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.

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 Spectral expansion method for analysis of a system with shifted Erlang and hyper-Erlang distributions

2021 ◽  
Vol 24 (2) ◽  
pp. 55-61
Author(s):  
Veniamin N. Tarasov ◽  
Nadezhda F. Bakhareva
Keyword(s):  
Waiting Time ◽  
Queuing Theory ◽  
Expansion Method ◽  
Spectral Expansion ◽  
Queuing System ◽  
Erlang Distribution ◽  
Modern Theory ◽  
Calculation Formula ◽  
Average Waiting Time ◽  
Erlang Distributions

In this paper, we obtained a spectral expansion of the solution to the Lindley integral equation for a queuing system with a shifted Erlang input flow of customers and a hyper-Erlang distribution of the service time. On its basis, a calculation formula is derived for the average waiting time in the queue for this system in a closed form. As you know, all other characteristics of the queuing system are derivatives of the average waiting time. The resulting calculation formula complements and expands the well-known unfinished formula for the average waiting time in queue in queuing theory for G/G/1 systems. In the theory of queuing, studies of private systems of the G/G/1 type are relevant due to the fact that they are actively used in the modern theory of teletraffic, as well as in the design and modeling of various data transmission systems.

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.

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.

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.

The general bulk queue as a matrix factorisation problem of the Wiener-Hopf type. Part II.

1975 ◽  
Vol 7 (3) ◽  
pp. 647-655 ◽  
Author(s):  
John Dagsvik
Keyword(s):  
Waiting Time ◽  
Service Time ◽  
Time Distribution ◽  
Single Server ◽  
Service Time Distribution ◽  
Time Process ◽  
Waiting Time Process ◽  
Bulk Queue ◽  
Erlang Distributions ◽  
Matrix Factorisation

In a previous paper (Dagsvik (1975)) the waiting time process of the single server bulk queue is considered and a corresponding waiting time equation is established. In this paper the waiting time equation is solved when the inter-arrival or service time distribution is a linear combination of Erlang distributions. The analysis is essentially based on algebraic arguments.

A queue with Markov-dependent service times

1968 ◽  
Vol 5 (02) ◽  
pp. 461-466
Author(s):  
Gerold Pestalozzi
Keyword(s):  
Steady State ◽  
Generating Function ◽  
Waiting Time ◽  
Service Time ◽  
Queueing System ◽  
Single Channel ◽  
Probability Generating Function ◽  
Average Waiting Time ◽  
Poisson Input ◽  
The Mean

A queueing system is considered where each item has a property associated with it, and where the service time interposed between two items depends on the properties of both of these items. The steady state of a single-channel queue of this type, with Poisson input, is investigated. It is shown how the probability generating function of the number of items waiting can be found. Easily applied approximations are given for the mean number of items waiting and for the average waiting time.

scholarly journals ANALISIS PELAYANAN COUNTER CHECK-IN CITILINK INDONESIA DENGAN MENGGUNAKAN METODE ANTRIAN DI ERA PANDEMI COVID-19

Jurnal IPTA ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 200
Author(s):  
Yudha Eka Nugraha ◽  
Yurisah Adiningsih Hau
Keyword(s):  
Service Quality ◽  
Waiting Time ◽  
Service Time ◽  
Queuing Theory ◽  
Air Travel ◽  
Research Results ◽  
Multiple Line ◽  
Ground Handling ◽  
Quantitative Descriptive ◽  
Descriptive Method

Tourism as one of the priority sectors that is developing in Indonesia has an impact on increasing number of air travelers in Indonesia. Air travel actors consist of various elements, one of which is tourists. During air travel, tourists need to go through the check-in process at the airport before they arrived in destination. This study aims to analyze the punctuality of the check-in counter service of Citilink Indonesia airline at Ground Handling company in PT Gapura Angkasa using the queuing theory and FIFO. Counter check-in services will refer to Ministerial Regulation Number 38, 2015. The quantitative descriptive method was chosen as the approach to this research. Services at Citilink Indonesia's check-in counters are analyzed using the multiple line queuing theory. In calculating the data, the queue observation is calculated, the queue time is recorded, and is equipped with an interview with the check-in counter frontliner. Based on the research results, Citilink Indonesia's counter check-in service at El-Tari Kupang Airport shows that: 1) The waiting time for air travelers in the queue is <20 minutes. 2) Service time for air travelers at the counter check at Citilink Indonesia per person is <2 minutes 30 seconds. The results of this study conclude that the Citilink Indonesia counter check-in service at El-Tari Airport, Kupang is in accordance with Ministerial Direction No. 38 of 2015. According to the result given, Check In Counter Citilink Indonesa has always been commited to give the best service quality for passenger, and obey health protocol while check-in process happens.

scholarly journals The Timeliness of the Reserved Service in the Cluster with the Regulation of the Time of Destruction of Overdue Requests in the Node Queues

2021 ◽  
Author(s):  
Vladimir Bogatyrev ◽  
Stanislav Bogatyrev ◽  
Anatoly Bogatyrev
Keyword(s):  
Distributed Computing ◽  
Real Time ◽  
Distributed Control ◽  
Waiting Time ◽  
Significant Influence ◽  
Queuing Systems ◽  
Time Limits ◽  
Average Waiting Time ◽  
Insufficient Time ◽  
Maximum Allowable Time

With the increasing complexity of distributed control tasks based on their intellectualization, there are problems of insufficient time and computing resources for functioning in real time. In this regard, there is a need to develop methods for organizing distributed real-time computer systems, based on the consolidation of distributed computing resources with their integration into clusters. The possibilities of increasing the probability of timely servicing of waiting- critical requests in the cluster as a result of query replication and controlling the time of destruction of potentially expired replicas in node queues are investigated. The cluster is represented as a group of queuing systems with infinite queues with a limited average waiting time. The effectiveness of the reserved service of a real-time request is determined by the probability of executing at least one of the generated copies of the request in the maximum allowable time without losing it due to errors and waiting time limits in the queues of cluster nodes. It is shown that there is an optimal multiplicity of query replication with a significant influence of the choice of restrictions on the waiting time for requests in queues before they are destroyed.

A Optimized Design of One Traffic Circle

2012 ◽  
Vol 588-589 ◽  
pp. 1632-1635
Author(s):  
Jun Yu Xiong ◽  
Xiao Hui Du ◽  
Jia Qi Wang ◽  
Hui Li Zhai
Keyword(s):  
Traffic Flow ◽  
Waiting Time ◽  
Queuing Theory ◽  
Original Method ◽  
Optimized Design ◽  
Average Waiting Time ◽  
Traffic Capacity ◽  
Signal Period ◽  
Pass Through

In this paper we use queuing theory to analysis the incoming traffic, developed an effective way to control the traffic of a circle by using stop signs and yield signs,and calculated the traffic capacity and average waiting time of this method. Then, we use signals to control the traffic and improve the original method by a analysis the ways the car can pass through the circle crossing. Taking into account of the traffic flow in the different time of a day, we got the light's signal period to adapt to the features of the traffic flow.

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