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.

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.

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.

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.

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.

A note on the convexity of performance measures of M/M/c queueing systems

1983 ◽  
Vol 20 (04) ◽  
pp. 920-923 ◽  
Author(s):  
Hau Leung Lee ◽  
Morris A. Cohen
Keyword(s):  
Performance Measures ◽  
Waiting Time ◽  
Queueing System ◽  
Queueing Systems ◽  
Arrival Rate ◽  
Control Problems ◽  
Traffic Intensity ◽  
Expected Number ◽  
Expected Waiting Time

Convexity of performance measures of queueing systems is important in solving control problems of multi-facility systems. This note proves that performance measures such as the expected waiting time, expected number in queue, and the Erlang delay formula are convex with respect to the arrival rate or the traffic intensity of the M/M/c queueing system.

scholarly journals FINITE TWO LAYERED QUEUEING SYSTEMS

2016 ◽  
Vol 30 (3) ◽  
pp. 492-513 ◽  
Author(s):  
Efrat Perel ◽  
Uri Yechiali
Keyword(s):  
Waiting Times ◽  
Queueing Systems ◽  
Service Rate ◽  
Dimensional System ◽  
Single Server ◽  
Matrix Analytic Methods ◽  
Birth And Death Processes ◽  
Operating Modes ◽  
Loss Probabilities ◽  
Steady State Probabilities

We study layered queueing systems comprised two interlacing finite M/M/• type queues, where users of each layer are the servers of the other layer. Examples can be found in file sharing programs, SETI@home project, etc. Let Li denote the number of users in layer i, i=1, 2. We consider the following operating modes: (i) All users present in layer i join forces together to form a single server for the users in layer j (j≠i), with overall service rate μjLi (that changes dynamically as a function of the state of layer i). (ii) Each of the users present in layer i individually acts as a server for the users in layer j, with service rate μj.These operating modes lead to three different models which we analyze by formulating them as finite level-dependent quasi birth-and-death processes. We derive a procedure based on Matrix Analytic methods to derive the steady state probabilities of the two dimensional system state. Numerical examples, including mean queue sizes, mean waiting times, covariances, and loss probabilities, are presented. The models are compared and their differences are discussed.

Generalised birth and death queueing processes: recent results

1977 ◽  
Vol 9 (1) ◽  
pp. 125-140 ◽  
Author(s):  
B. W. Conolly ◽  
J. Chan
Keyword(s):  
Waiting Time ◽  
Queueing Systems ◽  
System Size ◽  
Single Server ◽  
Traffic Intensity ◽  
Stable Regime ◽  
Mean Waiting Time ◽  
Service Rates ◽  
Queueing Processes ◽  
Effective Service

The systems considered are single-server, though the theory has wider application to models of adaptive queueing systems. Arrival and service mechanisms are governed by state (n)-dependent mean arrival and service rates λn and µn. It is assumed that the choice of λn and µn leads to a stable regime. Formulae are sought that provide easy means of computing statistics of effectiveness of systems. A measure of traffic intensity is first defined in terms of ‘effective’ service time and inter-arrival intervals. It is shown that the latter have a renewal type connection with appropriately defined mean effective arrival and service rates λ∗ and µ∗ and that in consequence the ratio λ∗/µ∗ is the traffic intensity, equal moreover to where is the stable probability of an empty system, consistent with other systems. It is also shown that for first come, first served discipline the equivalent of Little's formula holds, where and are the mean waiting time of an arrival and mean system size at an arbitrary epoch. In addition it appears that stable regime output intervals are statistically identical with effective inter-arrival intervals. Symmetrical moment formulae of arbitrary order are derived algebraically for effective inter-arrival and service intervals, for waiting time, for busy period and for output.

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