features common to semiconductor environments, such as batch processes, re-entrant flows, multiple product classes, and machine setups, are incorporated into the model. Comparisons with simulation show that the queueing and other approximations are reasonably accurate. Tests of the optimization routine demonstrate that it can find good solutions quickly.

Optimal design of stochastic production lines: A dynamic programming approach

We consider the problem of choosing the number and type of machines for each station in a new production line where the sequence of processes (i.e., manufacturing recipe) has already been established. We formulate a model to minimize cost (investment plus operating) subject to constraints on throughput and cycle time. Using queueing network approximations within a dynamic programming framework, we develop a line design algorithm that works in station-wise fashion. For computational tractability, we must discretize a continuous state space. However, we are able to compute bounds on the error in the cost function as a guide to the appropriate choice of grid size. We conclude by applying our algorithm to an industrial problem that motivated this work.

Generalized queueing network analysis of integrated supply chains

Supply chain networks are formed from complex interactions between several companies whose aim is to produce and deliver goods to the customers at specified times and places. Computing the total lead time for customer orders entering such a complex network of companies is an important exercise. In this paper we present analytical models for evaluating the average lead times of make-to-order supply chains. In particular, we illustrate the use of generalized queueing networks to compute the mean and variance of the lead time. We present four interesting examples and develop queueing network models for them. The first two examples consider pipeline supply chains and compute the variance of lead time using queueing network approximations available in the literature. This analysis indicates that for the same percentage increase in variance, an increase at the downstream facility has a far more disastrous effect than the same increase at an upstream facility. Through another example, we illustrate the point that coordinated improvements at all the facilities is important and improvements at individual facilities may not always lead to improvements in the supply chain performance. The existing literature on approximate methods of analysis of forkjoin queueing systems assumes heavy traffic and requires tedious computations. We present here two tractable approximate analytical methods for lead time computation in a class of fork-join queueing systems. Our method is based on the results presented by Clarke in 1961. For the case where the 'joining' servers of the queueing system are of the type D/N/1, we present an easy to use approximate method and illustrate its use in evaluating decisions regarding logistics (for instance, who should own the logistics fleet-the manufacturer or the vendor?) and computing simple upper bounds for delivery reliability, that is the probability that customer desired due dates are met.

Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.

Time-dependent queueing network approximations as simulation external control variates

A strategy for efficient evaluation of a complex stochastic model's performance is to use a simpler model's known performance as an approximation. Another strategy is to simulate using the simpler model's known performance as an external control variate. We combine these two strategies. We also relax the assumption that the simpler model's performance is known exactly by introducing mean-squared-error optimal control variates.

Approximations for Networks of Queues with Overtime

This paper presents simple approximations for networks of queues with overtime operation at some stations. This type of network is commonly encountered in several manufacturing applications. We provide bounds on the performance of the approximations for single and multiple machine stations. Our results suggest that the methods perform satisfactorily. These approximations can be used in conjunction with parametric decomposition methods to analyze queueing networks. The computational results indicate that the performance of the decomposition approach does not deteriorate when combined with the methods proposed in this paper.

The AMVA priority approximation

Most computer systems contain one or more system resources whose usage is controlled on the basis of workload priorities. Unfortunately, the exact analysis of queueing network models incorporating priority scheduling disciplines is usually infeasible. The MVA Priority Approximation has been proposed as a comparatively inexpensive, and yet reasonably accurate, approximation technique for queueing networks with priority scheduled service centers. Even this algorithm, however, is too expensive to apply to large networks with many classes of customers. In this paper, we show how the MVA Priority Approximation can be modified so that it utilizes approximate rather than exact Mean Value Analysis (MVA), without significant loss of accuracy. Extensive numerical experiments are performed to further assess the accuracy of the modified algorithm, termed here the AMVA Priority Approximation. These experiments utilize the parameter space mapping technique for studying ‘local’ queueing network approximations.

On Approximations for Queues, I: Extremal Distributions

Many approximations for queueing characteristics such as the mean equilibrium queue length are based on two moments of the interarrivai and service times. To evaluate these approximations, we suggest looking at the set of all possible values of the queueing characteristics given the specified moment parameters. This set-valued function is useful for evaluating the accuracy of approximations. For several models, such as the GI/M/1 queue, the set of possible values for the mean queue length given limited-moment information can be conveniently described by simple extremal distributions. Here we calculate the set of possible values for the mean queue length in a GI/M/1 queue and show how it depends on the traffic intensity and the second moment. We also use extremal distributions to compare alternative parameters for approximations. The results provide useful insights about approximations for non-Markov networks of queues and other complex queueing systems. The general procedure is widely applicable to investigate the accuracy of approximations.