Queueing Performance Measures Research Articles

Bayesian analysis of Markov modulated queues with abandonment

AbstractWe consider a Markovian queueing model with abandonment where customer arrival, service and abandonment processes are all modulated by an external environmental process. The environmental process depicts all factors that affect the exponential arrival, service, and abandonment rates. Moreover, the environmental process is a hidden Markov process whose true state is not observable. Instead, our observations consist only of customer arrival, service, and departure times during some period of time. The main objective is to conduct Bayesian analysis in order to infer the parameters of the stochastic system, as well as some important queueing performance measures. This also includes the unknown dimension of the environmental process. We illustrate the implementation of our model and the Bayesian approach by using simulated and actual data on call centers.

Analysis of Hetrogeneous Feedback Queue Model in Stochastic and in Fuzzy Environment Using L-R Method

In this paper, we analyse a feedback queue network in stochastic and in fuzzy environment. We consider a model with three heterogeneous servers which are commonly attached to a server in starting. At the initial stage, all queue performance measures are obtained in steady-state that is in stochastic environment. After that, work is extended to fuzzy environment because practically all characteristics of the system are not exact, they are uncertain in nature. In the present work we use probability generating function technique, triangular fuzzy numbers, classical formulae for the calculation of all queue characteristics and L-R method to calculate queue characteristics in fuzzy environment.

Technical note: A graphical approach to the analysis of travel times in an automated storage and retrieval system

AbstractOperating time is closely related to the performance of an order‐picking system. Knowing the probability distribution of operating time will greatly facilitate system performance analysis and optimization. When using an automated storage and retrieval system (AS/RS), travels of the storage and retrieval machine can be modeled using the Chebyshev metric. However, because of the dependence among the travel legs, analyzing the variance of travel time can be quite challenging. In this paper, a graphical approach based on contour lines is proposed for deriving the probability density function (PDF) of travel time in an AS/RS. With the PDF, the expected value and variance of travel time can be obtained. To demonstrate the use of the proposed approach in evaluating the performance of an AS/RS under stochastic conditions, we derive the PDF of travel time and apply it in a queueing analysis. Based on the queueing performance measures and an overall arrival rate, the minimum number of operation requests waiting or being served occurs when the arrival rates of storage and retrieval requests are approximately equal. Also, when arrival rates of storage and retrieval requests are equal, the number of operation requests waiting or being served is minimized when the rack is square‐in‐time.

Analysis of a single server batch arrival non-Markovian queuing model approach in classification of foot thermograms

In this article, a single server non-Markovian queuing model with Poisson arrivals in stacks of different sizes is discussed. The server provides clients with a general service at a time with a random duration setup time after completing the first level of service. Also, the setup time stage is introduced to carry out the preparatory work required for the stage 2 service and customers may renege during the setup phase. Additionally, the server is subject to random interruption during the third stage of service and it enters into the repair process immediately. The effect of the setup time stage, interruption and reneging in the defined queuing system are well analysed. Using the supplementary variable technique of queuing theory, the duration of the queue length of different states of the system in terms of probability generating functions and the queue performance measures are determined. The issue of characterised lines is very much clarified by a reasonable application and supported by a mathematical portrayal.

Analysis of Fuzzy Non-preemptive Priority Queuing Model with Unequal Service Rate

This article provides an effective method to analyze the performance measures of non-preemptive fuzzy priority queues with unequal service rates. Here the arrival rate and the service rate are in fuzzy numbers. Using a new ranking method, the fuzzy values are reduced to the crisp values. For that cause, both the Triangular Fuzzy Number (TFN) and Trapezoidal Fuzzy Number (TpFN) are chosen to establish the proposal's effectiveness. An illustration is given to find the efficiency of the performance measures of the fuzzy queuing model.

Bayesian Estimation of Queueing Model using Bivariate Prior

This article describes Bayes estimation of various queue characteristics such as queue parameters λ and μ, and queue performance measures like traffic intensity, expected waiting time in the queue, and expected queue size of model using Mckay’s bivariate gamma distribution as prior under squared error loss function as well as entropy loss function. Closed form expressions are obtained for the Bayes estimators of the queue parameters and various queue performance measures using the properties of confluent hyper geometric function and Gauss hyper geometric function. Bootstrap Bayes estimates and credible regions are computed using simulated data for different set of hyper parameter values. Also we apply Markov Chain Monte Carlo method and compute Bayes estimates and credible intervals of various queue characteristics using the same joint prior distribution and compare the values with bootstrap estimates.

Double-sided queues with marked Markovian arrival processes and abandonment

In this paper, we study a double-sided queueing model with marked Markovian arrival processes and finite discrete abandonment times. We apply the theory of multi-layer Markov modulated fluid flow (MMFF) processes to analyze the queueing model. First, we define three age processes for the queueing system and convert them into a multi-layer MMFF process. Then we analyze the multi-layer MMFF process to find queueing performance measures related to the age processes, matching rates/probabilities, waiting times, and queue lengths for both sides of the queueing system. We obtain a number of aggregate quantities as well as quantities for individual types of inputs, which can be useful for the analysis and design of, for examples, passenger-taxi service systems and organ transplantation systems.

Improving service in the presence of surge traffic: a K-policy voluntary flushing queueing system

Unexpected surge in online services causes poor performance and makes it challenging for the service provider to differentiate and serve genuine users. In this paper, we propose a novel approach using voluntary flushing to mitigate disruptions caused by surge traffic, such as distributed denial of service (DDoS) attacks on e-commerce websites. Voluntary flushing of customers has never been modeled because flushing was perceived negatively for customer service. We show that it may not be the case in today’s online services. We propose three queueing performance measures to evaluate the system and present optimal policies for flushing under various scenarios. We use DDoS attacks as use-case to present numerical analysis and discuss the implications of the flushing policies.

Condition-Based Maintenance for Queues With Degrading Servers

The integration of condition monitoring with queueing systems to support decision making is not well explored. This paper addresses the impact of condition monitoring of the server on the system-level performance experienced by entities in a queueing system. The system consists of a queue with a single-server subject to Markovian degradation. The model assumes a Poisson arrival process with service times and repair times according to general distributions. We develop stability conditions and perform steady-state analysis to obtain performance measures (average queue length, average degradation, and so on). We propose minimizing an objective function involving four types of costs: repair, catastrophic failure, quality, and holding. The queue performance measures derived from steady-state analysis are benchmarked and compared to those from a discrete event simulation model. After verifying the queuing model, a sensitivity analysis is performed to determine the relationships between system performance and model parameters. Results indicate that the total cost function is convex and, thus, subject to an optimal repair policy. The model is sensitive to service time, quality costs, and failure costs for late-stage policy repairs decisions and sensitive to expected repair times and repair costs for early stage policy repair decisions. Note to Practitioners —We propose condition-based maintenance for queues with degrading servers. The proposed model has been verified in simulation using degradation data of tools in a manufacturing facility. In order to implement, one must first model the underlying degradation process using the data from condition monitoring. We can get service and repair time distributions using data from the plant floor. Then, we need reasonable estimates of costs. These include costs of holding, quality, repair, and failure. They can be relative, prioritized or zero. The long-run average cost per unit time can then be evaluated. Then, the model provides the degradation state of the server at which maintenance is the most cost efficient. The degradation data can be used to continually obtain the optimal state of degradation at which the server can be taken offline with minimal costs.

Modelling the Mean Waiting Times for Queues in Selected Banks in Eldoret Town-Kenya

The mathematical study of waiting lines is mainly concerned with queue performance measures where several applications have been drawn in past studies. Among the vast uses and applications of the theory of queuing system in banking halls, is the main focus of this study where the theory has been used to solve the problem of long queues as witnessed in banks leads to resource waste. The study aims to model the waiting times for queues in selected banks within Eldoret town, Kenya. The latter component was put under D/D/1 framework and therein its mean derived while the stochastic component was put under the M/M/c framework. Harmonization of the moments of the deterministic and the stochastic components was done to come up with the mean of the overall bank queue traffic delay. The simulation was performed using MATLAB for traffic intensities ranging from 0.1 to 1.9. The results reveal that both deterministic and the stochastic delay components are compatible in modelling waiting time. The models also are applicable to real-time bank queue data whereupon simulation, both models depict fairly equal waiting times for server utilisation factors below 1 and an infinitely increasing delay at rho greater than 1. In conclusion, the models that estimate waiting time were developed and applied on real bank queue data. The models need to be implemented by the banks in their systems so that customers are in a position to know the expected waiting time to be served as soon as they get the ticket from the ticket dispenser.

A study on <i>M</i><SUP align="right"><i>X</i></SUP>/<i>G</i>(<i>a</i>, <i>b</i>)/<i>1</i> queue with server breakdown without interruption and controllable arrivals during multiple adaptive vacations

A single server model, after completion of a bulk service, if there is no breakdown with probability (1 − ψ) and queue length (queue) ≥ 'a', then the bulk service continues, otherwise, the server performs closedown work is analysed. At the end of bulk service, if there is a breakdown occurs with probability (ψ), then the server performs renovation process. After that, if the queue is ≥ 'a', then he performs bulk service otherwise the server perform closedown work follows a vacation. After that, if the queue is less than 'a', then he takes at most 'M' vacations successively. After 'M' vacations, if the queue is still less than 'a', then he remains in the system. However, the customers enter the service station with probability 'p' (0 ≤ p ≤ 1) during multiple adaptive vacations. The probability generating function (PGF) of queue size and performance measures are obtained and cost model is developed.

Analysis of strategic customer behavior in fuzzy queueing systems

<p style='text-indent:20px;'>This paper analyzes the optimal and equilibrium strategies in fuzzy Markovian queues where the system parameters are all fuzzy numbers. In this work, tools from both fuzzy logic and queuing theory have been used to investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. By Zadeh's extension principle and $α$-cut approach, we formulate a pair of parametric nonlinear programs to describe the family of crisp strategy. Then the membership functions of the strategies in single and multi-server models are derived. Furthermore, the grated mean integration method is applied to find estimate of the equilibrium strategy in the fuzzy sense. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and a sensitivity analysis is performed, which show the relationship of these strategies and social benefits. Our finding reveals that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the system managers could get more precise information.

Analysis of queuing mine-cars affecting shaft station radon concentrations in Quzhou uranium mine, eastern China

Abstract Shaft stations of underground uranium mines in China are not only utilized as waiting space for loaded mine-cars queuing to be hoisted but also as the principal channel for fresh air taken to working places. Therefore, assessment of how mine-car queuing processes affect shaft station radon concentration was carried out. Queuing network of mine-cars has been analyzed in an underground uranium mine, located in Quzhou, Zhejiang province of Eastern China. On the basis of mathematical analysis of the queue network, a MATLAB-based quasi-random number generating program utilizing Monte-Carlo methods was worked out. Extensive simulations were then implemented via MATALB operating on a DELL PC. Thereafter, theoretical calculations and field measurements of shaft station radon concentrations for several working conditions were performed. The queuing performance measures of interest, like average queuing length and waiting time, were found to be significantly affected by the utilization rate (positively correlated). However, even with respect to the “worst case”, the shaft station radon concentration was always lower than 200 Bq/m3. The model predictions were compared with the measuring results, and a satisfactory agreement was noted. Under current working conditions, queuing-induced variations of shaft station radon concentration of the study mine are not remarkable.

Performance and reliability analysis of a repairable discrete‐time Geo/G/1 queue with Bernoulli feedback and randomized policy

This paper is concerned with a discrete‐time Geo/G/1 repairable queueing system with Bernoulli feedback and randomized ‐policy. The service station may be subject to failures randomly during serving customers and therefore is sent for repair immediately. The ‐policy means that when the number of customers in the system reaches a given threshold value N, the deactivated server is turned on with probability p or is still left off with probability 1−p. Applying the law of total probability decomposition, the renewal theory and the probability generating function technique, we investigate the queueing performance measures and reliability indices simultaneously in our work. Both the transient queue length distribution and the recursive expressions of the steady‐state queue length distribution at various epochs are explicitly derived. Meanwhile, the stochastic decomposition property is presented for the proposed model. Various reliability indices, including the transient and the steady‐state unavailability of the service station, the expected number of the service station breakdowns during the time interval and the equilibrium failure frequency of the service station are also discussed. Finally, an operating cost function is formulated, and the direct search method is employed to numerically find the optimum value of N for minimizing the system cost. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Perturbation Analysis of the GI/M/s Queue

This paper investigates the M/M/s queueing model to predict an estimate for the proximity of the performance measures of queues with arrival processes that are slightly different from the Poisson process. Specifically, we use the strong stability method to obtain perturbation bounds on the effect of perturbing the arrival process in the M/M/s queue. Therefore, we build some algorithms based on strong stability method to predict stationary performance measures of the GI/M/s queue. Some numerical examples are sketched out to illustrate the accuracy of the proposed method.

Bayes estimation in M/M/1 queues with bivariate prior

Bayes estimator of different queueing performance measures are derived in steady state by recording system size from each of n iid M/M/1 queues. The Bayes estimators are obtained under both squared error loss function and precautionary loss function with a bivariate distribution beta stacy as prior, with natural restriction 0 < λ < µ where λ and µ are arrival rate and service rate respectively. A comprehensive simulation results are also shown at the last section.

A multi-objective fuzzy queuing priority assignment model

After disasters such as earthquakes or floods, the provision of necessary and vital services involves dealing with queues which have parameters and variables that depend on fuzzy decisions made by both service-providing personnel and service-needing people. In modeling these cases, fuzzy and stochastic queuing models are necessary. Although the interesting subject of fuzzy queues has been discussed by different researchers, none has considered fuzzy decisions made by service-providing personnel and service-needing people simultaneously. In this work, tools from both fuzzy logic and queuing theory have been used to address the fuzzy decisions made by service-providing personnel and service-needing people. The service-providing personnel make fuzzy decisions in order to manage the queues; the service-needing people also make fuzzy decisions for choosing among different queues. This study also highlights the application of fuzzy multi-criteria decision making for management of fuzzy queues. A discrete time fuzzy priority queue with partial buffer-sharing is modeled and analyzed where both priority assignment and buffer-control are subject to fuzzy decisions. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the multi-objective priority assignment problem is addressed by fuzzy Data Envelopment Analysis (FDEA). Based on the proposed fuzzy DEA, the properties of the fuzzy threshold-based space priority buffer management scheme are explored.

Queue performance measures in construction simulation models containing subjective uncertainty

Abstract In recent decades, discrete event simulation (DES) has been widely used for analyzing construction projects. Recently, fuzzy discrete event simulation (FDES), which is an integration of fuzzy set theory with DES, has been proposed for simulating construction projects. FDES provides a framework to consider subjective uncertainty (uncertainty due to vagueness, subjectivity, and linguistic expression of knowledge) in construction simulation models. Current FDES frameworks only calculate simulation time (e.g., project completion time) as the simulation output. However, queue performance measures (e.g., average queue length and waiting time)—though important simulation model outputs for decision making, finding bottlenecks, and optimizing construction resources—are not analyzed in current FDES methodologies. Using fuzzy logic to consider the subjective uncertainty of service time and the inter-arrival time of systems' queues may improve such simulation models by more realistically representing their results. This paper provides a novel methodology to consider subjective uncertainty in analyzing the fuzzy queues in construction FDES models. Incorporating fuzzy queuing theory with FDES methodology as proposed in this paper enhances the applicability of FDES in construction projects. The proposed methodology is validated through mathematically solved queueing examples, and its practical aspects are illustrated using an example of an asphalt paving operation.

Can Server Behavior and Service System Design Outweigh the Benefits of Pooling?

It is widely accepted that multi-server single-queue (SQ) systems outperform multi-server parallel-queue (PQ) systems due to the pooling effect. However, humans are not machines, and the assumption that servers perform identically regardless of the queueing system, is not necessarily true in practice. In this paper, we investigate the impact of two human server behaviors - social loafing and workload-dependent speedup - and one physical factor - walking time - on the performance of the SQ and PQ systems. We develop theoretical models and comparatively analyze them with respect to three queueing performance measures: expected wait time, pre-service time, and total system time. We propose functional forms for social loafing and speedup effects and evaluate the performance of the models numerically. We find that social loafing worsens the performance of the SQ system. Server speedup improves performance of the PQ system. Moreover, this improvement is greater than the direct impact of a comparable average increase in the server's speed. We derive thresholds beyond which PQ outperform SQ systems for all three performance measures. Our findings provide managers with guidance in the design of their queueing operations and serve as a foundation for future research on service systems with behavioral elements.

Efficient analysis of the MMAP[K]/PH[K]/1 priority queue

In this paper we consider the MMAP/PH/1 priority queue, both the case of preemptive resume and the case of non-preemptive service. The main idea of the presented analysis procedure is that the sojourn time of the low priority jobs in the preemptive case (and the waiting time distribution in the non-preemptive case) can be represented by the duration of the busy period of a special Markovian fluid model. By making use of the recent results on the busy period analysis of Markovian fluid models it is possible to calculate several queueing performance measures in an efficient way including the sojourn time distribution (both in the time domain and in the Laplace transform domain), the moments of the sojourn time, the generating function of the queue length, the queue length moments and the queue length probabilities.