# Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{X}/G (a, b)/1 queue system with multiple vacations [3]. In all queue models with vacations that have been studied in this context, the server remains on vacation, even whilst the queue period is excessive enough to initiate the primary provider. The modeling of actual-time systems can benefit from those vacation methods. Baba examined the M/PH/1 line with working vacations and interruptions [4]. The M

^{X}/G (a, b)/1 queue model with vacation interruption was also researched by Haridass and Arumuganathan [5]. Through real-time applications, they deduced diverse queue device functions. Gao and Liu investigated the M/G/1 queue under a Bernoulli agenda with a single working vacation and vacation interruption [6]. Customers arrive at a queue served by a single server, and their arrivals follow a Poisson process. The duration of the service provided by the server conforms to an exponential distribution. However, the server takes periodic vacations which are scheduled based on a Bernoulli process. During the vacations, the server can be interrupted and resume service if a customer arrives, preventing the vacation from being completed, as studied by Tao et al. [7]. A countless-buffer bulk arrival queue with a bulk-size-dependent provider was tested by Pradhan and Gupta [8]. Madan et al. studied the consistent state of two M

^{X}/M (a, b)/1 queue models with random breakdowns. In 2003, they took into account the reality that the repair time is deterministic for one version and exponential for every other. It has been cited in the literature on queue fashions with server breakdown that, excluding Madan et al., each creator who discusses server breakdown in the context of a bulk provider queue version deals with a server that can only serve one customer at a time. The server may also interrupt right away if trouble arises. Yet, in the majority of instances, it is impossible to disturb the server before it has completed providing its bulk services [9]. Jeyakumar and Senthilnathan also studied breakdown without carrier disruption in a bulk carrier queue model and a bulk arrival queue model. They developed a model using closedown time and constructed probability-generating functions for the completion epochs of services, vacations, and renewals [10]. Wu et al. investigated an M/G/1 queue system with an N-policy, a solitary vacation, an unreliable service station, and a replaceable repair facility [11,12].

^{X}/G (a, b)/1 queuing system with an optional additional service and numerous vacations, and setup time is examined by Ayyappan and Deepa [22]. Niranjan et al. analyzed a non-Markovian bulk queueing system with renewal and startup/shutdown times [23]. Nithya and Haridass conducted a maximum entropy analysis of a queueing system that controls both arrival and bulk service, incorporating breakdowns and multiple vacation periods [24]. In their article, Enogwe and Obiora-Ilouno explored the impacts of three pivotal factors—reneging, server breakdown, and server vacation—on various stages of a queueing system with bulk arrivals and a single server [25]. Khan IE and Paramasivam R analyzed the quality control policy for the Markovian model with feedback, balking, and maintaining reneged clients using an iterative method for the nth customer in the system [26]. Ammar, Rajadurai P analyzed an innovative type of retry queueing system with functional breakdown services presented in this inquiry. Priority and regular clients are two different categories that are taken into consideration [27]. Gabi Hanukov and Shraga Shoval introduce a compelling vacation queue model that accounts for dynamic server service rates affected by factors like human fatigue and machinery wear. The analysis employed multidimensional methods to explore the impact of different vacation policies on system efficiency. The findings underscore the significant positive effect of strategic vacation scheduling on mean customer waiting time, suggesting potential benefits even when switching to a temporary server during times of higher main-server service rates [28]. Xing et al. investigated traffic accident patterns in undersea tunnels, offering valuable insights into the evolution of vehicle congestion queuing. The authors present precise queue-length estimation models based on shockwave theory and real-time data input, demonstrating optimal accuracy with a 30 s time interval. The results highlight the effectiveness of the model, with an accuracy of 92.34% for the maximum queue-length estimation model and 83.50% for the real-time whole-process queue-length estimation model. The proposed approach outperforms the input–output model, indicating its potential for supporting timely and effective control measures in undersea tunnel traffic management [29]. Mohan Chaudhry et al. addresses a finite-space, single-server queueing system with a unique (a, b)-bulk service rule and finite-buffer capacity ‘N’. It introduces a novel approach utilizing embedded Markov chains, Markov renewal theory, and semi-Markov processes to derive probability distributions for queue lengths at post-departure and random epochs. This investigation establishes a functional relationship between the probability-generating function representing the queue-length distribution and the Laplace–Stieltjes transform (LST) of the queueing-time distribution. This connection facilitates the derivation of waiting-time distributions for individual random customers. The use of LSTs facilitates a comprehensive discussion of the probability density function of waiting times, emphasizing numerical implementations for practical applications [30]. Laurentiu Rece et al. introduce a novel approach using queueing theory models to optimize production department size, production costs, and provisioning. This method employs queuing mathematical models to form the basis for an experimental algorithm and a numerical approach. This research effectively employed these models in designing a practical industrial engineering unit, aligning with technological flow and equipment schemes. The focus on minimizing costs in terms of server count is addressed using the Monte Carlo method, showcasing the practicality of iterative methods like Jacobi and Gauss–Seidel in solving the associated linear system for Jackson queueing networks [31]. Mustafa Demircioglu et al. investigated the influence of disasters on a discrete-time single-server queueing system featuring general independent arrivals and service times. Disasters, modeled as a Bernoulli process, lead to the simultaneous removal of all customers. This study employs a two-dimensional Markovian state description, providing expressions for probability-generating functions, and average values, variances, and tail probabilities for both system content and customer sojourn time are analyzed under a first-come-first-served policy. The derivation of customer loss probability due to disaster occurrences is also addressed, with numerical illustrations enhancing the understanding of the proposed models [32]. In all the above queueing models, essential two-phase bulk service is not considered. It is mandatory to analyze many real-time systems such as communication systems, the manufacturing industry, production systems, network systems, etc. Multiple control policies and renewal of service stations in first-phase customer feedback are the technical terms introduced in this paper which will be useful the studying the performance analysis of DRX mechanisms in network systems.

## 2. Motivation

## 3. System Analysis

#### 3.1. Arrival Process

_{1}, as described in our work. The inter-arrival time adheres to a certain pattern. The distribution of group size follows a geometric distribution, while the distribution of the exponential follows an exponential distribution.

#### 3.2. Service Process

#### 3.3. Bernoulli Feedback

#### 3.4. Renewal Time

#### 3.5. Vacation

#### 3.6. Setup Time

#### 3.7. Model Description

_{1}. The bulk size distribution of the arrival is geometric. The bulk service process is split into two phases called FES and SES with minimum server capacity ${}^{\prime}{a}^{\prime}$ and maximum server capacity ${}^{\prime}{b}^{\prime}$ by Neuts introduction of the general rule for bulk service [33]. The server will be turned on only if the queue length reaches the value ‘a’. In the event of a server failure during the FES epoch, the service process persists without interruption for the ongoing group of customers, facilitated by technical precautions. The server is designed to deliver a pivotal two-phase service. In the initial FES phase, if the server experiences failure with a probability of ‘δ’, the renewal of the service station is triggered. Conversely, if there is no server failure with a probability of ‘1 – δ’, the server transitions to a successive phase called SES. Customers who require further service as feedback will be given priority and join at the head of the queue with probability $\beta $. On the contrary, the customer who does not require feedback may leave the system with a probability ${}^{\prime}1-{\beta}^{\prime}$. If the queue length is less than ${}^{\prime}{a}^{\prime}$ after SES, the server may leave for a single vacation with probability ${}^{\prime}1-{\beta}^{\prime}$. When the server finds an inadequate number of customers in the queue after vacation completion, the server becomes dormant. After vacation completion, the server requires some time to start service which is the setup time. The setup time will be initiated only when the queue length is at least ${}^{\prime}{a}^{\prime}$. Even after setup time completion, the service process will be started only with the queue length ‘N’ (N > b).

## 4. Notations

## 5. Steady-State Queue Size Distribution

## 6. Probability-Generating Function of the Queue Size at Any Time

#### 6.1. Steady-State Condition

#### 6.2. Result

**Theorem**

**1.**

**Proof**

**.**

## 7. Performance Measures

#### 7.1. Expected Number of Customers in the Queue

#### 7.2. Expected Waiting Time of a Customer in the Queue

#### 7.3. Expected Duration of the Dormant Period

#### 7.4. Duration of the Server’s Expected Busy Period

## 8. Cost Model

## 9. Numerical Illustration

Minimum server capacity | a |

Maximum server capacity | b |

Threshold value | N |

Startup cost | Rs. 4 |

Holding cost per customer | Rs. 3 |

Operating cost per unit time | Rs. 5 |

Reward per unit time due to vacation | Rs. 1 |

Renewal cost per unit time | Rs. 2 |

Setup time cost per unit time | Rs. 0.50 |

#### 9.1. Results and Discussion

#### 9.1.1. Impact of Arrival Rate on Performance Metrics

#### 9.1.2. Impact of Breakdown Probability on Performance Metrics

#### 9.1.3. Effects of Renewal Rate on Performance Measures

#### 9.1.4. Effects of Threshold Values ‘a’ and ‘N’ on Total Average Cost

#### 9.2. Optimal Cost

## 10. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Tian, N.; Zhang, Z.G. Vacation Queueing Models Theory and Applications; International Series in Operations Research & Management Science; Springer: Boston, MA, USA, 2006; Volume 93, ISBN 978-0-387-33721-0. [Google Scholar]
- Arumuganathan, R.; Jeyakumar, S. Steady state analysis of a bulk queue with multiple vacations, setup times with N-policy and closedown times. Appl. Math. Model.
**2005**, 29, 972–986. [Google Scholar] [CrossRef] - Jeyakumar, S.; Senthilnathan, B. Steady state analysis of bulk arrival and bulk service queueing model with multiple working vacations. Int. J. Math. Oper. Res.
**2016**, 9, 375–394. [Google Scholar] [CrossRef] - Baba, Y. The M/PH/1 queue with working vacations and vacation interruption. J. Syst. Sci. Syst. Eng.
**2010**, 19, 496–503. [Google Scholar] [CrossRef] - Haridass, M.; Arumuganathan, R. Analysis of a MX/G(a,b)/1 queueing system with vacation interruption. RAIRO-Oper. Res.
**2012**, 46, 305–334. [Google Scholar] [CrossRef] - Gao, S.; Liu, Z. An M/G/1 queue with single working vacation and vacation interruption under Bernoulli schedule. Appl. Math. Model.
**2013**, 37, 1564–1579. [Google Scholar] [CrossRef] - Tao, L.; Wang, Z.; Liu, Z. The GI/M/1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Appl. Math. Model.
**2013**, 37, 3724–3735. [Google Scholar] [CrossRef] - Pradhan, S.; Gupta, U.C. Modeling and analysis of an infinite-buffer batch-arrival queue with batch-size-dependent service: MX/Gn(a,b)/1. Perform. Eval.
**2017**, 108, 16–31. [Google Scholar] [CrossRef] - Madan, K.; Abu-Dayyeh, W.; Gharaibeh, M. Steady state analysis of two MX/Ma,b/1 queue models with random breakdowns. Int. J. Inf. Manag. Sci.
**2003**, 14, 37–51. [Google Scholar] - Jeyakumar, S.; Senthilnathan, B. A study on the behaviour of the server breakdown without interruption in a Mx/G(a,b)/1 queueing system with multiple vacations and closedown time. Appl. Math. Comput.
**2012**, 219, 2618–2633. [Google Scholar] [CrossRef] - Wu, W.; Tang, Y.; Yu, M. Analysis of an M/G/1 queue with N-policy, single vacation, unreliable service station and replaceable repair facility. Opsearch
**2015**, 52, 670–691. [Google Scholar] [CrossRef] - Niranjan, S.P.; Chandrasekaran, V.M.; Indhira, K. Stochastic modelling of a two phase bulk service queueing system with active bernoulli feedback, server loss and vacation. Int. J. Pure Appl. Math.
**2017**, 115, 433–445. [Google Scholar] - Sama, H.; Vemuri, V.; Talagadadeevi, S.; Bhavirisetti, S. Analysis of an N-policy MX/M/1 Two-phase Queueing System with State-dependent Arrival Rates and Unreliable Server. Ingénierie Syst. Inf.
**2019**, 24, 233–240. [Google Scholar] [CrossRef] - Rao, A.A.; Vedala, N.R.D.; Chandan, K. M/M/1 Queue with N-Policy Two-Phase, Server Start-Up, Time-Out and Breakdowns. Int. J. Recent. Technol. Eng.
**2019**, 8, 9165–9171. [Google Scholar] [CrossRef] - Enogwe, S.U.; Onyeagu, S.I.; Obiora-Ilouno, H.O. On single server batch arrival queueing system with balking, three types of heterogeneous service and Bernoulli schedule server vacation. Math. Theory Model.
**2021**, 11, 40. [Google Scholar] - GnanaSekar, M.M.N.; Kandaiyan, I. Analysis of an M/G/1 Retrial Queue with Delayed Repair and Feedback under Working Vacation policy with Impatient Customers. Symmetry
**2022**, 14, 2024. [Google Scholar] [CrossRef] - Niranjan, S.P. Managerial decision analysis of bulk arrival queuing system with state dependent breakdown and vacation. Int. J. Adv. Oper. Manag.
**2020**, 12, 351–376. [Google Scholar] [CrossRef] - Blondia, C. A queueing model for a wireless sensor node using energy harvesting. Telecommun. Syst.
**2021**, 77, 335–349. [Google Scholar] [CrossRef] - Merit, C.K.D.; Haridass, M. A simulation study on the necessity of working breakdown in a state dependent bulk arrival queue with disaster and optional re-service. Int. J. Ad Hoc Ubiquitous Comput.
**2022**, 41, 1–15. [Google Scholar] [CrossRef] - Deepa, V.; Haridass, M.; Selvamuthu, D.; Kalita, P. Analysis of energy efficiency of small cell base station in 4G/5G networks. Telecommun. Syst.
**2023**, 82, 381–401. [Google Scholar] [CrossRef] - Niranjan, S.P.; Chandrasekaran, V.M.; Indhira, K. Phase dependent breakdown in bulk arrival queueing system with vacation break-off. Int. J. Data Anal. Tech. Strateg.
**2020**, 12, 127–154. [Google Scholar] [CrossRef] - Ayyappan, G.; Deepa, T. Analysis of batch arrival bulk service queue with multiple vacation closedown essential and optional repair. Appl. Appl. Math.
**2018**, 13, 2. [Google Scholar] - Niranjan, S.P.; Komala Durga, B.; Thangaraj, M. Steady-State Analysis of Bulk Queuing System with Renovation, Prolonged Vacation and Tune-Up/Shutdown Times. In Proceedings of the 2nd International Conference on Mathematical Modeling and Computational Science, Surat, India, 5–6 February 2022; Peng, S.-L., Lin, C.-K., Pal, S., Eds.; Springer Nature: Singapore, 2022; pp. 35–48. [Google Scholar]
- Nithya, R.P.; Haridass, M. Cost optimisation and maximum entropy analysis of a bulk queueing system with breakdown, controlled arrival and multiple vacations. Int. J. Oper. Res.
**2020**, 39, 279–305. [Google Scholar] [CrossRef] - Enogwe, S.; Obiora-Ilouno, O.; Enogwe, S.; Obiora-Ilouno, H. Effects of Re-neging, Server Breakdowns and Vacation on a Batch Arrival Single Server Queueing System with Three Fluctuating Modes of Service. Open J. Optim.
**2020**, 9, 105–128. [Google Scholar] [CrossRef] - Khan, I.; Paramasivam, R. Reduction in Waiting Time in an M/M/1/N Encouraged Arrival Queue with Feedback, Balking and Maintaining of Reneged Customers. Symmetry
**2022**, 14, 1743. [Google Scholar] [CrossRef] - Ammar, S.; Rajadurai, P. Performance Analysis of Preemptive Priority Retrial Queueing System with Disaster under Working Breakdown Services. Symmetry
**2019**, 11, 419. [Google Scholar] [CrossRef] - Hanukov, G.; Shoval, S. A Model for a Vacation Queuing Policy Considering Server’s Deterioration and Recovery. Mathematics
**2023**, 11, 2640. [Google Scholar] [CrossRef] - Xing, R.; Cai, X.; Liu, Y.; Yang, Z.; Wang, Y.; Peng, B. Study on Queue Length in the Whole Process of a Traffic Accident in an Extra-Long Tunnel. Mathematics
**2023**, 11, 1773. [Google Scholar] [CrossRef] - Chaudhry, M.; Datta Banik, A.; Barik, S.; Goswami, V. A Novel Computational Procedure for the Waiting-Time Distribution (In the Queue) for Bulk-Service Finite-Buffer Queues with Poisson Input. Mathematics
**2023**, 11, 1142. [Google Scholar] [CrossRef] - Rece, L.; Vlase, S.; Ciuiu, D.; Neculoiu, G.; Mocanu, S.; Modrea, A. Queueing Theory-Based Mathematical Models Applied to Enterprise Organization and Industrial Production Optimization. Mathematics
**2022**, 10, 2520. [Google Scholar] [CrossRef] - Demircioglu, M.; Bruneel, H.; Wittevrongel, S. Analysis of a Discrete-Time Queueing Model with Disasters. Mathematics
**2021**, 9, 3283. [Google Scholar] [CrossRef] - Marcel, F. Neuts A General Class of Bulk Queues with Poisson Input. Ann. Math. Stat.
**1967**, 38, 759–770. [Google Scholar] [CrossRef] - Cox, D.R. The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Math. Proc. Camb. Philos. Soc.
**1955**, 51, 433–441. [Google Scholar] [CrossRef]

**Figure 7.**Threshold value vs. TAC for $\lambda =2,b=8,{\mathrm{\mu}}_{1}={\mathrm{\mu}}_{2}=2.0,\mathrm{\u1d93}=1,\gamma =3,\eta =4,\alpha =6$.

Cumulative Distribution Function | Probability Distribution Function | Laplace–Stieltjes Transform | Remaining Service Time | |
---|---|---|---|---|

First essential service | $S(y)$ | $s(y)$ | $\stackrel{~}{S}(\mathsf{\theta})$ | ${S}^{0}(y)$ |

Second essential service | ${S}_{2}(y)$ | ${s}_{2}(y)$ | ${\stackrel{~}{S}}_{2}(\mathsf{\theta})$ | ${{S}_{2}}^{0}(y)$ |

Vacation | V($y)$ | v$(y)$ | $\stackrel{~}{V}(\mathsf{\theta})$ | ${V}^{0}$$(y)$ |

Renewal time | R($y)$ | r$(y)$ | $\stackrel{~}{R}(\mathsf{\theta})$ | ${R}^{0}(y)$ |

Preparatory work | A$(y)$ | a$(y)$ | $\stackrel{~}{A}(\mathsf{\theta})$ | ${A}^{0}(y)$ |

Parameter | Distribution | Notations |
---|---|---|

Arrival rate | Poisson distribution | ${\lambda}_{1}$ |

First essential service | 2-Erlang distribution | ${\mu}_{1}$ |

Second essential service | 2-Erlang distribution | ${\mu}_{2}$ |

Vacation | Exponential distribution | ᶓ |

Renewal | Exponential distribution | η |

Setup time | Exponential distribution | $\alpha $ |

${\mathsf{\lambda}}_{1}$ | E(Q) | E(B) | E(I) | E(W) |
---|---|---|---|---|

3.0 | 3.7541 | 5.2123 | 2.5454 | 1.2212 |

3.4 | 4.5432 | 6.5216 | 2.1181 | 2.3321 |

4.0 | 5.6632 | 7.9934 | 1.5432 | 4.1254 |

4.5 | 7.1123 | 9.1211 | 1.1214 | 5.4313 |

5.0 | 8.8246 | 11.2321 | 0.5432 | 6.2242 |

5.6 | 9.6542 | 12.4532 | 0.3243 | 7.3232 |

6.0 | 10.4342 | 14.2321 | 0.1211 | 9.1214 |

$\mathsf{\delta}$ | E(Q) | E(B) | E(I) | E(W) |
---|---|---|---|---|

0.5 | 10.2343 | 8.5472 | 8.2231 | 9.1227 |

0.6 | 11.4532 | 9.2345 | 7.8535 | 11.2236 |

0.7 | 13.1231 | 11.4532 | 5.1232 | 13.5645 |

0.8 | 15.2941 | 12.6341 | 3.2212 | 15.6543 |

0.9 | 16.3987 | 13.9871 | 2.1545 | 16.6432 |

Renewal Rate | E(Q) | E(B) | E(I) | E(W) |
---|---|---|---|---|

2 | 6.932 | 6.6532 | 4.1534 | 5.2345 |

4 | 6.134 | 6.2543 | 5.6734 | 3.4564 |

6 | 5.513 | 5.2321 | 7.4765 | 1.5643 |

8 | 4.927 | 4.1431 | 8.2451 | 1.1457 |

10 | 3.623 | 2.9256 | 10.1543 | 0.4381 |

a | E(Q) | E(B) | E(I) | TAC |
---|---|---|---|---|

2 | 1.3563 | 6.2874 | 2.8963 | 1.2753 |

3 | 1.4389 | 6.6421 | 3.7524 | 0.8754 |

4 | 1.5129 | 7.7943 | 4.7327 | 0.6798 |

5 | 1.8319 | 8.4862 | 5.5639 | 0.9875 |

6 | 2.0345 | 13.5782 | 5.8032 | 0.9234 |

7 | 2.2427 | 18.4592 | 7.6731 | 0.9875 |

8 | 2.8193 | 24.5728 | 8.6932 | 0.9109 |

9 | 3.6738 | 26.8432 | 9.4589 | 1.5098 |

10 | 3.8921 | 30.1732 | 10.5821 | 1.4326 |

Threshold Value ′N′ to Start Bulk Service | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Threshold value ′a′ to start service | N | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |

a | TAC | ||||||||||

1 | 4.861 | 4.732 | 4.746 | 4.965 | 5.583 | 5.951 | 6.247 | 6.365 | 7.483 | ||

2 | 4.567 | 4.518 | 4.872 | 5.432 | 5.673 | 6.084 | 6.247 | 6.941 | |||

3 | 4.452 | 4.672 | 4.792 | 4.864 | 5.272 | 5.431 | 5.934 | ||||

4 | 4.295 | 4.643 | 4.821 | 5.093 | 5.187 | 5.531 | |||||

5 | 4.283 | 4.314 | 4.421 | 5.042 | 5.467 | ||||||

6 | 4.315 | 4.410 | 4.989 | 5.035 | |||||||

7 | 4.357 | 4.745 | 5.021 | ||||||||

8 | 4.578 | 4.995 | |||||||||

9 | 4.982 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Niranjan, S.P.; Devi Latha, S.; Mahdal, M.; Karthik, K.
Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation. *Mathematics* **2024**, *12*, 75.
https://doi.org/10.3390/math12010075

**AMA Style**

Niranjan SP, Devi Latha S, Mahdal M, Karthik K.
Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation. *Mathematics*. 2024; 12(1):75.
https://doi.org/10.3390/math12010075

**Chicago/Turabian Style**

Niranjan, S. P., S. Devi Latha, Miroslav Mahdal, and Krishnasamy Karthik.
2024. "Multiple Control Policy in Unreliable Two-Phase Bulk Queueing System with Active Bernoulli Feedback and Vacation" *Mathematics* 12, no. 1: 75.
https://doi.org/10.3390/math12010075