scholarly journals On a Batch Arrival Queuing System Equipped with a Stand-by Server during Vacation Periods or the Repairs Times of the Main Server

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Rehab F. Khalaf ◽  
Kailash C. Madan ◽  
Cormac A. Lukas
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Waiting Time ◽  
Generating Functions ◽  
Queuing System ◽  
Sudden Failure ◽  
Supplementary Variables ◽  
Service Times ◽  
Mathcad Software ◽  
Number Of Customers

We study a queuing system which is equipped with a stand-by server in addition to the main server. The stand-by server provides service to customers only during the period of absence of the main server when either the main server is on a vacation or it is in the state of repairs due to a sudden failure from time to time. The service times, vacation times, and repair times are assumed to follow general arbitrary distributions while the stand-by service times follow exponential distribution. Supplementary variables technique has been used to obtain steady state results in explicit and closed form in terms of the probability generating functions for the number of customers in the queue, the average number of customers, and the average waiting time in the queue while the MathCad software has been used to illustrate the numerical results in this work.

scholarly journals Analysis of M/M/1 Queuing System with Customer Reneging During Server Vacations Subject To Server Breakdown and Delayed Repair

2018 ◽  
Vol 7 (4.10) ◽  
pp. 552
Author(s):  
Ch. Swathi ◽  
V. Vasanta Kumar
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Detailed Analysis ◽  
Performance Measures ◽  
Queuing System ◽  
Multiple Vacation ◽  
Delayed Repair ◽  
Server Breakdown ◽  
Number Of Customers ◽  
Steady State Probabilities

In this paper, we consider an M/M/1 queuing system with customer reneging for an unreliable sever. Customer reneging is assumed to occur due to the absence of the server during vacations.  Detailed analysis for both single and multiple vacation models during different states of the server such as busy, breakdown and delayed repair periods is presented. Steady state probabilities for single and multiple vacation policies are obtained. Closed form expressions for various performance measures such as average number of customers in the system, proportion of customers served and reneged per unit time during single and multiple vacations are obtained.   

A single server Markovian queuing system with limited buffer and reverse balking

2021 ◽  
Vol 12 (7) ◽  
pp. 1774-1784
Author(s):  
Girin Saikia ◽  
Amit Choudhury
Keyword(s):  
Steady State ◽  
Performance Measures ◽  
Mutual Fund ◽  
System Size ◽  
Queuing System ◽  
Single Server ◽  
Insurance Company ◽  
Number Of Customers ◽  
Steady State Probabilities

The phenomena are balking can be said to have been observed when a customer who has arrived into queuing system decides not to join it. Reverse balking is a particular type of balking wherein the probability that a customer will balk goes down as the system size goes up and vice versa. Such behavior can be observed in investment firms (insurance company, Mutual Fund Company, banks etc.). As the number of customers in the firm goes up, it creates trust among potential investors. Fewer customers would like to balk as the number of customers goes up. In this paper, we develop an M/M/1/k queuing system with reverse balking. The steady-state probabilities of the model are obtained and closed forms of expression of a number of performance measures are derived.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (03) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

Development of a Heuristics Model for Simplifying a Multi-Channel Queuing System

2011 ◽  
Vol 367 ◽  
pp. 647-652
Author(s):  
B. Kareem ◽  
A. A. Aderoba
Keyword(s):  
Waiting Time ◽  
Channel Model ◽  
Single Channel ◽  
Queuing System ◽  
Queuing Model ◽  
Average Waiting Time ◽  
Queuing Models ◽  
Significant Difference ◽  
Number Of Customers ◽  
Proportionality Principle

Queuing model has been discussed widely in literature. The structures of queuing systems are broadly divided into three namely; single, multi-channel, and mixed. Equations for solving these queuing problems vary in complexity. The most complex of them is the multi-channel queuing problem. A heuristically simplified equation based on relative comparison, using proportionality principle, of the measured effectiveness from the single and multi-channel models seems promising in solving this complex problem. In this study, six different queuing models were used from which five of them are single-channel systems while the balance is multi-channel. Equations for solving these models were identified based on their properties. Queuing models’ performance parameters were measured using relative proportionality principle from which complexity of multi-channel system was transformed to a simple linear relation of the form = . This showed that the performance obtained from single channel model has a linear relationship with corresponding to multi-channel, and is a factor which varies with the structure of queuing system. The model was tested with practical data collected on the arrival and departure of customers from a cocoa processing factory. The performances obtained based on average number of customers on line , average number of customers in the system , average waiting time in line and average waiting time in the system, under certain conditions showed no significant difference between using heuristics and analytical models.

On the waiting-time and busy period distributions for a general birth-and-death queueing model

1975 ◽  
Vol 12 (03) ◽  
pp. 524-532 ◽  
Author(s):  
Bent Natvig
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Arbitrary Number ◽  
Busy Period ◽  
Second Order ◽  
Direct Approach ◽  
Queueing Model ◽  
Number Of Customers

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.

scholarly journals On the stationary distributions in some queueing systems

2021 ◽  
pp. 5-13
Author(s):  
Виталий Николаевич Соболев ◽  
Александр Евгеньевич Кондратенко
Keyword(s):  
Steady State ◽  
Generating Functions ◽  
Queueing Systems ◽  
Queuing System ◽  
Stationary Distributions ◽  
Probability Generating Functions

В статье рассматриваются стационарные распределения числа требований в системах массового обслуживания $M_{\lambda}|G|n|\infty$ и $GI_{\lambda}^{\nu}|M_{\mu}|1|\infty$, и показывается, что введение в данные системы массового обслуживания вспомогательных распределений с понятным вероятностным смыслом вместе с их производящими функциями позволяет упростить как доказательство так и его восприятие, а также приводит к новой записи полученных результатов. В первой системе рассматривается усечённое распределение искомого стационарного распределения для вложенной цепи Маркова. Данное усечение связано с количеством каналов $n$ и описывает вероятностные веса состояний системы, когда существует хотя бы один незанятый канал. Во второй системе для описания результатов используется распределение, связанное с распределением количества заявок во входящей группе требований: определяются вероятности хвостов описанного распределения, а потом для получения вспомогательного вероятностного распределения берётся их удельный вес между собой. This paper deals with two queuing system: $M_{\lambda}|G|n|\infty$ and $GI_{\lambda}^{\nu}|M_{\mu}|1|\infty$. The purpose is to find the steady-state results in terms of the probability-generating functions.

scholarly journals A finite capacity queue with Markovian arrivals and two servers with group services

1994 ◽  
Vol 7 (2) ◽  
pp. 161-178 ◽  
Author(s):  
S. Chakravarthy ◽  
Attahiru Sule Alfa
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Queue Length ◽  
Time Distribution ◽  
Arrival Process ◽  
Finite Capacity ◽  
Waiting Time Distribution ◽  
Phase Type ◽  
Stationary Waiting Time ◽  
Number Of Customers

In this paper we consider a finite capacity queuing system in which arrivals are governed by a Markovian arrival process. The system is attended by two exponential servers, who offer services in groups of varying sizes. The service rates may depend on the number of customers in service. Using Markov theory, we study this finite capacity queuing model in detail by obtaining numerically stable expressions for (a) the steady-state queue length densities at arrivals and at arbitrary time points; (b) the Laplace-Stieltjes transform of the stationary waiting time distribution of an admitted customer at points of arrivals. The stationary waiting time distribution is shown to be of phase type when the interarrival times are of phase type. Efficient algorithmic procedures for computing the steady-state queue length densities and other system performance measures are discussed. A conjecture on the nature of the mean waiting time is proposed. Some illustrative numerical examples are presented.

scholarly journals ANALISIS SISTEM ANTREAN PADA LOKET PENDAFTARAN POLIKLINIK SPESIALIS DI RS BALIMED

2021 ◽  
Vol 10 (2) ◽  
pp. 70
Author(s):  
KADEK DITA SUGIARI ◽  
I WAYAN SUMARJAYA ◽  
KETUT JAYANEGARA
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Queuing Theory ◽  
Queuing System ◽  
State Condition ◽  
Steady State Condition ◽  
Average Waiting Time ◽  
Number Of Patients ◽  
Measures Of Performance

Hospital is one of the service facilities that is not free from queue problem. One example of this hospital is Balimed Hospital. At certain times, especially in the morning, there is a lineup of patients at the Balimed Hospital’s Specialist Polyclinic. In order to maximize service, it is necessary to analyze the queuing system by applying the queuing theory. This study focuses on queues at the Balimed Hospital’s Specialist Polyclinic in Internal Disease. After conducting the research, it was found that the model used at the Specialist Polyclinic in Internal Disease is . With this model, the queuing system at Balimed Hospital's Specialist Polyclinic in Internal Disease is in steady state condition because ???? < 1. The measures of performance for queuing system at Balimed Hospital’s Specialist Polyclinic in Internal Disease is the average number of patients in queue  is 0,1 patient or it can be said that there is almost no patient in queue because the value of  is close to 0, the average number of patients in system  is 1 patient, the average waiting time for patients in queue  is 1 minute, and the average time spent by patients start from queuing until being served  is 2,5 minutes. The queuing system has been effective, it can be seen from the short waiting time for patients.

Realization probability in closed Jackson queueing networks and its application

1987 ◽  
Vol 19 (3) ◽  
pp. 708-738 ◽  
Author(s):  
X. R. Cao
Keyword(s):  
Steady State ◽  
Perturbation Analysis ◽  
Queueing Networks ◽  
Sample Path ◽  
Queueing Network ◽  
Jackson Network ◽  
The Mean ◽  
Service Times ◽  
A New Technique ◽  
Number Of Customers

Perturbation analysis is a new technique which yields the sensitivities of system performance measures with respect to parameters based on one sample path of a system. This paper provides some theoretical analysis for this method. A new notion, the realization probability of a perturbation in a closed queueing network, is studied. The elasticity of the expected throughput in a closed Jackson network with respect to the mean service times can be expressed in terms of the steady-state probabilities and realization probabilities in a very simple way. The elasticity of the throughput with respect to the mean service times when the service distributions are perturbed to non-exponential distributions can also be obtained using these realization probabilities. It is proved that the sample elasticity of the throughput obtained by perturbation analysis converges to the elasticity of the expected throughput in steady-state both in mean and with probability 1 as the number of customers served goes to This justifies the existing algorithms based on perturbation analysis which efficiently provide the estimates of elasticities in practice.

On the waiting-time and busy period distributions for a general birth-and-death queueing model

1975 ◽  
Vol 12 (3) ◽  
pp. 524-532 ◽  
Author(s):  
Bent Natvig
Keyword(s):  
Steady State ◽  
Waiting Time ◽  
Arbitrary Number ◽  
Busy Period ◽  
Second Order ◽  
Direct Approach ◽  
Queueing Model ◽  
Number Of Customers

A general birth-and-death queueing model is considered with N waiting positions (0 ≦ N ≦ ∞), s servers (1 ≦ s ≦ ∞) and a first-come-first-served queueing discipline. The first and second order moments of the steady-state waiting-time (excluding service) for a non-lost arriving customer are given. By setting the busy period equal to the time where at least one service is in progress, we obtain the first and second order moments of the length of a busy period and also the distribution of the number served during it, given an arbitrary number of customers present originally. Using a direct approach all expressions are given in explicit forms which, although being far from elegant, are suitable for evaluation on a computer.

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