Reliability and Sensitivity Analysis of a Batch Arrival Retrial Queue with k-Phase Services, Feedback, Vacation, Delay, Repair and Admission

Queueing theory is a way for real-world problems modeling and analyzing. In many processes, the input is converted to the desired output after several successive steps. But usually limitations and conditions such as Lack of space, feedback, vacation, failure, repair, etc. have a great impact on process efficiency.This article deals with the modeling the steady-state behavior of a retrial queueing system with phases of service. The arriving batches join the system with dependent admission due to the server state.If the customers find the server busy, they join the orbit to repeat their request. Although, the first phase of service is essential for all customers, any customer has three options after the completion of the phase . They may take the phase of service with probability , otherwise return the orbit with probability or leave the system with probability . Also, after each phase, the probabilistic failure, delay, repair and vacation are considered.In this article, after finding the steady-state distributions, the probability generating functions of the system and orbit size have been found. Then, some important performance measures of the system have been derived. Also, the system reliability has been defined. Eventually, to demonstrate the capability of the proposed model, the sensitivity analysis of performance measures via some model parameters (arrival/retrial/vacation rate) in different reliability levels have been investigated in a specific case of this model. Additionally, for optimizing the performance of system, some technical suggestions are presented. Keyword:Bernoulli vacation, Feedback, Performance measures, Retrial queue, State-dependent admission, Repair, Delay, Reliability


Introduction *
One way to identify the behavior of the systems in order to control and increase their productivity is to determine the model that they follow. On the other hand, without considering the priorities, real conditions and possible limitations for a system, the model fitting will not have the necessary efficiency.Today, increasing satisfaction of the customers is one of the most important priorities of dynamic systems.Sometimes, the customers arrive to the systems individually and sometimes in batches to receive services. For example, the sent products to the inspection test unit can be mentioned. In many systems, such as production of lines, achieving the desired result occurs after a multi-steps process. On the other hand, any system faces some limitations. One of these limitations is the lack of space for customers. In this case, the server is not ready to serve the customer at the * Corresponding Author Email: abdollahi6028@yahoo.com moment of arrival. Therefore, considering another space (called the orbit) for new customers to repeat their requests from there is one solution for this problem. This is known as the retrial phenomenon.Another constraint islimited resources and facilities.These systems have to impose restrictions on customer admission according to their conditionsdue to the state of the server (idle/busy). Also, dissatisfaction with the results of each step leads to incomplete process for reasons such as returning the customer to orbit for reservice or leaving the system. Sometimes the system has to be refreshed by going on a vacation. On the other hand, system failure, especially when it is not possible to repair the system immediately, and the increasing cost and time can't be neglected.Since reducing time and cost is one of the most important factors in customer satisfaction, overcoming these limitations is essential for the survival of the system.One of the useful techniques for modeling the systems and determining its performance measures (such as the mean of customers in the system and orbit and their waiting times) is the queueing theory. On the other hand, the reliability assessment is an effective approach to maintain and enhance the quality of the process output, increasing customer satisfaction and market share in the competitive world today. So far, many studies have been done by scientists in this regard. Some of them are as follows: The optimizing of the phoneconversations in a call center by Erlang [1] was the first experience of using the queueing theory.
Afterward, the retrial queueing models have been investigated by several researchers such as Falin and Templeton [2]. Also, a literature of the investigations about the retrial queues has been presented by Artalejo [3].
Besides, the batch arrival of the customers has been studied by many researchers. Falin [4], Kulkarni [5] and Yamamuro [6] are some examples of this subject.
Some of the studied retrial queueing models have several essential or optional phases of services. Some of the works in this area have been done in this area are Kumar [18] have considered the feedback assumption in their models.
On the other hand, depending on the situation, different systems face different types of vacations, such as general vacation by Senthikumar and Arumuganathan [19], modified vacation by Jain and Bhagat [20], Bernoulli vacation by Choudhury and Ke [21], working vacation by Azhagappan [22] and variant vacation by Ke [23].
Occurrence of the breakdown/failure isthe inevitable issue for any system. Therefore, in designing any system, preventive or corrective actions should be planned. So, thebreakdowns/failures repairis one of the most important topics in this program. Therefore, this issue has been considered in many of the studied systems such as V. G. Kulkarni and Bong Dae Choi [24] and P.Rajaduraia et al. [25].
But sometimes due to some limitations, these repairs are delayed.Madhu Jain and AmitaBhagat [26] and Choudhury and Ke [21] have been considered this issue in their model. Some systems aren't able to respond to all customers. So, they have to impose restrictions on customer admission according to their conditions due to the state of the server (idle/busy). In this relation, Choudhury and Deka [27][28] have considered the Bernoulli admission mechanism in their model.
Improving the system reliability is one way to achieve the secure system. The reliability of multi-component systems was studied by Birnbaun et al. [29]. Also, these subjects have been considered in the queueing models by some authors such as Li et al. [30], Tang [31], Wang et al. [32] and Achcar and Piratelli [33].
In this article, modeling and analyzing a‫ܯ‬ Ȁ‫ܩ‬Ȁ ͳretrial queue system with k-phases of heterogeneous services in succession with first essential and ݇ െ ͳ optional phases, and state-dependent admission have been studied. Also, after each phase, the probabilistic feedback, failure, delay, repair, and vacation have been considered. Also, by considering the successful delivery of all service stages as the system successful, the conception of reliability has been defined and the reliability analysis hasbeen done. Of course, there exist the other definitions of the concept of reliability for other models which can be referred to [34][35][36].
Despite many valuable studies, any system has not been studied with these conditions. The novelties of this article are considering all of the above conditions in a system together, modeling and obtaining the performance measures of the system and reliability and sensitivity analysis of a special case of it. In this relation, the queueingmethod for modeling and analysis of systems with process approaches hasbeen considered. This model is applicable in many processes such as telecommunication systems, telephone switching systems, computer networks, and inspection tests of products.
For this model, the steady-state distributions, the probability generating functions of the system and orbit size have been found. Then, the performance measures have been obtained by using the supplementary variable technique.
In summarizing, the main contributions of this article are as below: 1) Considering batch arrival, state-dependent admission and (after each phase) the probabilistic feedback, failure, delay, repair, and vacation conditions together in a k-phases retrial queueing system with first essential and ݇ െ ͳ optional phases, 2) Having three choices for customers after ݅ െ ‫݄ݐ‬phase ሺ݅ ൌ ͳǡʹǡ ǥ ǡ െ ͳሻ i. going to ሺ݅ ͳሻ െ ‫݄ݐ‬phase service with probability ߠ ሺߠ ൌ Ͳሻ, ii. going to orbit with probability‫‬ ǡ iii. leaving the system with probability ሺͳ െ ‫‬ െߠ ሻ, 3) Considering general assumptions such as arbitrary distributions of retrial/service times, different probabilities at each phase, and variable size of arrival batches to have a comprehensive model to contain different systems in special cases, 4) Considering the system reliability and its effect in sensitivity analysis,

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5)
Providing an applicable example in the engineering field with technical suggestions, 6) Using the queueing method for modeling and analysis of the systems with process approach.
This paper is organized as follows. The model description is given in section 2. Section 3 deals with the analysis of the system containing the definitions, steady-state equations, and PGF † s. The PGFs of the system and orbit size and some important performance measures are obtained in section 4. Eventually, in section 5, by some numerical examples, the sensitivity of some performance measures isinvestigated. Also, the conclusions are provided in the section 6.

Model Description
The considered retrial queue has the following assumptions: A. The customers arrive in batches from outside the system according to a Poisson process with arrival rate ߣ with admission depending to the state of server. So, the probabilities of arrival areߙ ଵ when the server is idle,ߙ ଶ when the server is busy, ߙ ଷ when the server is on vacation,ߙ ସ when the server is in delay andߙ ହ when the server is under repair. The probability mass function (p.m.f) of the size of batches is ܿ ൌ ሺܺ ൌ ݉ሻ Ǣ ݉ ͳ, withPGF ‫ܥ‬ሺ‫ݖ‬ሻ ൌ ‫ݖ‪ሾ‬ܧ‬ ሿand the first two factorial moments ‫ܥ‬ ሾଵሿ and‫ܥ‬ ሾଶሿ are finite. B. There is no waiting space and if the server is busy, the arriving batches enter a retrial group (orbit) to repeat the request for service with the FCFS discipline. Otherwise, one customer of the arriving batch takes the service and the others enter the orbit. C. If an arriving customer (primary or retrial) finds the server idle, then he/she/it enters immediately to take the first phase of service. The concept of the primary customer is the customer who has arrived to the system for the first time. D. The retrial times are generally distributed with distribution function ‫ܣ‬ሺ‫ݔ‬ሻ, density function ܽሺ‫ݔ‬ሻ and Laplace transform‫ܣ‬ ‫כ‬ ሺߠሻ. Also, the first and second moments of this distribution are finite. E. The server provides ݇ phases of heterogeneous services in succession. The first phase is essential for all customers. At the end of ݅ െ ‫݄ݐ‬ phase of service ሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ െ ͳሻ, the customer may take the ሺ݅ ͳሻ െ ‫݄ݐ‬ phase with probability Ʌ ୧ , return to orbit to repeat the † Probability generating functions retrial to take the service again with probability ୧ or depart the system with probability ͳ െ Ʌ ୧ െ ୧ ǤThe service times are independent and for ݅ െ ‫݄ݐ‬ phase denoted by the random variable ‫ܤ‬ with general distribution functions ‫ܤ‬ ሺ‫ݔ‬ሻ, density functions ܾ ሺ‫ݔ‬ሻ and Laplace transforms‫ܤ‬ ‫כ‬ ሺߠሻሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻ. Also, the first and second moments of these distributions are finite. F. At the end of ݅ െ ‫݄ݐ‬phase ሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻ, if the applicant does not go to ሺ݅ ͳሻ െ ‫݄ݐ‬ phase the server can have a vacation with probability ߬ ୧ or may continue the new service with probabilityͳ െ ߬ ୧ . Vacation times are random variables with general distribution functionܸ ሺ‫ݔ‬ሻ, density function ሺ‫ݔ‬ሻ and Laplace transformܸ ‫כ‬ ሺߠሻ. The first and second moments of this distribution are finite. G. The server may fail at any phase. The life of the equipment used for the ݅ െ ‫݄ݐ‬ phase service has an exponential distribution with parameter ͳ ܽ ୧ ൗ ǡ ሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻǤ H. The delay time that occurred from the time of failure of the system to the time of repair at ݅ െ ‫݄ݐ‬ phase ሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻ has a random variable ‫ܦ‬ ୧ with general distribution functions ‫ܦ‬ ሺ‫ݔ‬ሻ, density functions ݀ ሺ‫ݔ‬ሻ and Laplace transforms‫ܦ‬ ‫כ‬ ሺߠሻሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻ. Also the first and second moments of these distributions are finite. I. The repair time of the system faced with failure in the ݅ െ ‫݄ݐ‬ phase has a random variableܴ ୧ with general distribution functions ܴ ሺ‫ݔ‬ሻ, density functions ‫ݎ‬ ሺ‫ݔ‬ሻ and Laplace transformsܴ ‫כ‬ ሺߠሻሺ݅ ൌ ͳǡʹǡ ǥ ǡ ݇ሻ. Also the first and second moments of these distributions are finite.

Analysis of the model
To analyze the model, first, the state of the system is recognized. Since the distribution of the service times is unknown (general), thus, this model doesn't hasthe Markovian property. But, an embedded Markov chain can be defined.
For this, the state of the system at time t by the Markov process ܼሺ‫ݐ‬ሻ ൌ ሼ ܰሺ‫ݐ‬ሻǡ ܺሺ‫ݐ‬ሻሽ is considered, in which for ͳ ݅ ݇: and Also,ܰሺ‫ݐ‬ሻcorresponds the number of customers in the retrial queue at time t.
e) The repeated attempts (retrial) rate at time ‫ݔ‬ is

Definition2The
reliability function of the system/applicants is defined as: According to the stated points in this section, the theory of the model can be extended in the following.
First, the stationary distribution of the Markovprocessሼܼሺ‫ݐ‬ሻǢ ‫ݐ‬ Ͳሽ under the stability condition is found. For this goal, the bellow probabilities are defined: where ‫ܬ‬ሺ‫ݐ‬ሻ and ܰሺ‫ݐ‬ሻ have been defined before and the probability densities are: and for ݅ ൌ ͳǡ ǥ ǡ ݇ǡ Now, by using the supplementary variable technique the steady state equations can be obtained for ݅ ൌ ͳǡ ǥ ǡ ݇ as follow:

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System performance measures
In this section, some essential performance measures are derived. For this, the partial generating functions of the number of customers in the system and in orbit are introduced. Then by differentiating of these functions, the means of system and orbit size are obtained.

Proposition
a) The PGF of the number of customers in the system is The PGF of the number of customers in orbit is

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ܹሺ‫ݖ‬ሻ ൌ ܴ ିଵ ܺ ିଵ
The mean orbit size is: Then, the inspection re values of rel the following  , ͳ h