Maximum Entropy Approach for Un-Reliable Server
Vacation Queueing Model with Optional Bulk 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.

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A Taylor Series Approach for Service-Coupled Queueing Systems with Intermediate Load

Mathematical Problems in Engineering ◽

10.1155/2017/3298605 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 3

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%.

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Two parallel finite queues with simultaneous services and Markovian arrivals

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953397000439 ◽

1997 ◽

Vol 10 (4) ◽

pp. 383-405 ◽

Cited By ~ 2

Author(s):

S. R. Chakravarthy ◽

S. Thiagarajan

Keyword(s):

Markov Process ◽

Performance Measures ◽

System Performance ◽

Arrival Process ◽

Single Server ◽

Queueing Model ◽

Finite Capacity ◽

Numerical Examples ◽

Finite Queues ◽

Large State Space

In this paper, we consider a finite capacity single server queueing model with two buffers, A and B, of sizes K and N respectively. Messages arrive one at a time according to a Markovian arrival process. Messages that arrive at buffer A are of a different type from the messages that arrive at buffer B. Messages are processed according to the following rules: 1. When buffer A(B) has a message and buffer B(A) is empty, then one message from A(B) is processed by the server. 2. When both buffers, A and B, have messages, then two messages, one from A and one from B, are processed simultaneously by the server. The service times are assumed to be exponentially distributed with parameters that may depend on the type of service. This queueing model is studied as a Markov process with a large state space and efficient algorithmic procedures for computing various system performance measures are given. Some numerical examples are discussed.

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Steady-State Analysis of a Flexible Markovian Queue with Server Breakdowns

Entropy ◽

10.3390/e21030259 ◽

2019 ◽

Vol 21 (3) ◽

pp. 259 ◽

Cited By ~ 1

Author(s):

Messaoud Bounkhel ◽

Lotfi Tadj ◽

Ramdane Hedjar

Keyword(s):

Steady State ◽

Probability Generating Function ◽

Threshold Level ◽

System Size ◽

Single Server ◽

Markovian Queue ◽

Breakdown Rates ◽

Bulk Service ◽

Service Rates ◽

Number Of Customers

A flexible single-server queueing system is considered in this paper. The server adapts to the system size by using a strategy where the service provided can be either single or bulk depending on some threshold level c. If the number of customers in the system is less than c, then the server provides service to one customer at a time. If the number of customers in the system is greater than or equal to c, then the server provides service to a group of c customers. The service times are exponential and the service rates of single and bulk service are different. While providing service to either a single or a group of customers, the server may break down and goes through a repair phase. The breakdowns follow a Poisson distribution and the breakdown rates during single and bulk service are different. Also, repair times are exponential and repair rates during single and bulk service are different. The probability generating function and linear operator approaches are used to derive the system size steady-state probabilities.

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On Mx/(G1G2)/1/G(BS)/Vs vacation queue with two types of general heterogeneous service

Journal of Applied Mathematics and Decision Sciences ◽

10.1155/jamds.2005.123 ◽

2005 ◽

Vol 2005 (3) ◽

pp. 123-135 ◽

Cited By ~ 11

Author(s):

Kailash C. Madan ◽

Z. R. Al-Rawi ◽

Amjad D. Al-Nasser

Keyword(s):

Steady State ◽

Performance Measures ◽

Waiting Time ◽

Busy Period ◽

Single Server ◽

Queue Size ◽

Single Vacation ◽

The Mean ◽

Expected Waiting Time ◽

Heterogeneous Service

We analyze a batch arrival queue with a single server providing two kinds of general heterogeneous service. Just before his service starts, a customer may choose one of the services and as soon as a service (of any kind) gets completed, the server may take a vacation or may continue staying in the system. The vacation times are assumed to be general and the server vacations are based on Bernoulli schedules under a single vacation policy. We obtain explicit queue size distribution at a random epoch as well as at a departure epoch and also the mean busy period of the server under the steady state. In addition, some important performance measures such as the expected queue size and the expected waiting time of a customer are obtained. Further, some interesting particular cases are also discussed.

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The single server vacation queue with geometric abandonments and Bernoulli’s feedbacks

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i2.21.11861 ◽

2018 ◽

Vol 7 (2.21) ◽

pp. 172

Author(s):

V Vijayalakshmi ◽

K Kalidass

Keyword(s):

Performance Measures ◽

Numerical Experiments ◽

The State ◽

Single Server ◽

Server Vacation

In this article the behaviour of a single server vacation queue with geometric abandonments and Bernoulli’s feedbacks is carried out and various important performance measures are derived. Some numerical experiments are presented to study how the parameters of the model influence the state of the system.

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The single server vacation queueing model with geometric abandonments

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2011.03.010 ◽

2011 ◽

Vol 141 (8) ◽

pp. 2863-2877 ◽

Cited By ~ 9

Author(s):

Spiros Dimou ◽

Antonis Economou ◽

Demetrios Fakinos

Keyword(s):

Single Server ◽

Queueing Model ◽

Server Vacation

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Analysis of a Single Server Retrial Queue with Server Vacation and Two Waiting Buffers Based on ATM Networks

Mathematical Problems in Engineering ◽

10.1155/2019/4193404 ◽

2019 ◽

Vol 2019 ◽

pp. 1-14

Author(s):

Shan Gao ◽

Xianchao Wang

Keyword(s):

Performance Measures ◽

Generating Function ◽

Cost Optimization ◽

Retrial Queue ◽

Atm Networks ◽

Single Server ◽

Supplementary Variable ◽

Single Vacation ◽

Extensive Analysis ◽

Queue Lengths

This paper presents a discrete-time Geo/G/1 retrial queue with two waiting buffers to model an ATM network, in which the server begins a single vacation in cases where the system is empty at the instant of a service completion. New arriving customer who finds the server being on vacation can decide to either enter the retrial buffer with some probability p or leave the system with complementary probability 1−p. But the new arriving customer can begin its service immediately if he finds the server idle and join the original buffer if he finds the server busy. We first carry out an extensive analysis of the model by using the supplementary variable method and the generating function approach, and give some performance measures, such as server’s state probabilities and mean queue lengths in the original buffer, retrial buffer, and in the system. Secondly, we give the generating function of the sojourn time of a customer in the system and prove that Little’s law still holds in our model. Sensitivity analysis and cost optimization are finally given for illustrative purposes.

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A feedback queue with additional optional batch service

Malaysian Journal of Fundamental and Applied Sciences ◽

10.11113/mjfas.v6n2.194 ◽

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.

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