Analyzing the Two-Phase Heterogeneous and Batch Service Queuing System with Breakdown in Two-Phases, Feedback, and Vacation

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Introduction
Several academicians have tested queue systems with vacations and their numerous combos.Some of those studies are queue structures with vacation queue fashions via Tian & Zhang 1 .In many actual applications, there can be a couple of arrivals into the machine, these types of systems are labeled as bulk arrival queue structures.For batch carrier queue structures with multiple vacations, Arumuganathan & Jeyakumar 2 obtained consistent nation conditions for numerous performance measures and value optimization.The steady-state queue size distribution for the M X /G (a, b)/1 queue machine with numerous operating vacations changed into lately acquired by way of Jeyakumar & Senthilnathan 3 .In all queue models with vacations that have been studied inside the beyond, 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 fashions.Baba 4 examined the M/PH/1 line with working vacations and interruptions.M X /G (a, b)/1 queue model with vacation interruption changed into researched through Haridass & Arumuganathan 5 .
Through the real-time application, the researchers were in a position to deduce diverse queue device functions.Gao & Liu 6 investigated the M/G/1 queue under a Bernoulli agenda with working vacation and vacation interruption.When looking back at 2013, Tao et al., 7 blanketed the GI/M/1 queue

Abstract
Customers are arriving at the system in bulk according to the Poisson process with rate λ1.The batch service process is split into two phases called first essential service (FES) and second essential service (SES) with minimum server capacity ′′ and maximum server capacity ′′.Customer who requires feedback after SES completion will be taken immediately for service by the server with probability .After completion of SES, if there is no feedback with probability 1 − , the queue length is greater than or equal to ′′ then the server begins FES, or if the queue length is less than ′′ then the server leaves for vacation.Here we address utilization of renewal period of the server after completion of FES or SES with probability  or  respectively.When there is no server failure after FES completion, the server will provide SES with probability1 − .Similarly, if there is no server failure after SES the server may go for FES or vacation with probability1 − .After completing a vacation if the queue length is greater than or equal to ′′ then the server begins FES.On the other hand, if the number of customers waiting in the queue is smaller than′′, then the server remains idle until the queue length reaches the value′′.For the designed model probability-generating function of the queue size at an arbitrary time is obtained by using the supplementary variable technique and suitable analytical results are derived with numerical examples.Sama et al., 12 introduced an unstable server and delayed repair for a bulk arrival Markovian queueing system with state-dependent arrival and Npolicy.The M/M/1 vacation two-phase queueing model with server start-up, time-out, and breakdowns was examined by Rao et al., 13 .Enogwe et al., 14 analysed a non-Markovian queue with two types of service balking and Bernoulli server vacation.The model is referred to as a bivariate Markov process and includes the elapsed service time and vacation time as supplementary variables.In the article of Sekar & Kandaiyan 15 investigated a single-server retrial queue with delayed repair and feedback under working breaks and vacations.If the essential and sufficient requirements are viable, the system can be stabilized.Server breakdown based on service modes was introduced by Niranjan 16 .In this paper, the researcher used supplementary variable techniques to derive important performance measures.Blondia 17 analyzed energy harvesting in the queueing model for a wireless sensor node.Merit & Haridass 18 worked on a simulation analysis of the bulk queueing system with a working breakdown.An application of queueing system in 4G/5G networks was given by Deepa et al., 19 .Niranjan et al., 20 introduced two types of breakdowns with two Phases of service in a bulk queueing system.A M x /G (a, b)/1 queuing system with optional additional services, numerous vacations, and setup time is examined by Ayyappan & Deepa 21 .
Niranjan et al., 22 analysed a non-markovian bulk queueing system with renewal and startup/shutdown times.Maximum entropy analysis of the control of the arrival and batch service queueing system with breakdown and multiple vacations was discussed by Nithya & Haridass 23 .The impacts of reneging, server breakdown, and server vacation on the various stages of the batch arrivals queueing system with a single server serving clients in three varying modes are investigated in the article by Enogwe & Obiora-Ilouno 24 .Khan & Paramasivam 25 analyzed the quality control policy for the Markovian model with feedback, balking, and maintaining reneged clients using an iterative method to the nth customer in the system..In business management, Moussa et al., 29 analyzed the service time characteristics of fast-food outlets using an M/M/1 queuing model.In addition, Abdelmawgoud et al., 30 determined how long service wait times affect customers' satisfaction in five-star hotels.

Motivation:
Cloud computing has played a vital role in many practical applications.A type of cloud computing that allows to access applications on the internet.Cloud computing takes a minimum of 20 documents and a maximum of 100 documents.Also, iCloud takes the data from the cloud and transfers it directly without any interruption.
Simple Mail Transfer Protocol (SMTP) is used to deliver data across the internet using file transfer data to clients.The next process of attachments is to transfer the data to clients.During the data transfer sometimes the system may be affected by the virus, https://dx.doi.org/10.21123/bsj.2024.9126P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal the service will not be interrupted immediately, and it will be continued for the current batch of data attached or else transferred by doing some technical precaution arrangements.After the service is completed, the antivirus is activated and detects the problems in the system.If there are no issues the process would be continued.After data transmission, if there is no data available for processing the server will be assigned to some secondary works such as system updating, and cleaning temporary files etc., This can be modeled as 'Analysis of two-phase heterogeneous batch service queue system with a breakdown in two phases, feedback and vacation'.

Model Description:
Customers are arriving at the system in bulk according to the Poisson process with rate λ1.The batch service process is split into two phases called FES and SES with minimum server capacity ′′ and maximum server capacity ′′ according to the general bulk service rule 31 .When the server fails, during FES or SES the service process will not be interrupted.It performs continuously for the current batch by doing some technical precaution arrangements.The server will be repaired after the service completion of FES or SES with probability  or  respectively during the renewal period of the server.When there is no server failure after FES completion, the server will provide SES with probability1 − .Simi\larly, if there is no server failure after SES the server may go for FES or vacation with probability1 − .Customer who requires feedback after SES completion will be taken immediately for service by the server with probability.After completion of SES, if there is no feedback with probability 1 −  queue length is greater than or equal to ′′ then the server begins FES, or if the queue length is less than ′′ then the server leaves for vacation.On SES completion, if the queue length is greater than or equal to ′′ then the server goes for FES.After completing a vacation if the queue length is greater than or equal to ′′ then the server begins FES.On the other hand, if the number of customers waiting in the queue is smaller than ′′.The server remains idle after the vacation completion if the queue size is smaller than ′′.This proposed system is schematically represented in Fig 1 Notations: Let X be the group size random variable of the arrival, λ1 be the Poisson arrival rate,  be the probability that k customers arrive in a batch, () be the probability generating function (PGF) of X.Let  () be the number of customers under the service at time t and q () be the number of customers waiting for service at time t.,  be the breakdown probability of FES and SES respectively,  be the feedback probability.Notations are given in Table 1 below.()= 0-when the server is busy with FES,1-when the server is busy with SES, 2-when the server is on vacation, 3-when the server is on repair, 4-when the server is on a dormant period.
The State probabilities are defined as follows;

Probability Generating Function (PGF)
To obtain the PGF of the queue size at an arbitrary time epoch, the following PGF are defined.By multiplying from Eq.13 to Eq.22 with suitable powers of   and summing over j=0 to ∞, then by using Eq.23, z n ] 27 Let P (z) be the probability-generating function of the queue size at an arbitrary time epoch.Then Where

Theorem 1:
The unknown constants   involved in   are expressed in terms of   as,   =∑  −  =0   , n =0, 1, 2… a-1, where   is the probability that 'i' customers arrive during the vacation.

Theorem 2:
Let   be the collection of a set of positive integers (not necessarily distinct) A, such that, the sum of elements in A is j, then,  =

Proof:
From the Eq 11 and Eq 12, ) Hence the proof.

Steady State Condition
The probability-generating property P (z) must satisfy P (1) = 1.Using the L Hospitals rule to evaluate limz→1 P (z) and equating the expression to one, it can be deduced that 1 is the condition to be satisfied for the existence of a consistent nation for the model under study, in which https://dx.doi.org/10.21123/bsj. Where Expected waiting time in the queue

Expected length of busy period
Let B be the busy period random variable.Let T be the residence time that the server is rendering (FES) or under repair and M the residence time that the server is rendering (SES) or under renewal.Then E

Expected length of idle period
Between the start of a vacation and the start of a busy period is the definition of an idle period.For the idle period, let I serve as the random variable.), and E (W) increase whereas E (I) decreases.In Table 3 and 4 and Fig. 3 and 4, an effect of performance measures for different failure rates and repair times are presented, it is observed that E(Q) and E(W) will be increased whenever the failure rate increases and E(Q) and E(W) will be decreased whenever the repair time increases.(For a=2, b=4, ᶓ=10, η=8, δ=0.2,β = 0.5) E (Q) -Expected queue length E (B) -Expected length of busy period E (W) -Expected waiting time in the queue E (I) -Expected length of idle period

Published
Online First: January, 2024 https://dx.doi.org/10.21123/bsj.2024.9126P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal with Bernoulli-scheduled vacation and vacation interruption.A countless-buffer batch-arrival queue with a batch-size-dependent provider was tested with the aid of Pradhan & Gupta 8 .The consistent state takes a look at two M X /M (a, b)/1 queue models with random breakdowns changed into examined using Madan et al., 9 they took into account the reality that the repair time is deterministic for one version and exponential for every other.It has been cited inside the literature on queue fashions with server breakdown that, excluding Madan et al. each creator who discusses server breakdown inside the context of a bulk provider queue version deals with a server that can simply 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 batch of services.Jeyakumar & Senthilnathan 10 also look at breakdown without carrier disruption in a batch 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 renovations.An M/G/1 queue with an N-policy, a single vacation, an unreliable service station, and a replaceable repair facility was examined by Wu et al., 11 .
Priority and regular clients are two different categories that are taken into consideration.The following article presents that combination with a cryptosystem which has several Substitution Cipher Algorithms along with the Circular queue data structure.Homophonic Substitution Cipher and Polyalphabetic Substitution Cipher are the two different substitution techniques Ibraheem & Hasan 27 .The Public Network Channel Transferring their data by Secured Based Steganography and Cryptography Techniques, Naser et al.,

Table 1 . Notations
The https://dx.doi.org/10.21123/bsj.2024.9126P-ISSN: 2078-8665 -E-ISSN: 2411-7986 Baghdad Science Journal Effects of arrival rate on various performance measures are presented in Table 2 and Fig 2. Effects of breakdown probability on various performance measures are presented in Table 3 and Fig 3. Effects of renovation rate on various performance measures are presented in Table 4 and Fig 4. Impacts of various performance measures for fixed threshold values are presented.From Tables 2 and Fig 2, it is clear that, if λ 1 increases, E (Q), E (B