In this study, we investigate a single server architecture in which the customer service system is not aware of the length of the line that was sent to them. Thus, the system-controlling equations here become what are known as the vacation differential equations. Here, we argue specifically that this vacation is what is causing the oscillations in the performance measurements of the system. The symbolic structure of the differential equation of the non-Markovian queuing problem is introduced in this study. A procedure of maintenance work is included in this model in terms of the vacation stage to support this minimal non-interrupted service system. The findings offer a thorough analysis of the system that enables it to operate more profitably even if it is interrupted by any associated activities. The supplemental variable method solves the queuing problem caused by the aforementioned subsequent outcomes. Estimates are made for the queue size, server idle time, use, and probability generating factors for each operating method. Numerical analysis was performed on specific examples using mathematical software. This strategy is perfectly acceptable because it is regularly employed and makes use of a statistical demarcation method. The graphical representation of this perspective provides precise calculations of the apparent constraints.

## I. INTRODUCTION

The mathematical theory of traffic, waiting lanes, and queues is known as queuing theory. The word “customer” can apply to many different things, although it typically refers to an individual. In this study, we focus on the ordinary differential-difference equation method to the M^{X}/G/1 model of a non-Markovian system. The birth–death study procedure is used to introduce the differential algebraic equation (DAE) of the defined system before the initial and boundary conditions of the system are framed. The Laplace transform, differentiation, integration, and additional variable methods are used to solve the system of equations. A few properties of the performance measures are discussed along with the system results. Finally, numerical examples demonstrate the approximation accuracy of our suggested strategies.

In our paper, we present a technique that enables the analytical solution of moments of any order as well as probability distributions from a single equation system. For moments of random order varieties, a description of the differential equation generating process is given. Based on the use of generating functions, this approach works well because it enables the solution of moments without the need for intricate probability calculations. It is crucial in empirical studies of complex systems with multiple components.

A queueing model is created to anticipate line lengths and wait durations. Because the results are frequently used to make business decisions about the types of resources needed to deliver a service, queuing theory is commonly considered a subset of operations research. Agner Krarup Erlang’s study, in which he constructed models to characterize the Copenhagen Telephone Exchange Company’s system, is credited with the beginnings of queueing theory. Telecommunications, traffic engineering, computers, and application in a wide range of engineering, factory, shop, office, and hospital design, as well as project management, have all benefited from the principles.

Management must thoroughly analyze a queuing system in order to make effective decisions. The input pattern, service time pattern, service discipline, server count, system capacity, and service stages are the six fundamental qualities that make up a queuing system.

The mean time spent in the wait, the mean time spent in the system, the mean number of customers in the queue, the average number of customers in the system, and the utilization factor are also among the most important performance indicators in the analysis of queuing models. These performance metrics give the service center the ability to establish the values of the proper effectiveness measures within the system and create the best possible system.

A number of authors have made brief observations on the non-Markovian model. Long waits, or queues, are an aspect of queuing theory. In a non-Markovian queuing model, Vignesh *et al.*^{1} investigated a restricted admissibility service in stages. Under a Bernoulli schedule, Rajadurai *et al.*^{2} investigated an unreliable retrial G-queue with working vacations and vacation interruptions. Srinivas *et al.*^{3} used a queueing model that accounted for server outages, maintenance, vacations, and backup servers. Singhal *et al.*^{4} looked at the customer’s waiting time value for general queues. Arqub and Abo-Hammour^{5} explained the concept of a second-order boundary value problem using a continuous genetic algorithm. Sankara Reddy *et al.*^{6} analyzed business data attacks using the Hadoop Framework for Big Data Analytics. Tayebi *et al.*^{7} developed performance measures using the cubic B-spline interpolation method for numerical point solutions of conformable boundary value problems. Due to server downtime, Gao and Zhang^{8} provided a performance and sensitivity analysis of an M/G/1 queue with retry clients. The GI/M/1 Queue with Multiple Service Phases and Vacations was the focus of Li and Liu’s^{9} research. Kim *et al.*^{10} used a differential equations approach to investigate multi-server queues with removable servers. Shanmugasundaram and Sivaram^{11} analyzed the M/G/1 Feedback Queue when the server is off and on vacation. Maragathasundari and Dhanalakshmi^{12} investigated the problem of mobile ad hoc networks using a queueing strategy. Maragathasundari *et al.*^{13} studied the queuing strategy in deep-ocean tsunami assessment and reporting. Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems using the kernel algorithm were discussed in detail by Abu Arqub.^{14} A F-DEMATEL Method to Evaluate Criteria for Affecting Productivity in HP Valve Manufacturing Industries was studied by Maragathasundari *et al.*^{15} Maragathasundari *et al.*^{16} investigated the issue of the limited acceptability of customers in non-Markovian queues. An investigation of the dynamics of the queuing system in the thermopack process was written by Vanalakshmi and Maragathasundari.^{17} Abu Arqub *et al.*^{18} solved the singular two-point boundary value problems and determined the performance measures of the system. Delay differential equations in queues are studied by Pender *et al.*^{19} Maragathasundari *et al.*^{20} examined the queuing system with an established time and a second discretionary administration time.

The queuing system’s extra variable technique is used to address the queuing problem. Vacations in this context refer to the server maintenance tasks that must be done during such time to prevent server downtime or guarantee that the server operates continuously. This maintenance effort helps the system as much as it can to deliver a smooth service. The problem is solved by considering all the system parameters as additional variables. This queuing framework displays peculiar phenomena as a result of the settlement of discretionary funding. The length of the state line, the number of customers in the system, the amount of time that consumers wait in line and also on the spot, the amount of time that people spend in the association, and worker idle time are all established in the enthusiastic state line check spread.

## II. THE STUDY OF QUEUING STRATEGY IN TERMS OF DIFFERENTIAL EQUATIONS

### A. Overview of the stated queuing problem (model description)

Queuing problem is defined as follows:

The queue structure that the system specifies belongs to a non-Markovian model. Customer arrivals follow a Poisson process, but service times follow a generic distribution. A single server serves each client after they join up for the service. Customers wait in line for the second step of service after completing the first round of administration. The server takes a brief vacation to complete the necessary server maintenance after providing the first stage of service to all arriving clients and if there are no clients waiting in line for the first level of service. The server takes a long vacation that is split into two halves after the short vacation is over. However, the first step of extended vacation is necessary, although the second stage is not. The above description of the process’ maintenance requirements is in-depth. Additionally, in order to reduce lengthy waits for procedures during brief vacations, the concept of restricted admissibility is established. Additionally, a backup server is accessible in case the server takes a second stage of an extended vacation. To prevent the line from growing too long, this is a necessary step in the process. As a result, a standby server will supply the second stage of service if the primary server enters the second stage of the prolonged vacation procedure. The defined queuing problem is given as a design of the non-markovian queuing system with a slew of vacation styles and service stages in Fig. 1.

### B. Diagrammatic representation of the model

Stages of service: Before leaving the system, the consumers can go through one stage or numerous stages to get their service done. By using a multistage queuing system, the consumer enters one queue, waits for service, is serviced, and then leaves the service station to join another queue for yet another service. Recycling or feedback may be permitted in some multistage queuing systems; this is typical in industrial operations where defective items are returned for further processing.

Vacation: A vacation is a time whenever the servers are not accessible to just provide service in a queuing situation. Only once the server has returned from vacation can deliveries that were made during that time be put into service. A server vacation can result from a variety of circumstances, including equipment failure, network management, and periodic workstations. In this model, the vacation is given in two stages, short vacation and long vacation. The bulk of the maintenance work will be done when the servers are on vacation. During the brief vacation time, the system’s essential maintenance will be completed. The first stage of the long vacation will involve significant maintenance work. A long vacation’s second phase is optional as well. Stage 2 of the long vacation will be taken into consideration if more maintenance work needs to be done on the system.

Restricted admissibility: The admission to the system is controlled, and some arrivals may be denied admission. Not all the arriving customers are allowed to get into the system for the service. Based on the environment of the system, the input process takes place.

Standby server: Several queueing models have practical uses, such as serving as a backup server when the primary server is unavailable (due to absences like vacations and server failures). These backup server services can be thought of as the primary server operating while on leave or experiencing a malfunction. The backup server is assumed as the primary server operating (at a lower rate) while off-site or undergoing maintenance.

The comprehensive queuing system is examined using one of the most basic operational research approaches, the additional variable methodology. The performance measures are calculated utilizing Laplace Stieltje’s transform and boundary value criteria, and they are backed up by numerical analysis and graphical depiction.

The above queuing issue is transformed into a mathematical issue that is thoroughly explained and resolved in the following sections.

### C. Representations and assertions of the mathematical modeling

#### 1. Arrival pattern

The interval of time between two successive arrivals defines an arrival. The inter-arrivals are considered to be independent of one another and to follow a common distribution. The input patterns describe the behaviors of the clients when they arrive at the service system. Some traditions may be patient and wait for a long time, while others are less so and depart quickly.

#### 2. Service form

The way the service is provided is determined by the pattern of service times. It is described in terms of how long it takes to finish service. Deterministic and exponential distributions are the most often seen service time distributions. The queue length may also affect how quickly a service is provided.

In this Non Markovian model, clients are admitted to the structure in groups of different sizes, based on a Poisson distribution with just an arrival rate *λ*_{k} > 0, and service follows a general distribution.

Notations . | Description . | Distribution function . | Density function . |
---|---|---|---|

$Hn1(x)$ | First stage of service: The probability that there are n ≥ 1 customers in the line excluding the one being served and the elapsed service time for this customer is x when there are n numbers of customers in the first stage of service. | $D1*x$ | d_{1}(x) |

$\epsilon 1x=d1(x)1\u2212D1*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

$Inx$ | Short vacation: (Restricted admissibility): It denotes the probability of n ≥ 1 consumers in the queue and the server is on a short vacation with passed short vacation time x. | $D2*x$ | d_{2}(x) |

Let $\theta x$ be the conditional probability of a completed short vacation, which is calculated as follows: | |||

$\theta x=d2(x)1\u2212D2*xandd2x=\theta xe\u2212\u222b0x\theta tdt$ | |||

$Jn1x$ | Long vacation stage—I: (standby server): Just like a short vacation, it corresponds to the probability of happening stage 1 of the long vacation. | $D3*x$ | d_{3}(x) |

$\gamma 1x=d3(x)1\u2212D3*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

φ | Standby server: A standby server is used in place of the primary server to ensure that clients receive uninterrupted service. During the long vacation I, a standby server is used. The arrival rate of standby servers is φ > 0. It is based on the Poisson distribution. | ⋯ | ⋯ |

$Jn2(x)$ | Long vacation stage—II: (Optional): | $D4*x$ | d_{4}(x) |

$\gamma 2x=d4(x)1\u2212D4*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

$Hn2(x)$ | Second stage of service: | $D5*x$ | d_{5}(x) |

$\epsilon 2x=d5(x)1\u2212D5*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ |

Notations . | Description . | Distribution function . | Density function . |
---|---|---|---|

$Hn1(x)$ | First stage of service: The probability that there are n ≥ 1 customers in the line excluding the one being served and the elapsed service time for this customer is x when there are n numbers of customers in the first stage of service. | $D1*x$ | d_{1}(x) |

$\epsilon 1x=d1(x)1\u2212D1*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

$Inx$ | Short vacation: (Restricted admissibility): It denotes the probability of n ≥ 1 consumers in the queue and the server is on a short vacation with passed short vacation time x. | $D2*x$ | d_{2}(x) |

Let $\theta x$ be the conditional probability of a completed short vacation, which is calculated as follows: | |||

$\theta x=d2(x)1\u2212D2*xandd2x=\theta xe\u2212\u222b0x\theta tdt$ | |||

$Jn1x$ | Long vacation stage—I: (standby server): Just like a short vacation, it corresponds to the probability of happening stage 1 of the long vacation. | $D3*x$ | d_{3}(x) |

$\gamma 1x=d3(x)1\u2212D3*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

φ | Standby server: A standby server is used in place of the primary server to ensure that clients receive uninterrupted service. During the long vacation I, a standby server is used. The arrival rate of standby servers is φ > 0. It is based on the Poisson distribution. | ⋯ | ⋯ |

$Jn2(x)$ | Long vacation stage—II: (Optional): | $D4*x$ | d_{4}(x) |

$\gamma 2x=d4(x)1\u2212D4*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ | |||

$Hn2(x)$ | Second stage of service: | $D5*x$ | d_{5}(x) |

$\epsilon 2x=d5(x)1\u2212D5*xandd1x=\epsilon 1xe\u2212\u222b0x\epsilon 1tdt$ |

Determining the defined modeling and forming the system of equations, preliminary, and boundary conditions.

A birth–death (BD) process is a Markov process with a discrete state space whose states can be enumerated with index *i* = 0, 1, 2, …, such that state transitions can occur only between neighboring states, $i\u2192i+1ori\u2192i\u22121$.

Transition rates:

The process of discrete randomness X(t) called the birth and death process is represented by the number of individuals present at time t in a population in which two types of events occur—one representing birth that contributes to its increase and the other representing death that contributes to its decrease. The following are the postulates that govern birth and death:

Using the above process of study, the following differential equations are framed based on the model defined:

Boundary condition,

## III. SOLVING OF THE MATHEMATICAL PROBLEM

The supplementary variable strategy must be used to overcome the problems mentioned above. This technique is elaborated on below:

This method, first presented by Cox in 1955, involves adding extra variable elements to the system in order to make it Markovian. A straightforward description of this tactic is as follows: The M/M/1 queue’s inter-arrival and service periods are exponentially distributed; hence, a Markov process can be used to describe the queue size process N (t). We cannot use a Markov process to mimic the queue size process N if the arrival and/or service processes are not memory efficient (t). The queuing issue was handled via a non-Markovian model. The usage of the additional variable technique results from this. Cox was the first to use this strategy (1955). To demonstrate the supplemental variable method, we will suppose that the service hours are distributed using a general probability density function. The following queue quality controls are determined at this point: the length of the line, the number of clients in the frame, the customers’ wait times on the line and in the system, the time entirely dedicated to the clients for service, and the server’s idleness. Partial integration, the Laplace form, and the L’Hopital rule are used to fix the queue condition.

We use the notion of supplemental variable approach to calculate the preceding Eqs. (1)–(16) for a linear solution,

Now, integrating Eq. (17) from 0 to x yields

Integrating Eq. (18) by parts, we get

where $D1*a=e\u2212\lambda c\u2212\lambda cMzxdD1(x)$ is the Laplace stieltjes of the service time $D1x$.

Multiplying Eq. (18) by *ɛ*_{1}(*x*) on both sides, we get

Similarly, for the other parameters, we have the following:

Using the supplementary variable process in (12), we get

The above information is used to calculate the queue size’s probability generating function. This serves as the foundation for calculating all queue performance metrics.

Let *Y*_{q}(*z*) be the probability generating function,

## IV. PERFORMANCE MEASURES OF THE MODEL AND ANALYSIS

This section derives all execution metrics, including the number of customers in the queue and the system, the number of customers waiting in the line and the system, and the server efficiency factor.

Idle period and utilization factor

Q the idle period is determined using the normalization condition $Yqz+T=1$,Utilization factor$Q=1\u2212\lambda cEG1+pER+ES+EG5+mEG41\u2212\lambda cEG1+pER+ES+EG5+mEG4+m\lambda cEG4+2m(EG1+pER+E(S)).$*ρ*is calculated using*ρ*= 1 –*Q*.Number of clients (Inputs) in the line ($Lq$

*L*_{q}, the steady-state average queue length is given by$Lq=1\u2212\lambda c[E(G1)+pE(R)+E(S)+E(G5)+mE(G4)]\u2212\lambda c2EG12+pE(R2)+E(S2)EG52mEG42+2E(G1)E(R)+E(G1)E(G5)+pE(R)(E(G1)+E(G5)+E(R))+2\lambda cmE(G4)[E(G1)+pE(R)+E(S)+E(G5)]+2\lambda c2mE(G4)E(G1)+pE(R)+E(S)+E(G5)\u2212m\lambda cE(G4)+2mE(G1)+pE(R)+E(S)\lambda c2E(G12)+pE(R2)+E(S2)+E(G52)+mE(G4)E(G1)+pE(R)+E(S)+E(G5)+E(G4)+2pE(G1)E(R)+E(G1)E(S)E(G1)E(G5)+pE(R)E(S)+pE(R)E(G5)+E(S)E(G5)21\u2212\lambda c[E(G1)+pE(R)+E(S)+E(G5)+mE(G4])2.$Average client wait time in both the queue and the system, as well as the total number of customers (inputs)

Last but not least, Little’s method can be used to determine the average length of time a customer waits in line, in the system, and elsewhere.

### A. Particular cases

#### 1. Case (i) No optional long vacation

If we assume there is no optional long vacation, this means that m = 0.

The number of clients in the line is indicated by

#### 2. Case (ii) No restricted admissibility

If restricted admissibility is not taken into consideration, then p = 0.

Idle time and utilization factors are given by

### B. Numerical analysis of the modeling

#### 1. Analysis through the R tool

##### a. Case (i) analysis on the effect of arrival rate.

Table I presents that as the rate of customer arrival rises and they join the queue, the length of the queue increases. This increases the number of clients in the queue as well as the total number of clients in the system. Even as the queue and utilization factors increase, the load factor decreases. Even though the concept of restricted admissibility is introduced in this defined queuing system, as long as the arrival rate of the customers is consistent throughout, the length of the queue automatically gets expanded. The queue, the duration that customers must wait in the queue, and the system all get longer as a result of increased client arrival rates, as clearly illustrated in Fig. 2.

λ_{c}
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

3 | 0.8130 | 0.1869 | 0.9084 | 0.0954 | 0.4542 | 0.0477 |

4 | 0.8025 | 0.1975 | 0.9335 | 0.1310 | 0.4667 | 0.0655 |

5 | 0.7956 | 0.2044 | 0.9797 | 0.1841 | 0.4898 | 0.0920 |

6 | 0.7907 | 0.2093 | 1.0313 | 0.2406 | 0.5156 | 0.1203 |

λ_{c}
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

3 | 0.8130 | 0.1869 | 0.9084 | 0.0954 | 0.4542 | 0.0477 |

4 | 0.8025 | 0.1975 | 0.9335 | 0.1310 | 0.4667 | 0.0655 |

5 | 0.7956 | 0.2044 | 0.9797 | 0.1841 | 0.4898 | 0.0920 |

6 | 0.7907 | 0.2093 | 1.0313 | 0.2406 | 0.5156 | 0.1203 |

##### b. Case (ii) effect of the optional long vacation.

The stated queue mechanism introduces a lengthy vacation period to carry out significant maintenance tasks. The length of the queue and the amount of waiting time reduce as optional vacation time grows, as presented in Table II. The server’s idle time is also decreased. If the server does not take their elective vacation, their working hours will increase. Although considering the idea of a long vacation is optional, doing so makes the system work more efficiently and fast, which lowers all queue performance metrics. This concept is enlightened in Fig. 3.

m
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.2 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

0.4 | 0.7149 | 0.2851 | 0.8686 | 0.0631 | 0.4343 | 0.0320 |

0.6 | 0.6302 | 0.3697 | 0.8240 | 0.0601 | 0.4018 | 0.0285 |

0.8 | 0.5657 | 0.4343 | 0.7163 | 0.0552 | 0.3624 | 0.0131 |

1 | 0.5149 | 0.4817 | 0.6751 | 0.0498 | 0.2409 | 0.0088 |

m
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.2 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

0.4 | 0.7149 | 0.2851 | 0.8686 | 0.0631 | 0.4343 | 0.0320 |

0.6 | 0.6302 | 0.3697 | 0.8240 | 0.0601 | 0.4018 | 0.0285 |

0.8 | 0.5657 | 0.4343 | 0.7163 | 0.0552 | 0.3624 | 0.0131 |

1 | 0.5149 | 0.4817 | 0.6751 | 0.0498 | 0.2409 | 0.0088 |

##### c. Case (iii) consequence of restricted admissibility.

Due to maintenance tasks completed over the brief vacation, there is a restricted admittance. Table III demonstrates that line length and wait time reduce as restricted admissibility rises. The amount of idle time decreases if the server keeps running. The utilization factor rises as well. Admission is prohibited during brief breaks. Hence, the queue’s length is decreased. Figure 4 gives the measures of the defined queuing system in course of a brief vacation when the idea of restricted admissibility is introduced. This reduces idle time and raises the utilization factor.

p
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.8 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

1.2 | 0.8229 | 0.1761 | 0.7717 | 0.0502 | 0.3858 | 0.0264 |

1.6 | 0.8176 | 0.1824 | 0.7035 | 0.0341 | 0.3176 | 0.0171 |

2 | 0.8121 | 0.1879 | 0.6263 | 0.0258 | 0.2813 | 0.0102 |

2.4 | 0.8070 | 0.1930 | 0.5098 | 0.0139 | 0.1474 | 0.0030 |

p
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.8 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

1.2 | 0.8229 | 0.1761 | 0.7717 | 0.0502 | 0.3858 | 0.0264 |

1.6 | 0.8176 | 0.1824 | 0.7035 | 0.0341 | 0.3176 | 0.0171 |

2 | 0.8121 | 0.1879 | 0.6263 | 0.0258 | 0.2813 | 0.0102 |

2.4 | 0.8070 | 0.1930 | 0.5098 | 0.0139 | 0.1474 | 0.0030 |

##### d. Case (iv) outcome of the standby server rate.

The impact of raising the standby server frequency is then presented in Table IV. In the course of the protracted vacation, a backup server is setup. Service will be terminated if the primary server is not present. As a result, a lengthy line will develop. So, throughout the extended vacation period, a standby server fills in to prevent the growth of waits. As a result, all incoming clients receive prompt service before leaving the system. There are now fewer patrons and shorter queue waits as a result. The long-vacation procedure is actually divided into two stages, with Stage I seeing an overview of standby servers, which increases queue system idle time and is justified in Fig. 5.

φ
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

6 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

6.5 | 0.8322 | 0.1677 | 0.8783 | 0.0460 | 0.4391 | 0.0230 |

7 | 0.8333 | 0.1667 | 0.8709 | 0.0454 | 0.4302 | 0.0227 |

7.5 | 0.8342 | 0.1657 | 0.8692 | 0.0442 | 0.4253 | 0.0224 |

8 | 0.8351 | 0.1649 | 0.8647 | 0.0438 | 0.4239 | 0.0221 |

φ
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

6 | 0.8310 | 0.1689 | 0.9013 | 0.0702 | 0.4506 | 0.0351 |

6.5 | 0.8322 | 0.1677 | 0.8783 | 0.0460 | 0.4391 | 0.0230 |

7 | 0.8333 | 0.1667 | 0.8709 | 0.0454 | 0.4302 | 0.0227 |

7.5 | 0.8342 | 0.1657 | 0.8692 | 0.0442 | 0.4253 | 0.0224 |

8 | 0.8351 | 0.1649 | 0.8647 | 0.0438 | 0.4239 | 0.0221 |

### C. Analysis by Python programming

The expansion in the client arrival rate, which follows a Poisson process, lengthens the queue, and enhances the server’s utilization rate. By using numerical analysis, Table V offers justification for this.

λ_{c}
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.923 339 | 0.076 611 | 6.581 074 | 6.781 074 | 3.290 537 | 3.390 537 |

5 | 0.859 517 | 0.140 483 | 11.354 04 | 11.554 04 | 5.677 022 | 5.777 022 |

10 | 0.803 45 | 0.196 55 | 20.478 3 | 20.678 3 | 10.239 15 | 10.339 15 |

15 | 0.766 302 | 0.233 698 | 29.737 83 | 29.937 83 | 14.868 92 | 14.968 92 |

20 | 0.724 474 | 0.275 526 | 39.026 98 | 39.226 98 | 19.513 49 | 19.613 49 |

25 | 0.666 471 | 0.333 529 | 48.327 39 | 48.527 39 | 24.163 69 | 24.263 69 |

30 | 0.602 142 | 0.397 858 | 57.633 26 | 57.833 26 | 28.816 63 | 28.916 63 |

35 | 0.574 256 | 0.425 744 | 66.942 2 | 67.142 2 | 33.471 1 | 33.571 1 |

40 | 0.519 945 | 0.480 055 | 76.253 03 | 76.453 03 | 38.126 51 | 38.226 51 |

45 | 0.459 971 | 0.540 029 | 85.565 11 | 85.765 11 | 42.782 56 | 42.882 56 |

50 | 0.390 742 | 0.609 258 | 94.878 06 | 95.078 06 | 47.439 03 | 47.539 03 |

λ_{c}
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.923 339 | 0.076 611 | 6.581 074 | 6.781 074 | 3.290 537 | 3.390 537 |

5 | 0.859 517 | 0.140 483 | 11.354 04 | 11.554 04 | 5.677 022 | 5.777 022 |

10 | 0.803 45 | 0.196 55 | 20.478 3 | 20.678 3 | 10.239 15 | 10.339 15 |

15 | 0.766 302 | 0.233 698 | 29.737 83 | 29.937 83 | 14.868 92 | 14.968 92 |

20 | 0.724 474 | 0.275 526 | 39.026 98 | 39.226 98 | 19.513 49 | 19.613 49 |

25 | 0.666 471 | 0.333 529 | 48.327 39 | 48.527 39 | 24.163 69 | 24.263 69 |

30 | 0.602 142 | 0.397 858 | 57.633 26 | 57.833 26 | 28.816 63 | 28.916 63 |

35 | 0.574 256 | 0.425 744 | 66.942 2 | 67.142 2 | 33.471 1 | 33.571 1 |

40 | 0.519 945 | 0.480 055 | 76.253 03 | 76.453 03 | 38.126 51 | 38.226 51 |

45 | 0.459 971 | 0.540 029 | 85.565 11 | 85.765 11 | 42.782 56 | 42.882 56 |

50 | 0.390 742 | 0.609 258 | 94.878 06 | 95.078 06 | 47.439 03 | 47.539 03 |

An understanding of all the defined performance measurements of the system provided by the Python program and the graphical framework for the effect of arrival rate are revealed in Fig. 6.

Table VI exemplifies the consequence of an optional long vacation, which makes the idle time of the server to increase and the utilization factor to decline.

m
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.1 | 0.196 26 | 0.803 74 | 7.966 113 526 | 8.166 113 526 | 3.983 056 763 | 4.083 056 763 |

0.2 | 0.471 405 | 0.528 595 | 7.141 092 212 | 7.341 092 212 | 3.570 546 106 | 3.670 546 106 |

0.3 | 0.549 609 | 0.450 391 | 6.532 072 243 | 6.732 072 243 | 3.266 036 122 | 3.366 036 122 |

0.4 | 0.646 148 | 0.353 852 | 6.064 113 939 | 6.264 113 939 | 3.032 056 97 | 3.132 056 97 |

0.5 | 0.679 317 | 0.320 7 | 5.693 328 528 | 5.893 328 528 | 2.846 664 264 | 2.946 664 264 |

0.6 | 0.721 133 | 0.278 867 | 5.392 313 437 | 5.592 313 437 | 2.696 156 719 | 2.796 156 719 |

0.7 | 0.793 333 | 0.206 667 | 5.143 079 223 | 5.343 079 223 | 2.571 539 612 | 2.671 539 612 |

0.8 | 0.854 893 | 0.145 107 | 4.933 328 891 | 5.133 328 891 | 2.466 664 446 | 2.566 664 446 |

0.9 | 0.921 457 | 0.078 543 | 4.754 372 138 | 4.954 372 138 | 2.377 186 069 | 2.477 186 069 |

1 | 0.987 838 | 0.012 162 | 4.599 893 798 | 4.799 893 798 | 2.299 946 899 | 2.399 946 899 |

m
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.1 | 0.196 26 | 0.803 74 | 7.966 113 526 | 8.166 113 526 | 3.983 056 763 | 4.083 056 763 |

0.2 | 0.471 405 | 0.528 595 | 7.141 092 212 | 7.341 092 212 | 3.570 546 106 | 3.670 546 106 |

0.3 | 0.549 609 | 0.450 391 | 6.532 072 243 | 6.732 072 243 | 3.266 036 122 | 3.366 036 122 |

0.4 | 0.646 148 | 0.353 852 | 6.064 113 939 | 6.264 113 939 | 3.032 056 97 | 3.132 056 97 |

0.5 | 0.679 317 | 0.320 7 | 5.693 328 528 | 5.893 328 528 | 2.846 664 264 | 2.946 664 264 |

0.6 | 0.721 133 | 0.278 867 | 5.392 313 437 | 5.592 313 437 | 2.696 156 719 | 2.796 156 719 |

0.7 | 0.793 333 | 0.206 667 | 5.143 079 223 | 5.343 079 223 | 2.571 539 612 | 2.671 539 612 |

0.8 | 0.854 893 | 0.145 107 | 4.933 328 891 | 5.133 328 891 | 2.466 664 446 | 2.566 664 446 |

0.9 | 0.921 457 | 0.078 543 | 4.754 372 138 | 4.954 372 138 | 2.377 186 069 | 2.477 186 069 |

1 | 0.987 838 | 0.012 162 | 4.599 893 798 | 4.799 893 798 | 2.299 946 899 | 2.399 946 899 |

Python program and the graphical representation of the growth in the long vacation rate clearly show that the system is made more effective, and the performance measurements are reduced by the maintenance work done during the long vacation process. These concepts are well picturized in Fig. 7.

Table VII gives an idea of restricted admissibility that causes the server’s idle time in this system to increase while the customers’ wait times to decrease.

p
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.8 | 0.083 915 | 0.916 085 | 13.091 82 | 13.291 82 | 6.545 909 | 6.645 909 |

1.3 | 0.114 553 | 0.885 447 | 15.181 13 | 15.381 13 | 7.590 565 | 7.690 565 |

1.8 | 0.285 066 | 0.714 934 | 16.508 19 | 16.708 19 | 8.254 097 | 8.354 097 |

2.3 | 0.316 63 | 0.683 37 | 17.341 6 | 17.541 6 | 8.670 799 | 8.770 799 |

2.8 | 0.438 051 | 0.561 949 | 17.847 45 | 18.047 45 | 8.923 727 | 9.023 727 |

3.3 | 0.512 816 | 0.487 184 | 18.131 91 | 18.331 91 | 9.065 956 | 9.165 956 |

3.8 | 0.563 482 | 0.436 518 | 18.264 66 | 18.464 66 | 9.132 332 | 9.232 332 |

4.3 | 0.600 083 | 0.399 917 | 18.292 53 | 18.492 53 | 9.146 265 | 9.262 646 |

4.8 | 0.627 763 | 0.372 237 | 18.475 57 | 18.547 56 | 9.237 783 | 9.327 783 |

5.3 | 0.649 428 | 0.350 572 | 18.520 44 | 18.952 04 | 9.307 602 | 9.376 022 |

p
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

0.8 | 0.083 915 | 0.916 085 | 13.091 82 | 13.291 82 | 6.545 909 | 6.645 909 |

1.3 | 0.114 553 | 0.885 447 | 15.181 13 | 15.381 13 | 7.590 565 | 7.690 565 |

1.8 | 0.285 066 | 0.714 934 | 16.508 19 | 16.708 19 | 8.254 097 | 8.354 097 |

2.3 | 0.316 63 | 0.683 37 | 17.341 6 | 17.541 6 | 8.670 799 | 8.770 799 |

2.8 | 0.438 051 | 0.561 949 | 17.847 45 | 18.047 45 | 8.923 727 | 9.023 727 |

3.3 | 0.512 816 | 0.487 184 | 18.131 91 | 18.331 91 | 9.065 956 | 9.165 956 |

3.8 | 0.563 482 | 0.436 518 | 18.264 66 | 18.464 66 | 9.132 332 | 9.232 332 |

4.3 | 0.600 083 | 0.399 917 | 18.292 53 | 18.492 53 | 9.146 265 | 9.262 646 |

4.8 | 0.627 763 | 0.372 237 | 18.475 57 | 18.547 56 | 9.237 783 | 9.327 783 |

5.3 | 0.649 428 | 0.350 572 | 18.520 44 | 18.952 04 | 9.307 602 | 9.376 022 |

A rise in the effect of restricted admissibility provides insight into the defined classical queuing model’s performance indicators and is well demarcated in Fig. 8.

The increase in the effect of the standby server in the queuing system during the long vacation process is numerically demonstrated in Table VIII.

φ
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.725 988 | 0.274 012 | 7.966 114 | 8.166 114 | 3.983 057 | 4.083 057 |

4 | 0.640 878 | 0.359 122 | 6.277 805 | 6.477 805 | 3.138 902 | 3.238 902 |

6 | 0.595 604 | 0.404 396 | 5.501 693 | 5.701 693 | 2.750 846 | 2.850 846 |

8 | 0.514 129 | 0.485 871 | 5.055 856 | 5.255 856 | 2.527 928 | 2.627 928 |

10 | 0.485 942 | 0.514 058 | 4.766 487 | 4.966 487 | 2.383 244 | 2.483 244 |

12 | 0.402 206 | 0.597 794 | 4.563 522 | 4.763 522 | 2.281 761 | 2.381 761 |

14 | 0.364 474 | 0.635 526 | 4.413 29 | 4.613 29 | 2.206 645 | 2.306 645 |

16 | 0.317 327 | 0.682 673 | 4.297 605 | 4.497 605 | 2.148 803 | 2.248 803 |

18 | 0.256 458 | 0.743 542 | 4.205 78 | 4.405 78 | 2.102 89 | 2.202 89 |

20 | 0.214 281 | 0.785 719 | 4.131 123 | 4.331 123 | 2.065 562 | 2.165 562 |

φ
. | Q
. | ρ
. | L_{q}
. | L
. | W_{q}
. | W
. |
---|---|---|---|---|---|---|

2 | 0.725 988 | 0.274 012 | 7.966 114 | 8.166 114 | 3.983 057 | 4.083 057 |

4 | 0.640 878 | 0.359 122 | 6.277 805 | 6.477 805 | 3.138 902 | 3.238 902 |

6 | 0.595 604 | 0.404 396 | 5.501 693 | 5.701 693 | 2.750 846 | 2.850 846 |

8 | 0.514 129 | 0.485 871 | 5.055 856 | 5.255 856 | 2.527 928 | 2.627 928 |

10 | 0.485 942 | 0.514 058 | 4.766 487 | 4.966 487 | 2.383 244 | 2.483 244 |

12 | 0.402 206 | 0.597 794 | 4.563 522 | 4.763 522 | 2.281 761 | 2.381 761 |

14 | 0.364 474 | 0.635 526 | 4.413 29 | 4.613 29 | 2.206 645 | 2.306 645 |

16 | 0.317 327 | 0.682 673 | 4.297 605 | 4.497 605 | 2.148 803 | 2.248 803 |

18 | 0.256 458 | 0.743 542 | 4.205 78 | 4.405 78 | 2.102 89 | 2.202 89 |

20 | 0.214 281 | 0.785 719 | 4.131 123 | 4.331 123 | 2.065 562 | 2.165 562 |

The standby server acts during the first stage of a long vacation, and it makes all the performance measures to increase and reduces idle time. This phenomenon is well explicated in Fig. 9.

## V. CONCLUSION

In this paper, we examine a non-Markovian boundary value problem. The queuing issue that arises with the aforementioned structural processes is converted into a mathematical problem by means of the differential difference equations approach and solved using the supplemental variable technique. For the specified problem, various execution strategies are developed for the queue length, server throughput time, utilization, and probabilistic generating functions. Several rare examples were examined using numerical analysis. In differential processes with uncertain stochastic processes, we have examined the solutions. We have demonstrated that one uniqueness of the use of generating functions is the ability to discover not only moments but also unknown functions. The concept of probability and stochastic phenomena has this benefit over other mathematical theories. More information about process studies can be gleaned through analytical solutions than from approximative or numerical approaches. Researchers and others who design and build fast computing facilities with hundreds of computational nodes will find the presented model and its solutions interesting. They are helpful for wireless network studies as well. They are used in logistics, economics, and the evaluation of how well various companies operate.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**K. Karthikeyan**: Conceptualization (equal); Methodology (equal). **S. Maragathasundari**: Conceptualization (equal); Methodology (equal). **M. Kameshwari**: Supervision (equal); Validation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.