Comparing Short-Memory Charts to Monitor the Traffic Intensity of Single Server Queues

2018 ◽  
Vol 33 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Marta Santos ◽  
Manuel Cabral Morais ◽  
António Pacheco
Keyword(s):  
Quality Control ◽  
Control Charts ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Service Rate ◽  
Single Server ◽  
Traffic Intensity ◽  
Statistical Process ◽  
Single Server Queues ◽  
Short Memory

Abstract This paper describes the application of simple quality control charts to monitor the traffic intensity of single server queues, a still uncommon use of what is arguably the most successful statistical process control tool. These charts play a vital role in the detection of increases in the traffic intensity of single server queueing systems such as the {M/G/1} , {GI/M/1} and {GI/G/1} queues. The corresponding control statistics refer solely to a customer-arrival/departure epoch as opposed to several such epochs, thus they are termed short-memory charts. We compare the RL performance of those charts under three out-of-control scenarios referring to increases in the traffic intensity due to: a decrease in the service rate while the arrival rate remains unchanged; an increase in the arrival rate while the service rate is constant; an increase in the arrival rate accompanied by a proportional decrease in the service rate. These comparisons refer to a broad set of interarrival and service time distributions, namely exponential, Erlang, hyper-exponential, and hypo-exponential. Extensive results and striking illustrations are provided to give the quality control practitioner an idea of how these charts perform in practice.

Comparing Short and Long-Memory Charts to Monitor the Traffic Intensity of Single Server Queues

2019 ◽  
Vol 34 (1) ◽  
pp. 9-18
Author(s):  
Marta Santos ◽  
Manuel Cabral Morais ◽  
António Pacheco
Keyword(s):  
Long Memory ◽  
Control Charts ◽  
Performance Metrics ◽  
Waiting Times ◽  
Queueing Systems ◽  
Transportation Systems ◽  
Single Server ◽  
Traffic Intensity ◽  
Cusum Charts ◽  
Short Memory

Abstract The traffic intensity (ρ) is a vital parameter of queueing systems because it is a measure of the average occupancy of a server. Consequently, it influences their operational performance, namely queue lengths and waiting times. Moreover, since many computer, production and transportation systems are frequently modelled as queueing systems, it is crucial to use control charts to detect changes in ρ. In this paper, we pay particular attention to control charts meant to detect increases in the traffic intensity, namely: a short-memory chart based on the waiting time of the n-th arriving customer; two long-memory charts with more sophisticated control statistics, and the two cumulative sum (CUSUM) charts proposed by Chen and Zhou (2015). We confront the performances of these charts in terms of some run length related performance metrics and under different out-of-control scenarios. Extensive results are provided to give the quality control practitioner a concrete idea about the performance of these charts.

scholarly journals Traffic Intensity Estimation in Finite Markovian Queueing Systems

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Frederico R. B. Cruz ◽  
Márcio A. C. Almeida ◽  
Marcos F. S. V. D’Angelo ◽  
Tom van Woensel
Keyword(s):  
Queueing Systems ◽  
Arrival Rate ◽  
Research Area ◽  
Service Rate ◽  
Intermediate Step ◽  
Single Server ◽  
Traffic Intensity ◽  
Finite Sample ◽  
Jeffreys Prior ◽  
Accurate Performance

In many everyday situations in which a queue is formed, queueing models may play a key role. By using such models, which are idealizations of reality, accurate performance measures can be determined, such as traffic intensity (ρ), which is defined as the ratio between the arrival rate and the service rate. An intermediate step in the process includes the statistical estimation of the parameters of the proper model. In this study, we are interested in investigating the finite-sample behavior of some well-known methods for the estimation of ρ for single-server finite Markovian queues or, in Kendall notation, M/M/1/K queues, namely, the maximum likelihood estimator, Bayesian methods, and bootstrap corrections. We performed extensive simulations to verify the quality of the estimators for samples up to 200. The computational results show that accurate estimates in terms of the lowest mean squared errors can be obtained for a broad range of values in the parametric space by using the Jeffreys’ prior. A numerical example is analyzed in detail, the limitations of the results are discussed, and notable topics to be further developed in this research area are presented.

scholarly journals A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

2017 ◽  
Vol 2017 ◽  
pp. 1-10 ◽  
Author(s):  
Ekaterina Evdokimova ◽  
Sabine Wittevrongel ◽  
Dieter Fiems
Keyword(s):  
Performance Measures ◽  
Queueing Systems ◽  
Poisson Processes ◽  
Series Expansions ◽  
Service Rate ◽  
Single Server ◽  
Queueing Model ◽  
State Space Explosion ◽  
Finite Queues ◽  
Service Rates

This paper investigates the performance of a queueing model with multiple finite queues and a single server. Departures from the queues are synchronised or coupled which means that a service completion leads to a departure in every queue and that service is temporarily interrupted whenever any of the queues is empty. We focus on the numerical analysis of this queueing model in a Markovian setting: the arrivals in the different queues constitute Poisson processes and the service times are exponentially distributed. Taking into account the state space explosion problem associated with multidimensional Markov processes, we calculate the terms in the series expansion in the service rate of the stationary distribution of the Markov chain as well as various performance measures when the system is (i) overloaded and (ii) under intermediate load. Our numerical results reveal that, by calculating the series expansions of performance measures around a few service rates, we get accurate estimates of various performance measures once the load is above 40% to 50%.

scholarly journals Large Deviations for the Single-Server Queue and the Reneging Paradox

2021 ◽  
Author(s):  
Rami Atar ◽  
Amarjit Budhiraja ◽  
Paul Dupuis ◽  
Ruoyu Wu
Keyword(s):  
Large Deviations ◽  
Decay Rate ◽  
Sample Path ◽  
Arrival Rate ◽  
Rate Function ◽  
Service Rate ◽  
Single Server ◽  
Large Deviations Principle ◽  
Long Run ◽  
The Individual

For the M/M/1+M model at the law-of-large-numbers scale, the long-run reneging count per unit time does not depend on the individual (i.e., per customer) reneging rate. This paradoxical statement has a simple proof. Less obvious is a large deviations analogue of this fact, stated as follows: the decay rate of the probability that the long-run reneging count per unit time is atypically large or atypically small does not depend on the individual reneging rate. In this paper, the sample path large deviations principle for the model is proved and the rate function is computed. Next, large time asymptotics for the reneging rate are studied for the case when the arrival rate exceeds the service rate. The key ingredient is a calculus of variations analysis of the variational problem associated with atypical reneging. A characterization of the aforementioned decay rate, given explicitly in terms of the arrival and service rate parameters of the model, is provided yielding a precise mathematical description of this paradoxical behavior.

Statistical Process and Quality Control

2000 ◽  
pp. 233-244
Keyword(s):  
Quality Control ◽  
Quality Improvement ◽  
Quality Management ◽  
Total Quality Management ◽  
Statistical Process Control ◽  
Control Charts ◽  
Statistical Testing ◽  
Total Quality ◽  
Statistical Process ◽  
Sampling Plans

Abstract This chapter provides an introduction to statistical process control and the concept of total quality management. It begins with a review of quality improvement efforts in the extrusion industry and the considerations involved in developing sampling plans and interpreting control charts. It then lays out the steps that would be followed in order to implement statistical testing for billet casting, die performance, or any other process or variable that impacts extrusion quality. The chapter concludes with an overview of the fundamentals of total quality management.

scholarly journals Simple and Yet Efficient Estimators for Markovian Multiserver Queues

2018 ◽  
Vol 2018 ◽  
pp. 1-7 ◽  
Author(s):  
Emilio Suyama ◽  
Roberto C. Quinino ◽  
Frederico R. B. Cruz
Keyword(s):  
Maximum Likelihood ◽  
Arrival Rate ◽  
Research Area ◽  
Service Rate ◽  
Traffic Intensity ◽  
Moment Estimator ◽  
Likelihood Estimator ◽  
Future Developments ◽  
Multiserver Queues

Estimators for the parameters of the Markovian multiserver queues are presented, from samples that are the number of clients in the system at arbitrary points and their sojourn times. As estimation in queues is a recognizably difficult inferential problem, this study focuses on the estimators for the arrival rate, the service rate, and the ratio of these two rates, which is known as the traffic intensity. Simulations are performed to verify the quality of the estimations for sample sizes up to 400. This research also relates notable new insights, for example, that the maximum likelihood estimator for the traffic intensity is equivalent to its moment estimator. Some limitations of the results are presented along with a detailed numerical example and topics for future developments in this research area.

Extremal properties of the FIFO discipline in queueing networks

1992 ◽  
Vol 29 (4) ◽  
pp. 967-978 ◽  
Author(s):  
Rhonda Righter ◽  
J. George Shanthikumar
Keyword(s):  
Likelihood Ratio ◽  
Service Time ◽  
Queueing Networks ◽  
Service Rate ◽  
Service Discipline ◽  
Single Server ◽  
Single Server Queues ◽  
First Results ◽  
Service Times ◽  
Number Of Customers

We show that using the FIFO service discipline at single server stations with ILR (increasing likelihood ratio) service time distributions in networks of monotone queues results in stochastically earlier departures throughout the network. The converse is true at stations with DLR (decreasing likelihood ratio) service time distributions. We use these results to establish the validity of the following comparisons:(i) The throughput of a closed network of FIFO single-server queues will be larger (smaller) when the service times are ILR (DLR) rather than exponential with the same means.(ii) The total stationary number of customers in an open network of FIFO single-server queues with Poisson external arrivals will be stochastically smaller (larger) when the service times are ILR (DLR) rather than exponential with the same means.We also give a surprising counterexample to show that although FIFO stochastically maximizes the number of departures by any time t from an isolated single-server queue with IHR (increasing hazard rate, which is weaker than ILR) service times, this is no longer true for networks of more than one queue. Thus the ILR assumption cannot be relaxed to IHR.Finally, we consider multiclass networks of exponential single-server queues, where the class of a customer at a particular station determines its service rate at that station, and show that serving the customer with the highest service rate (which is SEPT — shortest expected processing time first) results in stochastically earlier departures throughout the network, among all preemptive work-conserving policies. We also show that a cµ rule stochastically maximizes the number of non-defective service completions by any time t when there are random, agreeable, yields.

Inspection in Process Control

1998 ◽  
Vol 42 (16) ◽  
pp. 1170-1174
Author(s):  
Somchart Thepvongs ◽  
Brian M. Kleiner
Keyword(s):  
Quality Control ◽  
Process Control ◽  
Signal Detection ◽  
Control Chart ◽  
Control Charts ◽  
Signal Detection Theory ◽  
Detection Theory ◽  
Statistical Quality Control ◽  
Total Quality ◽  
Statistical Process

Consistent with the precepts of total quality control and total quality management, there has been a resource shift from incoming and outgoing inspection processes to statistical quality control of processes. Furthermore, process control operators are responsible for their own quality, necessitating the in-process inspection of components. This study treated the statistical process control task of “searching” control charts for out-of-control conditions as an inspection task and applied the Theory of Signal Detection to better understand this behavior and improve performance. Twelve subjects participated in a research study to examine how the portrayal of control chart information affected signal detection theory measures. The type of display did not have a significant effect on the sensitivity and response criterion of subjects. These results are discussed in terms of the applicability of Signal Detection Theory in control chart decision making as well as implications on display design.

The convexity of a general performance measure for multiserver queues

1987 ◽  
Vol 24 (03) ◽  
pp. 725-736 ◽  
Author(s):  
Arie Harel ◽  
Paul Zipkin
Keyword(s):  
Queueing Systems ◽  
Arrival Rate ◽  
Performance Measure ◽  
Service Rate ◽  
Service Time Distribution ◽  
Multiserver Queues ◽  
General Performance ◽  
Production Models ◽  
Convexity Properties ◽  
Special Case

This paper examines a general performance measure for queueing systems. This criterion reflects both the mean and the variance of sojourn times; the standard deviation is a special case. The measure plays a key role in certain production models, and it should be useful in a variety of other applications. We focus here on convexity properties of an approximation of the measure for the M/G/c queue. For c ≧ 2 we show that this quantity is convex in the arrival rate. Assuming the service rate acts as a scale factor in the service-time distribution, the measure is convex in the service rate also.

scholarly journals Assessment of process stability and capability in a manufacturing organization: a case study

2021 ◽  
Vol 343 ◽  
pp. 05011
Author(s):  
Carmen Simion
Keyword(s):  
Quality Control ◽  
Control Charts ◽  
Statistical Quality Control ◽  
Statistical Tool ◽  
Statistical Process ◽  
Monitoring Process ◽  
Capability Analysis ◽  
Process Output ◽  
Manufacturing Organization

Quality is considered asthe principal factor that determines the long-termsuccess or failure of any organization. Organizations perform quality control by monitoring process output using Statistical Quality Control, performed as part of the production process (Statistical Process Control, SPC) or as a final quality control check (Acceptance Sampling).SPC is a major quality management statistical tool and its instruments (control charts and capability analysis) are applied to virtually any type of organization (manufacturing, services or transactions - for example, those involving data, communications, software, or movement of materials). The aim of this paper is to present a case study, realized in a manufacturing organizationfrom Sibiu, for a new product used in the automotive industry to check its conformance to designed requirements. The output data were analyzed using statistical analysis software Minitab.

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