An M/M/1/N Queue With Working Breakdowns and Vacations-an Approach to Cost Analysis

Deepa, B.; Kalidass, K.; Vijayalakshmi, V.

doi:10.1088/1742-6596/1850/1/012069

Cited by 5 publications

(7 citation statements)

References 6 publications

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“…(2018) Fundamentals of Queueing Theory explained the steady-state equations (19) . Deepa and Kalidass (2018) discussed the breakdown concept, steady-state equations, and their explanations (2) . It seems that only…”

Section: Steady-state Equationsmentioning

confidence: 99%

“…During the long and uncertain breakdown time, it is commonly considered in the majority of the prior studies that the server entirely stops its service. Deepa and Kalidass discussed the M/M/1/N queue with working breakdowns and vacations (2) . The idea of controlling the speed of arrival times in certain queueing models was discovered by A. Srinivasan and M. Thiagarajan in their study on controllable arrival rates in a variety of queueing models (3) .…”

Section: Introductionmentioning

confidence: 99%

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Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates

Therasal,

Thiagarajan

2024

IJST

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Objectives: This study aims at (i) introducing the finite capacity of the interdependent queueing model with breakdown and controllable arrival rates, (ii) calculating the average number of clients in the system, and identifying the expected waiting period of the clients in the system, (iii) dealing with the model descriptions, steady-state equations, and characteristics, which are expressed in terms of , and (iv) analyzing the probabilities of the queueing system and its characteristics with numerical verification of the obtained results. Methods: While providing the input, the arrival rates through faster and slower arrival rates are controlled using the Poisson process. Also, the service provides an exponential distribution. The server provides the service on an FCFS basis. In this article, two types of models are used: and which are the system’s conditions, where represents the number of units present in the queue in which their probability is and . All probabilities are distributed based on the speed of advent using this concept. Then, the steady-state probabilities are computed using a recursive approach. Findings: This paper discovers the number of clients using the system on average and the expected number of clients in the system using the probability of the steady-state calculation. Little’s formula is used to derive the expected waiting period of the clients in the system. Novelty: There are articles connected to the finite capacity of failed service in functioning and malfunctioning, but this takes the initiative to provide a link in connection with the rates of the controllable arrivals and interdependency in the arrival and service processes. Mathematics Subject allocation: 60K25, 68M20, 90B22. Keywords: M/M/1/K Queue Model, Finite Capacity, Breakdown, Controllable Arrival rates, FCFS Queue Discipline

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Section: Steady-state Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates

Therasal,

Thiagarajan

2024

IJST

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“…Similarly, if the system is in the level state i ≥ 2, the average amount of time to enter the absorbing state ( * ) is represented by the last two terms of equation ( 14). On differentiating (10), (11), (12) and substitute s = 0, we get…”

Section: Expected Waiting Timementioning

confidence: 99%

“…A single server queueing system with batch size service, multiple working vacations, and the server being subject to breakdowns has been investigated by Jeyakumar and Senthilkumar (2017). Deepa and Kalidass (2018) explored an M/M/1/N queueing system with two types of server vacations and working breakdowns. Rajadurai et al (2018) investigated a non-Markovian retrial priority queueing system with fluctuating modes of service and working vacation.…”

Section: Introductionmentioning

confidence: 99%

Analysis of MAP/PH(1),PH(2),PH(3)/2 Queueing System with Three Fluctuating Modes of Service, Standby Server, Working Breakdown, Impatient Behaviour of Customers, Vacations under Bernoulli Schedule and Phase Type Repairs

Govindan¹,

Karthikeyan²

2023

Preprint

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In this model, we consider the customers arrive according to Markovian Arrival Process (MAP) and their three fluctuating modes of service and repair process of both servers are based on phase-type (PH) distributions. While the main server-1 is offering any one of the modes of service who may affect by breakdown immediately go for phase-type repair process at that moment the standby server will interrupt the main server-1 and carry over the service process up to the phase-type repair completion of main server-1. Likewise, the main server-2 also caused by breakdown while offering any one of the modes of service to customers but the main server-2 has a capacity to provide service at the slower rate for the interrupted customer only before move to the phase-type repair process for rejuvenation due to the concept of working breakdown. At every service completion epoch, the main server-1 has an option to move forward to vacation or else remain in the system to offer service to next customer who is next in the queue. If both the server’s are unavailable in the system, the waiting customers in the queueing line may renege from the system because of impatient behaviour. Using matrix analytic method, we examine the steady-state probability vector, busy period and waiting time distribution. Furthermore, we illustrated some of the numerical and graphical representations of the proposed model.

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“…Doshi (4) produced a fantastic overview study on vacation. Additional information about the vacation model is studied from the other articles (5)(6)(7)(8)(9)(10)(11) . All these authors have studied the vacation model but have not included the controllable arrival rates.…”

Section: Introductionmentioning

confidence: 99%

M/M/1 Interdependent Queuing Model with Vacation and Controllable Arrival Rates

Subhapriya,

Thiagarajan

2024

IJST

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Objectives: Rather than working nonstop in the service area, servers take vacations when they have no clients. To determine the probability and features of the queuing system, this study introduces controllable arrival rates and interdependency in the system's service and arrival processes. It also performs a numerical verification of the results. Methods: A recursive method is employed to solve the steady-state probability equations, yielding explicit iterative formulas under the assumption that a single server provides services to all clients. Here, customer arrivals are controlled as either faster or slower, with Poisson assumed by default. Findings: For this model, steady-state solutions and characteristics are derived and explored, and some numerical analysis is carried out using MATLAB. All the probabilities are expressed in terms of , which indicates the system when empty. The movement of the average number of customers in the system and the expected waiting time, and respectively, of the customers in the system is investigated through a graph. and decrease when dependence service rate, and faster arrival rate increase. Additionally, increases and decreases when the slower arrival rate increases. Novelty: Although there have been studies on vacation in queuing theory, this new approach aims to bridge the gap between vacation and interdependency in the arrival and service process, as well as controllable arrival rates. When vacations with predictable arrival rates are utilised advantageously for the benefit of both the server and the client, waiting times may be minimised and the most practical, economical service can be provided. Keywords: Markovian Queuing System, Vacation, Loss and Delay, Finite Capacity, Interdependent Arrival and Service Rates, Varying Arrival Rates, Bivariate Poisson Process

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An M/M/1/N Queue With Working Breakdowns and Vacations-an Approach to Cost Analysis

Abstract: In this paper, we consider an M/M/1/N queue with working breakdown and two types of server vacations. Sensitivity analysis of the model is performed to aissue with the point of streamlining service rates by limiting the normal expense per unit time using the direct search method.

Cited by 5 publications

References 6 publications

Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates

Poisson Input and Exponential Service Time Finite Capacity Interdependent Queueing Model with Breakdown and Controllable Arrival Rates

Analysis of MAP/PH(1),PH(2),PH(3)/2 Queueing System with Three Fluctuating Modes of Service, Standby Server, Working Breakdown, Impatient Behaviour of Customers, Vacations under Bernoulli Schedule and Phase Type Repairs

M/M/1 Interdependent Queuing Model with Vacation and Controllable Arrival Rates

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