scholarly journals DELAY MODELS BASED ON SYSTEMS WITH USUAL AND SHIFTED HYPEREXPONENTIAL AND HYPERERLANGIAN INPUT DISTRIBUTIONS

2021 ◽  
pp. 56-64
Author(s):  
V. N. Tarasov ◽  
N. F. Bakhareva
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Waiting Time ◽  
Service Time ◽  
Spectral Decomposition ◽  
Flow Distribution ◽  
Input Flow ◽  
Average Delay ◽  
Spectral Expansions ◽  
Wide Range

Context. In the queueing theory, the study of systems with arbitrary laws of the input flow distribution and service time is relevant because it is impossible to obtain solutions for the waiting time in the final form for the general case. Therefore, the study of such systems for particular cases of input distributions is important. Objective. Getting a solution for the average delay in the queue in a closed form for queuing systems with ordinary and with shifted to the right from the zero point hyperexponential and hypererlangian distributions in stationary mode. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average delay for two systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used. Results. For the first time, a spectral decomposition of the solution of the Lindley integral equation for systems with ordinary and with shifted hyperexponential and hyperelangian distributions is obtained, which is used to derive a formula for the average delay in a queue in closed form. Conclusions. It is proved that the spectral expansions of the solution of the Lindley integral equation for the systems under consideration coincide; therefore, the formulas for the mean delay will also coincide. It is shown that in systems with a delay, the average delay is less than in conventional systems. The obtained expression for the waiting time expands and complements the wellknown incomplete formula of queuing theory for the average delay for systems with arbitrary laws of the input flow distribution and service time. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average waiting time, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the waiting time from its average value, the jitter can be determined through the variance of the waiting time.

scholarly journals MATHEMATICAL DELAY MODEL BASED ON SYSTEMS WITH HYPERERLANGIAN AND ERLANGIAN DISTRIBUTIONS

2021 ◽  
Vol 1 (1) ◽  
pp. 87-96
Author(s):  
V. N. Tarasov
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Service Time ◽  
Spectral Decomposition ◽  
Queuing Theory ◽  
Input Flow ◽  
Average Delay ◽  
Spectral Expansions ◽  
Wide Range ◽  
Two Systems

Context. Studies of G/G/1 systems in queuing theory are relevant because such systems are of interest for analyzing the delay of data transmission systems. At the same time, it is impossible to obtain solutions for the delay in the final form in the general case for arbitrary laws of distribution of the input flow and service time. Therefore, it is important to study such systems for particular cases of input distributions. We consider the problem of deriving a solution for the average queue delay in a closed form for two systems with ordinary and shifted hypererlangian and erlangian input distributions. Objective. Obtaining a solution for the main characteristic of the system – the average delay of requests in the queue for two queuing systems of the G/G/1 type with ordinary and with shifted hypererlangian and erlangian input distributions. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of the Lindley integral equation. This method allows to obtaining a solution for the average delay for systems under consideration in a closed form. The method of spectral decomposition of the solution of the Lindley integral equation plays an important role in the theory of systems G/G/1. For the practical application of the results obtained, the well-known method of moments of probability theory is used. Results. For the first time, spectral expansions of the solution of the integral Lindley equation for two systems are obtained, with the help of which calculation formulas for the average delay in a queue in a closed form are derived. Thus, mathematical models of queuing delay for these systems have been built. Conclusions. These formulas expand and supplement the known queuing theory formulas for the average delay G/G/1 systems with arbitrary laws distributions of input flow and service time. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters. In addition to the average delay, such an approach makes it possible to determine also moments of higher orders of waiting time. Given the fact that the packet delay variation (jitter) in telecommunications is defined as the spread of the delay from its average value, the jitter can be determined through the variance of the delay.

scholarly journals COMPARISON OF TWO FORMS OF ERLANGIAN DISTRIBUTION LAW IN QUEUING THEORY

2021 ◽  
pp. 48-56
Author(s):  
V. N. Tarasov
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Waiting Time ◽  
Service Time ◽  
Queuing Theory ◽  
Time Shift ◽  
Queuing Systems ◽  
Average Waiting Time ◽  
Coefficients Of Variation ◽  
Erlang Distributions

Context. For modeling various data transmission systems, queuing systems G/G/1 are in demand, this is especially important because there is no final solution for them in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary system with erlangian input distributions of the second order and for the same system with shifted to the right distributions is considered. Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing systems of type G/G/1 with usual and shifted erlangian input distributions. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows one to obtain a solution for average the waiting time for systems under consideration in a closed form. For the practical application of the results obtained, the well-known method of moments of the theory of probability was used. Results. For the first time, spectral expansions of the solution of the Lindley integral equation for systems with ordinary and shifted Erlang distributions are obtained, with the help of which the calculation formulas for the average waiting time in the queue for the above systems in closed form are derived. Conclusions. The difference between the usual and normalized distribution is that the normalized distribution has a mathematical expectation independent of the order of the distribution k, therefore, the normalized and normal Erlang distributions differ in numerical characteristics. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration turns them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the coefficient of variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The system with usual erlangian input distributions of the second order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope of these systems. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters.

Queuing Systems with a Time Lag

2021 ◽  
Vol 27 (6) ◽  
pp. 291-298
Author(s):  
V. N. Tarasov ◽  
Keyword(s):  
Integral Equation ◽  
Waiting Time ◽  
Zero Point ◽  
Queuing Systems ◽  
Average Waiting Time ◽  
Average Value ◽  
Special Cases ◽  
Wide Range ◽  
Distribution Laws ◽  
The Right

The article discusses various queuing systems (QS) formed by four laws of probability distributions: exponential, hyperexponential, Erlang and hyper-Erlang of the second order. These four laws form sixteen different QS. In contrast to the classical theory, this article considers QS with distribution laws shifted to the right from the zero point. Such QS are of type G/G/1 with arbitrary laws of the distribution of intervals between the requirements of the input flow and the service time. As you know, for such systems it is impossible to obtain solutions for the main characteristic of QS the average waiting time in the general case. Therefore, studies of such systems are important for special cases of distribution laws. The article provides an overview of the author's results for the average waiting time in a queue in a closed form for systems with input distributions shifted to the right from the zero point. To solve this problem, the spectral decomposition method for solving the Lindley integral equation was used. In the course of solving the problem, spectral decompositions of the solution of the Lindley integral equation for eight systems were obtained and with their help calculation formulas were derived for the average waiting time in the queue. It is shown that in systems with delay, the average waiting time is shorter than in conventional systems. The obtained calculation formulas for the average waiting time expand and complement the well-known incomplete formula of the queuing theory for the average waiting time for G/G/1 systems. The proposed approach allows us to calculate the average value and moments of higher orders of waiting time for these systems in mathematical packages for a wide range of changes in traffic parameters. Given the fact that the variation in packet delay (jitter) in the telecommunications standard is defined as the spread of waiting time around its average value, the jitter can be determined through the variance of the waiting time.

SET-VALUED PERFORMANCE APPROXIMATIONS FOR THE QUEUE GIVEN PARTIAL INFORMATION

2020 ◽  
pp. 1-23
Author(s):  
Yan Chen ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Decay Rate ◽  
Waiting Time ◽  
Partial Information ◽  
Upper And Lower Bounds ◽  
Interarrival Time ◽  
Limited Information ◽  
Wide Range ◽  
The Mean ◽  
Third Moment

In order to understand queueing performance given only partial information about the model, we propose determining intervals of likely values of performance measures given that limited information. We illustrate this approach for the mean steady-state waiting time in the $GI/GI/K$ queue. We start by specifying the first two moments of the interarrival-time and service-time distributions, and then consider additional information about these underlying distributions, in particular, a third moment and a Laplace transform value. As a theoretical basis, we apply extremal models yielding tight upper and lower bounds on the asymptotic decay rate of the steady-state waiting-time tail probability. We illustrate by constructing the theoretically justified intervals of values for the decay rate and the associated heuristically determined interval of values for the mean waiting times. Without extra information, the extremal models involve two-point distributions, which yield a wide range for the mean. Adding constraints on the third moment and a transform value produces three-point extremal distributions, which significantly reduce the range, producing practical levels of accuracy.

Closed-Form Correlation of Buildings Energy Use With Key Design Parameters Calibrated Using a Genetic Algorithm

2011 ◽  
Vol 133 (4) ◽  
Author(s):  
Raed I. Bourisli ◽  
Adnan A. AlAnzi
Keyword(s):  
Genetic Algorithm ◽  
Closed Form ◽  
Energy Use ◽  
Building Design ◽  
Office Buildings ◽  
Design Parameters ◽  
Wide Range ◽  
Variable Function ◽  
Real Coded Genetic Algorithm ◽  
The Difference

This work aims at developing a closed-form correlation between key building design variables and its energy use. The results can be utilized during the initial design stages to assess the different building shapes and designs according to their expected energy use. Prototypical, 20-floor office buildings were used. The relative compactness, footprint area, projection factor, and window-to-wall ratio were changed and the resulting buildings performances were simulated. In total, 729 different office buildings were developed and simulated in order to provide the training cases for optimizing the correlation’s coefficients. Simulations were done using the VisualDOE TM software with a Typical Meteorological Year data file, Kuwait City, Kuwait. A real-coded genetic algorithm (GA) was used to optimize the coefficients of a proposed function that relates the energy use of a building to its four key parameters. The figure of merit was the difference in the ratio of the annual energy use of a building normalized by that of a reference building. The objective was to minimize the difference between the simulated results and the four-variable function trying to predict them. Results show that the real-coded GA was able to come up with a function that estimates the thermal performance of a proposed design with an accuracy of around 96%, based on the number of buildings tested. The goodness of fit, roughly represented by R2, ranged from 0.950 to 0.994. In terms of the effects of the various parameters, the area was found to have the smallest role among the design parameters. It was also found that the accuracy of the function suffers the most when high window-to-wall ratios are combined with low projection factors. In such cases, the energy use develops a potential optimum compactness. The proposed function (and methodology) will be a great tool for designers to inexpensively explore a wide range of alternatives and assess them in terms of their energy use efficiency. It will also be of great use to municipality officials and building codes authors.

scholarly journals Tails in generalized Jackson networks with subexponential service-time distributions

2005 ◽  
Vol 42 (2) ◽  
pp. 513-530 ◽  
Author(s):  
François Baccelli ◽  
Serguei Foss ◽  
Marc Lelarge
Keyword(s):  
Closed Form ◽  
Service Time ◽  
Large Deviation ◽  
Exact Asymptotics ◽  
Fluid Limits ◽  
Jackson Networks ◽  
Single Station ◽  
Service Times

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

scholarly journals Analysis of queue change of visitors and performace system in the Department of Population and Civil Regristation of Semarang City

2020 ◽  
Vol 202 ◽  
pp. 15005
Author(s):  
Sugito ◽  
Alan Prahutama ◽  
Dwi Ispriyanti ◽  
Mustafid
Keyword(s):  
Customer Satisfaction ◽  
Waiting Time ◽  
Public Service ◽  
Service Time ◽  
Service Sector ◽  
General Distribution ◽  
Queuing Systems ◽  
Queuing Model ◽  
The Public ◽  
Second Period

The Population and Civil Registry Office in Semarang city is one of the public service units. In the public service sector, visitor / customer satisfaction is very important. It can be identified by the length of the queue, the longer visitors queue this results in visitor dissatisfaction with the service. Queue analysis is one of the methods in statistics to determine the distribution of queuing systems that occur within a system. In this study, a queuing analysis as divided into two periods. The first period lasts from 2-13 March 2015, while the second period lasts November 16th to December 20th 2019. The variables used are the number of visitors and the service time at each counter in intervals of 30 minutes. The results obtained are changes in the distribution and queuing model that is at counter 5/6 and counter 10. The queuing model obtained at the second perideo for the number of visitors and the time of service with a General distribution. The average number of visitors who come in 30 minute intervals in the second period is more than the first period, this indicates an increase in visitors. The opportunity for service units is still small, the waiting time in the queue is getting smaller. This shows that the performance of the queuing system at the Semarang Population and Civil Registry Office is getting better.

scholarly journals Decomposition of M/M/1 With Unreliable Service and a Working Vacation

2020 ◽  
Vol 9 (1) ◽  
pp. 63
Author(s):  
Joshua Patterson ◽  
Andrzej Korzeniowski
Keyword(s):  
Closed Form ◽  
Failure Rate ◽  
Waiting Time ◽  
Queue Length ◽  
Service Failure ◽  
Working Vacation ◽  
Stationary Queue Length ◽  
Stationary Waiting Time ◽  
Relationship Of ◽  
The Relationship

We use the stationary distribution for the M/M/1 with Unreliable Service and aWorking Vacation (M/M/1/US/WV) given explicitly in (Patterson & Korzeniowski, 2019) to find a decomposition of the stationary queue length N. By applying the distributional form of Little's Law the Laplace-tieltjes Transform of the stationary customer waiting time W is derived. The closed form of the expected value and variance for both N and W is found and the relationship of the expected stationary waiting time as a function of the service failure rate is determined.

On the Nature of Critical Heat Flux in Microchannels

2005 ◽  
Vol 127 (1) ◽  
pp. 101-107 ◽  
Author(s):  
A. E. Bergles ◽  
S. G. Kandlikar
Keyword(s):  
Heat Flux ◽  
Critical Heat Flux ◽  
Flow Boiling ◽  
Flow Distribution ◽  
High Heat ◽  
Operating Conditions ◽  
High Heat Flux ◽  
Clear Understanding ◽  
Wide Range ◽  
Parallel Microchannels

The critical heat flux (CHF) limit is an important consideration in the design of most flow boiling systems. Before the use of microchannels under saturated flow boiling conditions becomes widely accepted in cooling of high-heat-flux devices, such as electronics and laser diodes, it is essential to have a clear understanding of the CHF mechanism. This must be coupled with an extensive database covering a wide range of fluids, channel configurations, and operating conditions. The experiments required to obtain this information pose unique challenges. Among other issues, flow distribution among parallel channels, conjugate effects, and instrumentation need to be considered. An examination of the limited CHF data indicates that CHF in parallel microchannels seems to be the result of either an upstream compressible volume instability or an excursive instability rather than the conventional dryout mechanism. It is expected that the CHF in parallel microchannels would be higher if the flow is stabilized by an orifice at the entrance of each channel. The nature of CHF in microchannels is thus different than anticipated, but recent advances in microelectronic fabrication may make it possible to realize the higher power levels.

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