Approximation Methods for the Standard Deviation of Flow Times in the G/G/s Queue

We provide approximation methods for the standard deviation of flow time in system for a general multi-server queue with infinite waiting capacity (G / G / s ). The approximations require only the mean and standard deviation or the coefficient of variation of the inter-arrival and service time distributions, and the number of servers. These approximations are simple enough to be implemented in manual or spreadsheet calculations, but in comparisons to Monte Carlo simulations have proven to give good approximations (within ±10%) for cases in which the coefficients of variation for the interarrival and service times are between 0 and 1. The approximations also have the desirable properties of being exact for the specific case of Markov queue model M / M / s, as well as some imbedded Markov queuing models ( Ek / M / 1 and M / Eα / 1). The practical significance of this research is that (1) many real world queuing problems involve the G / G / s queuing systems, and (2) predicting the range of variation of the time in the system (rather than just the average) is needed for decision making. For example, one job shop facility with which the authors have worked, guarantees its customers a nine day turnaround time and must determine the minimum number of machines of each type required to achieve nine days as a “worst case” time in the system. In many systems, the “worst case” value of flow time is very relevant because it represents the lead time that can safely be promised to customers. To estimate this we need both the average and standard deviation of the time in system. The usefulness of our results stems from the fact that they are computationally simple and thus provide quick approximations without resorting to complex numerical techniques or Monte Carlo simulations. While many accurate approximations for the G / G / s queue have been proposed previously, they often result in algebraically intractable expressions. This hinders attempts to derive closed-form solutions to the decision variables incorporated in optimization models, and inevitably leads to the use of complex numeric methods. Furthermore, actual application of many of these approximations often requires specification of the actual distributions of the inter-arrival time and the service time. Also, these results have tended to focus on delay probabilities and average waiting time, and do not provide a means of estimating the standard deviation of the time in the system. We also extend the approximations to computing the standard deviation of flow times of each priority class in the G / G / s priority queues and compare the results to those obtained via Monte Carlo simulations. These simulation experiments reveal good approximations for all priority classes with the exception of the lowest priority class in queuing systems with high utilization. In addition, we use the approximations to estimate the average and the standard deviation of the total flow time through queuing networks and have validated these results via Monte Carlo Simulations. The primary theoretical contributions of this work are the derivations of an original expression for the coefficient of variation of waiting time in the G / G / s queue, which holds exactly for G / M / s and M / G /1 queues. We also do some error sensitivity analysis of the formula and develop interpolation models to calculate the probability of waiting, since we need to estimate the probability of waiting for the G / G / s queue to calculate the coefficient of variation of waiting time. Technically we develop a general queuing system performance predictor, which can be used to estimate all kinds of performances for any steady state, infinite queues. We intend to make available a user friendly predictor for implementing our approximation methods. The advantages of these models are that they make no assumptions about distribution of inter-arrival time and service time. Our techniques generalize the previously developed approximations and can also be used in queuing networks and priority queues. Hopefully our approximation methods will be beneficial to those practitioners who like simple and quick practical answers to their multi-server queuing systems. Key words and Phrases: Queuing System, Standard Deviation, Waiting Time, Stochastic Process, Heuristics, G / G/ s, Approximation Methods, Priority Queue, and Queuing Networks

Tandem queues with impatient customers for blood screening procedures

We study a blood testing procedure for detecting viruses like HIV, HBV and HCV. In this procedure, blood samples go through two screening steps. The first test is ELISA (antibody Enzyme Linked Immuno-Sorbent Assay). The portions of blood which are found not contaminated in this first phase are tested in groups through PCR (Polymerase Chain Reaction). The ELISA test is less sensitive than the PCR test and the PCR tests are considerably more expensive. We model the two test phases of blood samples as services in two queues in series; service in the second queue is in batches, as PCR tests are done in groups. The fact that blood can only be used for transfusions until a certain expiration date leads, in the tandem queue, to the feature of customer impatience. Since the first queue basically is an infinite server queue, we mainly focus on the second queue, which in its most general form is an S-server M=G[k;K]=S + G queue, with batches of sizes which are bounded by k and K. Our objective is to maximize the expected profit of the system, which is composed of the amount earned for items which pass the test (and before their patience runs out), minus costs. This is done by an appropriate choice of the decision variables, namely, the batch sizes and the number of servers at the second service station. As will be seen, even the simplest version of the batch queue, the M=M[k;K]=1 + M queue, already gives rise to serious analytical complications for any batch size larger than 1. These complications are discussed in detail. In view of the fact that we aim to solve realistic optimization problems for blood screening procedures, these analytical complications force us to take recourse to either a numerical approach or approximations. We present a numerical solution for the queue length distribution in theM=M[k;K]=S+M queue and then formulate and solve several optimization problems. The power-series algorithm, which is a numerical-analytic method, is also discussed

Practical Extensions to Cycle Time Approximations for the G=G=m-Queue With Applications

Abstract-Approximate closed form expressions for the mean cycle time in a -queue often serve as practical and intuitive alternatives to more exact but less tractable analyses. However, the -queue model may not fully address issues that arise in practical manufacturing systems. Such issues include tools with production parallelism, tools that are idle with work in process, travel to the queue, and the tendency of lots to defect from a failed server and return to the queue even after they have entered production. In this paper, we extend popular approximate mean cycle time formulae to address these practical manufacturing issues. Employing automated data extraction algorithms embedded in software, we test the approximations using parameters gleaned from production tool groups in IBM's 200 mm semiconductor wafer fabricator. Note to Practitioners-We develop extensions to intuitive closed-form approximations for the mean cycle time in queueing networks. Such approximations can be used to analyze the tradeoffs between equipment utilization and cycle time in a manufacturing facility. The extensions incorporate issues of practical import that have not been modeled in the literature and were motivated by the inability of existing models to accurately describe the performance of manufacturing in IBM's 200 mm semiconductor wafer fabricator. The utility of our extensions is that, using automated data collection systems, we are able to well model production tools and elucidate the sources of cycle time. Index Terms-Production management, queueing analysis, semiconductor device manufacture

Heavy and light traffic regimes for M|G|infinity traffic models

The $M|G|infty$ busy server process provides a class of structural models for communication network traffic. In this dissertation, we study the asymptotic behavior of a network multiplexer, modeled as a discrete-time queue, driven by an $M|G|infty$ correlated arrival stream. The asymptotic regimes considered here are those of heavy and light traffic. In heavy traffic, we show that the arising limits are described in terms of the classical Brownian motion and the $alpha$--stable L'{e}vy motion, under short- and long-range dependence, respectively. Salient features are then effectively captured by the exponential distribution and the Mittag-Leffler special function. In light traffic, the analysis reveals the effect of two aspects of the $M|G|infty$ process, i.e., the session duration distribution $G$ and the gradual nature of the arrivals, as opposed to the instantaneous inputs of a standard $GI|GI|1$ queue. We exploit these asymptotic results to construct interpolation approximations for system quantities of interest, applicable to all traffic intensities

Stochastic Approximations and Monotonicity of a Single Server Feedback Retrial Queue

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for an M/G/1 retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters

Inferring finite-time performance in the M/G/1 queueing model

A single server is approached by a stream of Poisson arrivals with known arrival rate. The service times are assumed to be independent identically distributed with unknown distribution. One has available a finite sample of service times obtained by observing the system. A nonparametric approach is taken towards estimating the expected waiting time encountered by a new arriving customer at a finite time t, EWt both for stable and unstable systems. The estimator uses approximations to EWt and an empirical version of the well known Laplace transform of EWt for the M/G/1 queue. Empirical transform; Laplace transform of the virtual waiting of the M/G/1 queue; Exponential approximation; Brownian motion with drift. (jes)Prepared for: Office of Naval Research Arlington, VAhttp://archive.org/details/inferringfinitet00jacoOffice of Naval ResearchApproved for public release; distribution is unlimited