scholarly journals A feedback queue with additional optional batch service

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
R. Kalayanaraman ◽  
S. Sumathy
Keyword(s):  
Queue Length ◽  
Probability Generating Function ◽  
Arrival Process ◽  
Single Server ◽  
Queuing Model ◽  
Steady State Probability ◽  
Poisson Arrival ◽  
Optional Service ◽  
Essential Service ◽  
Feedback Queue

A single server infinite capacity queuing system with Poisson arrival process along with Bernoulli feedback decision process is considered wherein the server provides two types of service. The first essential service is rendered one by one to all the customers and second optional service is given in batches of fixed size b. For this model the steady state probability generating function for the queue length process has been obtained and average queue length has been found explicitly. Results for particular cases are obtained and some numerical results are presented to test the feasibility of the queuing model.

scholarly journals TheN×D-BMAP/G/1 Queueing Model: Queue Contents and Delay Analysis

2011 ◽  
Vol 2011 ◽  
pp. 1-31 ◽  
Author(s):  
Bart Steyaert ◽  
Joris Walraevens ◽  
Dieter Fiems ◽  
Herwig Bruneel
Keyword(s):  
Closed Form ◽  
Customer Service ◽  
Probability Generating Function ◽  
Mean Value ◽  
Arrival Process ◽  
Single Server ◽  
Steady State Probability ◽  
Absolute Minimum ◽  
Number Of Customers ◽  
Customer Delay

We consider a single-server discrete-time queueing system with N sources, where each source is modelled as a correlated Markovian customer arrival process, and the customer service times are generally distributed. We focus on the analysis of the number of customers in the queue, the amount of work in the queue, and the customer delay. For each of these quantities, we will derive an expression for their steady-state probability generating function, and from these results, we derive closed-form expressions for key performance measures such as their mean value, variance, and tail distribution. A lot of emphasis is put on finding closed-form expressions for these quantities that reduce all numerical calculations to an absolute minimum.

Transient and steady-state analysis of a queuing system having customers’ impatience with threshold

2019 ◽  
Vol 53 (5) ◽  
pp. 1861-1876 ◽  
Author(s):  
Sapana Sharma ◽  
Rakesh Kumar ◽  
Sherif Ibrahim Ammar
Keyword(s):  
Steady State ◽  
Probability Generating Function ◽  
Threshold Value ◽  
Steady State Solution ◽  
Function Technique ◽  
Single Server ◽  
Queuing Model ◽  
Queuing Models ◽  
Special Cases ◽  
Number Of Customers

In many practical queuing situations reneging and balking can only occur if the number of customers in the system is greater than a certain threshold value. Therefore, in this paper we study a single server Markovian queuing model having customers’ impatience (balking and reneging) with threshold, and retention of reneging customers. The transient analysis of the model is performed by using probability generating function technique. The expressions for the mean and variance of the number of customers in the system are obtained and a numerical example is also provided. Further the steady-state solution of the model is obtained. Finally, some important queuing models are derived as the special cases of this model.

scholarly journals A discrete single server queue with Markovian arrivals and phase type group services

1995 ◽  
Vol 8 (2) ◽  
pp. 151-176 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
K. Laurie Dolhun ◽  
S. Chakravarthy
Keyword(s):  
Steady State ◽  
Queueing System ◽  
State Probability ◽  
Arrival Process ◽  
Single Server ◽  
Probability Vector ◽  
Phase Type Distribution ◽  
Steady State Probability ◽  
Phase Type ◽  
Number Of Customers

We consider a single-server discrete queueing system in which arrivals occur according to a Markovian arrival process. Service is provided in groups of size no more than M customers. The service times are assumed to follow a discrete phase type distribution, whose representation may depend on the group size. Under a probabilistic service rule, which depends on the number of customers waiting in the queue, this system is studied as a Markov process. This type of queueing system is encountered in the operations of an automatic storage retrieval system. The steady-state probability vector is shown to be of (modified) matrix-geometric type. Efficient algorithmic procedures for the computation of the rate matrix, steady-state probability vector, and some important system performance measures are developed. The steady-state waiting time distribution is derived explicitly. Some numerical examples are presented.

scholarly journals The BMAP/G/1 vacation queue with queue-length dependent vacation schedule

1998 ◽  
Vol 40 (2) ◽  
pp. 207-221 ◽  
Author(s):  
Yang Woo Shin ◽  
Chareles E. M. Pearce
Keyword(s):  
Queue Length ◽  
Length Distribution ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Server Vacation ◽  
Batch Markovian Arrival Process ◽  
Queue Length Distribution ◽  
Special Cases ◽  
Number Of Customers

AbstractWe treat a single-server vacation queue with queue-length dependent vacation schedules. This subsumes the single-server vacation queue with exhaustive service discipline and the vacation queue with Bernoulli schedule as special cases. The lengths of vacation times depend on the number of customers in the system at the beginning of a vacation. The arrival process is a batch-Markovian arrival process (BMAP). We derive the queue-length distribution at departure epochs. By using a semi-Markov process technique, we obtain the Laplace-Stieltjes transform of the transient queue-length distribution at an arbitrary time point and its limiting distribution

scholarly journals Analysis of Queue-Length Dependent Vacations and P-Limited Service in BMAP/G/1/N Systems: Stationary Distributions and Optimal Control

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
A. D. Banik
Keyword(s):  
Queue Length ◽  
Minimum Cost ◽  
Arrival Process ◽  
Service Discipline ◽  
Single Server ◽  
Finite Buffer ◽  
Batch Markovian Arrival Process ◽  
Special Cases ◽  
Number Of Customers ◽  
Limited Service

We consider a finite-buffer single server queueing system with queue-length dependent vacations where arrivals occur according to a batch Markovian arrival process (BMAP). The service discipline is P-limited service, also called E-limited with limit variation (ELV) where the server serves until either the system is emptied or a randomly chosen limit of L customers has been served. Depending on the number of customers present in the system, the server will monitor his vacation times. Queue-length distributions at various epochs such as before, arrival, arbitrary and after, departure have been obtained. Several other service disciplines like Bernoulli scheduling, nonexhaustive service, and E-limited service can be treated as special cases of the P-limited service. Finally, the total expected cost function per unit time is considered to determine locally optimal values N* of N or a maximum limit L^* of L^ as the number of customers served during a service period at a minimum cost.

scholarly journals Maximum Entropy Approach for Un-Reliable Server Vacation Queueing Model with Optional Bulk Service

2012 ◽  
Vol 23 (1) ◽  
pp. 89-113
Author(s):  
Madhu Jain, Madhu Jain,
Keyword(s):  
Maximum Entropy ◽  
Performance Measures ◽  
Probability Generating Function ◽  
Single Server ◽  
Queueing Model ◽  
Time Service ◽  
Server Vacation ◽  
Single Vacation ◽  
Bulk Service ◽  
Essential Service

In this study, we consider a single server vacation queueing model with optional bulk service and an un-reliable server. A single server provides first essential service (FES) to all arriving customers one by one; apart from essential service, he can also facilitate the additional phase of optional service (OS) in batches of fixed size b( ≥ 1), in case when the customers request for it. The server may take a single vacation whenever he finds no customers waiting in the queue to be served. Moreover, the server is subjected to unpredictable breakdown while providing the first essential service. The vacation time, service time and repair time of the server are exponentially distributed. The steady state results are obtained in terms of probability generating function for queue size distributions. By using the maximum entropy analysis (MEA), we derive various system performance measures. A comparative study is performed between the exact and approximate waiting time of the system. By taking the numerical illustrations, the sensitivity analysis is done to explore the effect of different descriptors on various performance measures.

scholarly journals Transient performance analysis of a single server queuing model with retention of reneging customers

2018 ◽  
Vol 28 (3) ◽  
pp. 315-331 ◽  
Author(s):  
Rakesh Kumar ◽  
Sapana Sharma
Keyword(s):  
Performance Analysis ◽  
Probability Generating Function ◽  
Time Dependent ◽  
Function Technique ◽  
Single Server ◽  
Transient Solution ◽  
Transient Performance ◽  
Queuing Model ◽  
Special Cases ◽  
Mean And Variance

In this paper, we study a single server queuing model with retention of reneging customers. The transient solution of the model is derived using probability generating function technique. The time-dependent mean and variance of the model are also obtained. Some important special cases of the model are derived and discussed. Finally, based on the numerical example, the transient performance analysis of the model is performed.

scholarly journals A bulk queueing system under N-policy with bilevel service delay discipline and start-up time

1993 ◽  
Vol 6 (4) ◽  
pp. 359-384 ◽  
Author(s):  
David C. R. Muh
Keyword(s):  
Queue Length ◽  
Queueing System ◽  
Probability Generating Function ◽  
Single Server ◽  
Numerical Examples ◽  
Queueing Process ◽  
Poisson Input ◽  
Service Delay ◽  
Start Up ◽  
Bulk Arrival

The author studies the queueing process in a single-server, bulk arrival and batch service queueing system with a compound Poisson input, bilevel service delay discipline, start-up time, and a fixed accumulation level with control operating policy. It is assumed that when the queue length falls below a predefined level r(≥1), the system, with server capacity R, immediately stops service until the queue length reaches or exceeds the second predefined accumulation level N(≥r). Two cases, with N≤R and N≥R, are studied.The author finds explicitly the probability generating function of the stationary distribution of the queueing process and gives numerical examples.

An N-policy discrete-time Geo/G/1 queue with modified multiple server vacations and Bernoulli feedback*

2019 ◽  
Vol 53 (2) ◽  
pp. 367-387
Author(s):  
Shaojun Lan ◽  
Yinghui Tang
Keyword(s):  
Discrete Time ◽  
Queue Length ◽  
Length Distribution ◽  
Probability Generating Function ◽  
Renewal Theory ◽  
Function Technique ◽  
Single Server ◽  
Bernoulli Feedback ◽  
Queue Length Distribution ◽  
Special Cases

This paper deals with a single-server discrete-time Geo/G/1 queueing model with Bernoulli feedback and N-policy where the server leaves for modified multiple vacations once the system becomes empty. Applying the law of probability decomposition, the renewal theory and the probability generating function technique, we explicitly derive the transient queue length distribution as well as the recursive expressions of the steady-state queue length distribution. Especially, some corresponding results under special cases are directly obtained. Furthermore, some numerical results are provided for illustrative purposes. Finally, a cost optimization problem is numerically analyzed under a given cost structure.

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