#### Abstract

The orthodox way of tile printing results in mulish production. To tame this, digital inkjet printing technology was imposed. In this article, a mathematical prototype is designed exclusively for the tile printing wing. The various plants coming under printing are correlated to the proposed prototype. The article throws light on the functioning of the prototype. The prototype turns up subservient to bulk service queue models with two types of vacations. The prototype is resolved under the supplementary variable technique. The prototype with peculiar conditions is proved to be the existing model. Anticipated line distance, awaited active duration, awaited standby time, and expected idle period were computed. Thus, the total average cost is acquired for diverse vacation values. The main intention of the pattern is to downside the total average cost.

#### 1. Introduction

As we all know, queues are a typical everyday encounter. When resources are guarded, there queues form. Having queues makes financial sense in reality. One of the major issues within the examination of any activity system is the investigation of delay. Delay is a more unpretentious concept. Queuing theory deals with consequences which include queuing (or holding up). Typical instances might be banks, supermarkets, computers, public transport, etc. In general, all queuing frameworks can be broken down into individual subsystems such as arrival process, service mechanism, queue characteristics, which are briefly discussed in [1].

Real-life systems barely ever reach a consistent state. Instead of this, a simple queuing formula can allow us a few understandings of how a system might carry on very rapidly. One factor that’s of note is traffic intensity equals (entry rate)/(departure rate) where entry-rated notes total of entries every time and departure rate represent the tally of departures per time. Traffic intensity may be a measure of the blockage of the system. In case it is close to zero, there’s exceptionally small lining in common, as the traffic intensity escalate (to approach 1 or indeed more than 1) the amount of queue upsurge. Before the implementation of digital printing over tiles, the manufacturers faced difficulties for design setting, because if the same design is asked it is tedious to bring back the same design accurately and definitely, there will be few moderations. The main advantage of using digital technology is one can change the colours as they need and even fine-tune designs and patterns if requested by customers. One more advantage is the designs can be stored for future use. A mathematical model is proposed for the digital printing technology. Here, the model considered is the bulk service queue model where the arrival of tiles and printing (service) is done in batches. Before printing the tiles, it has to be washed and cleaned which is mentioned as setup work. Apart from printing designs, the server is utilized for two more secondary jobs which are referred to as type 1 and type 2 jobs.

#### 2. Literature Survey

Arumuganathan and Jeyakumar [2] investigated the queuing framework for *N*-policy vacations, as well as setup and closedown times. The server does a closedown job and takes a vacation in case the waiting line length is fewer to “*a*.” The attendant goes on another vacation after the service is completed. In case the queue distance is smaller than *N*, setup work begins, followed by service to “*b*” customers. In addition, the authors discussed a few performance metrics. Sasikala and Kandaiyan [3] devised a bulk queuing system that included setup time, balking, multiple vacations, and close down. Arumuganathan and Malliga [4] looked at a bulk queue system with setup time and several vacations. *G*-Queues, along with server breakdown and reservice, have been considered by Ayyappan and Supraja [5]. Karpagam [6] has considered a queue system which has more application in industries. Ali and Reza[7] have considered a queue model with *k* sequential heterogeneous service steps and vacations.

Arumuganathan and Ramaswami [8] investigated a bulk queuing system along with a fast and moderate service rate, as well as multiple vacations. In addition, the same researchers looked at a state-dependent arrival, which means that while the server is busy, the rate of arrival would progressively grow, but when it is on vacation, the arrival rate will gradually fall. Ghosh et al. [9] have analysed a finite-buffer bulk service queue system where arrival service is correlated. The queue system was analysed by Laxmi and Gupta [10]. Singh et al. [11] have analysed a bulk service queue system with a modified bulk service rule. Nithya and Haridass [12] have devised a system that divides bulk service into two stages. Niranjan et al. [13] investigated an actual issue including a retrial model with a threshold, nondisruptive service when the server failed and avail various vacations. Singh et al. [14] have analysed the operation of a washing machine under queueing system and its performance measures. In addition to setup time, Krishnareddy et al. [15] examined a batch queuing framework with *N*-policy.

Jeyakumar along with Senthilnathan [16] have also studied a queuing system including setup time, closedown time, and various vacations, and it does cause any. Renovation of the service station was also explored by the authors. Kavitha et al. [17] have developed a queue model which is used for mobile wallet system by applying blockchain. Goswami and Mohan [18] determine an explicit form of sojourn-time distribution and evaluate the distribution function for any specific time. Haridass and Arumuganathan [19] have considered a mass service queue system with two types of vacations. The server performs three types of jobs namely service, two types of secondary jobs type1 and type 2, respectively.

As an extension to the above-mentioned model in this article, a mathematical model is developed by introducing setup time. In this model, the service is given in bulk, and setup work is carried out after the first batch gets service. In the proposed model, the server performs four types of jobs namely setup work, service, and two types of secondary jobs type I vacation and type II vacation.

#### 3. Model Description

The motivation for this mathematical model is materialized from the tile manufacturing and design sector. The traditional way of designing tile has multifold drawbacks such as long setup times, patterns repeat frequently, difficulty in colour management in case of repeated orders, and inflexible production planning, and only flat tiles can be decorated. To sweep off these difficulties, digital inkjet printing technology was imposed on tile designing. The mathematical model is created purely for designing part of the manufacturing unit. After the tiles are manufactured, the tiles are grated (if required), washed with cleaning agents, and dried. The tiles are moved to the printing area, and the dimensions of tiles are ensured. Meanwhile, the designs for printing were kept ready. We refer to the above process as setup work in our model. Tiles have to coat with base coat before printing the designs and have to dry for the next process, and this is termed as service, and it is done for *n* number of tiles at once. Hence, the suitable queuing model for the process is bulk service. The server is idle until the printed tiles are dried. To utilize the idle time, the server will perform two types of work: cleaning the dried tiles and applying the top coat which makes the tiles look glossy. And the second job is final grinding and side coating. In this model, the former work is mentioned as type 1 vacation, and the later work is quoted as type 2 vacation. The major goal of this model is to lower manufacturing costs. Table 1 is a classification of the procedure described as follows.

The method described above is developed as a bulk framework having two different types of various vacations and setup time. Consumers arrive in bulk in this scenario, and the server starts the service after “*a*” customers arrive. At the service terminus epoch, if the total clients are under “*a*,” type 1 vacation is available. If the queue volume is more than that of “*a*,” service will begin for the first *b* customers. Finally, after the type 1 vacation completion epoch, the server checks the number of customers; if the number is less than “*a*,” it continues type 1 vacation. Alternatively, if the system has a capacity greater than or equal to “*N*,” setup work is required. Alternatively, if the majority of customers have a capacity more than or equal to “*N*,” setup work begins. The server starts type 2 vacations if the total number of consumers is more than “a” but less than “*N*.” If the server detects less than “*N*” clients during type 2 vacation completion stage, it will either continue type 2 vacation or begin setup work. After the completion of setup work, service is provided to a set of clients with the smallest size “*a*” and the largest size “*b*.” Figure 1 depicts the service flow diagrammatically.

The process of tile printing through inkjet printer is shown in Figure 2.

#### 4. Notation

The arrival’s group size random variable is denoted as . Let *X*(*z*) be the probability generating function of . The chance that “*k*” customers appear in a group is symbolized as . *λ* be a sign of the arrival rate. (Hereafter throughout the article, *ε* represents arrival rate). indicate the count of customers in the line and customers availing service. Also,

Let , represents the service provider is on vacation of type 1 and type 2, respectively. Service time, setup time, secondary job time of type 1 and type 2 have a cumulative distribution function as The Laplace-Stieltjes transform of be , , , . Also, , , , be probability density functions of setup time, service time, type 1vacation, type 2 vacation. , , , designate the enduring period of service of the batch in service and type 1’s, type 2’s excess vacation time and remaining setup time randomly. The left over service time, setup time, and vacation time of the service provider are considered as supplementary variables. The steady-state difference differential equations for the system are addressed as follows.

The probabilities of each state are

The SVT (supplementary variable technique) is utilized to construct, and the subsequent equations for the queueing process are as follows:

#### 5. Queue Size Distribution

By treating the remaining service time, remaining setup time, and remaining vacation time of the service provider as supplementary variables, the steady state difference differential equations for the system are addressed as

Applying the Laplace-Stieltjes transform to two sides of equations (4) to (16), we obtain

Let

Multiplying the above equation by *z*^{0} and by *z*^{n} and aggregating and utilizing (18), we got

Replacing in equations (19) to (25), we get

Substituting the above values in their corresponding equation, we have

Consider (30)

Substituting value in equation, we obtain

Let

Therefore,

From (28) and substituting the values, we get

Substituting the above value in , we getwhere

If *P* (*z*) is the probability-generating function of queue size at any time period, then

On substituting the values, we get

*P*(*z*) need to fulfil *P*(1) = 1. To qualify this, the L’Hospital rule is applied, and the formulation is compared with 1. For the existence of a steady state, the criteria to be convinced is where and *E*(*p*) is the expected service time.

#### 6. Computational Aspects

Equation (40) has *N* + *b* unknowns. The three accompanying theorems have been shown to represent in the form of , and as a result, the numerator has *b* constant. Equation (40) now shows a probability-generating function containing only “*b*” unknowns. Using Rouché’s theorem, we can prove that there are *b* − 1 zero inside and one zero on the region . The above-said equations are solved using MATLAB software.

Theorem 1. *The constant in are communicated in terms of as where and is the chance that “ s” clients arrive during a vacation session.*

*Proof. *We know that equation (26) demonstratesFromWe getwhere

Theorem 2. *The constant in P(z) are pointed in terms of as where where represent the chance that “ c” clients enter the system during type 1secondary job.*

*Proof. *On both sides, the coefficients of are equatedSubstituting from Theorem 1, we get

Theorem 3. *Let where present in P (z) can be conveyed with reference to as , , and also where is the chance that “f” customers enter the system during type two job.*

*Proof. *ConsiderEquating the coefficients of Solving the above equation for , we get , ,

#### 7. Particular Case

*Case 1. *In the event that there is only one secondary job (i.e., ) and setup time is , the equation (40) reduces toThis result coincides with the distribution discussed in [15].

*Case 2. *If there is no setup time, that is, , we getis the distribution of bulk queuing system with vacation. The outcome coincides with the result stated in [6].

#### 8. Performance Measures

A few performance measures are accomplished in this part, which are applicable in building up the cost model and furthermore to correlate the framework with several features of secondary jobs.

##### 8.1. Awaited Queue Length (EQL)

The awaited line length *E*(*Q*) is obtained asi.e.,where