Particle Swarm Optimization and Maximum Entropy Results for M/G/1 Retrial G-Queue with Delayed Repair

This paper examines an M/G/1 retrial model with negative customers including the concepts of working vacation, Bernoulli feedback, delayed repair, state-dependent and multi-optional services. Such a queue is quite relevant in real world, for instance in computer systems, manufacturing organisations, packet-switching networks, telecommunication systems, etc. The arrival pattern of customers is according to Poisson distribution. The service is such that first essential service (FES) is provided to every customer and second optional service (SOS) is provided in k phases to those who wants to opt for the same. The negative customers may arrive during the time when the server is busy in serving a positive customer. This leads to breakdown of the server and thus the server has to be restored (repair) by the repair man. Some moments may be taken by the repair man to initiate the repair process, leading to delay in repair. In our work, firstly we have calculated performance measures like long run probabilities and orbit size along with some reliability indices. Then a relative study between the exact expected waiting time and approximate expected waiting time of the system is presented via maximum entropy approach. Also we perform cost optimization using particle swarm optimization (PSO method). Few numerical results are also provided. KeywordsRetrial, Multi-optional service, Queue length, Maximum entropy principle, Particle swarm optimisation.


Introduction
This study deals with bulk arrival retrial G-queue where the server is subjected to state dependent and multi-optional services. In G-queues, a negative customer arrives only when server is occupied with a positive customer and forces it to leave the system thereby causing an interruption in the service. A very common example is virus attacking a system. Banks, hospitals and shopping malls are some of the places where such queues can be observed. The concept was first given by Gelenbe (1991). Yang et al. (2013) have provided reliability measures and queueing measures for a batch arrival G-queue under general retrial times, breakdown and single vacation policy. Recently, Upadhyaya (2020b) investigated a single server retrial G-queue with second phase service by applying supplementary variable technique and also discussed its application to local area networks.
Our research also involves recurrent (retrial) queues wherein if the service provider is unavailable at the time of incoming customer then the latter enters a pond of waiting customers called 'orbit' and retries repeatedly to obtain that service. Such a situation is quite evident in our day to day life where blocked customers temporarily departs from the service zone and retries for the same service after an arbitrary interval of time. For instance, in computer networking if a message or packet is lost then later on it is retransmitted via Transmission Control Protocol (TCP). From past few decades, a lot of research has been done and is still going on in retrial queues. Authors like Artelajo (1994) and many more have contributed a lot in this field. Upadhyaya (2014) and Rajadurai et al. (2018) both have calculated all the beneficial performance indices for single server retrial system by applying supplementary variable technique. The former has considered a batch arrival pattern following modified vacation policy while the latter has included the concept of server breakdown, repair and multiple vacation policy. Later on, Upadhyaya (2018) has calculated all the necessary results for a non-continuous retrial model with geometric arrival following J-vacation policy.
In this model, the service is provided in two stages which include first essential service (FES) and second optional service (SOS). The SOS is in k stages which depend on the customer to demand for it or not. Further, whenever the server is free, it goes on working vacation. In working vacation state, server does not stop service completely rather it provides service at a slower rate. However, in this working vacation phase if customers are present at any service fulfilment, then it may immediately stop that phase and return to normal busy state i.e. there can be a possibility of vacation interruption. Sethi et al. (2019) have worked on an M/M/1 model under working vacation and further performed a comparative analysis of their results via Runge-Kutta method and ANFIS (Adaptive Neuro Fuzzy Interface System). The research done by Jain et al. (2019) and  on systems with vacation and server breakdown is worth mentioning.
The main motive of our investigation is to obtain maximum entropy results and present a relative study between the exact expected waiting time and approximate expected waiting time of the system. Jain and Upadhyaya (2012) have analysed a bulk retrial queue under N-policy using maximum entropy principle (MEP). Pandey and Tripathi (2014) derived approximate waiting time of a model with multi-optional services and made a comparison of maximum entropy results with the exact results. In recent years, Parkash and Mukesh (2016) have depicted the relation between this method and queueing theory in a very detailed and explanatory manner. A comparative study by applying MEP on the results obtained and exact results of M X /(G1, G2)/1 system with two stages of services has been presented by Chauhan (2018).
As per the literature survey, no work has been done on finding maximum entropy results and optimal parameters for an M X /G/1 retrial G-queue with multi-optional and state dependent service under Bernoulli feedback, working vacations as well as delayed repair. This work is very much relevant and helpful in the areas with high congestion and traffic intensity. We have also carried out cost analysis of the considered model by implementing PSO. This technique has become quite renowned in last few years due to the fact that the candidate solution gets better iteratively and thus the system provides us the best solution available. Whether we have to maximize performance indices or minimize the total cost, this method can be applicable in queueing analysis for obtaining fruitful results. Work done by Zhou et al. (2018) and Wang et al. (2019) in applying particle swarm optimization is worthy of praise. Former has investigated equilibrium strategies for constant retrial queue while the latter has worked on machine repair problem with retrial phenomenon and working breakdown. Recently authors like Sengupta et al. (2019) and Upadhyaya (2020a) have also worked on this technique.

Explanation of the Model
This work involves a single server system where customers come in batches and server provides multi-optional service and undergo delayed repair. The total customers present at times t is denoted by N(t). The detailed description is as follows: Arrival pattern-Arrival occurs in batches with respect to compound Poisson process and it is dependent upon the server as given below  λ0 -for server being idle either during working vacation or during regular service mode.  λ1 -for server being occupied with FES or SOS at time t.  λ2 -for server being occupied and on working vacation at time t.  λ3 -for server waiting for repair or under repair.
A random variable X is defined to represent the batch size and P(X=k) = Ck is the probability distribution of batch size with 1 1     k k C . Also we consider the random variable ϐ(t) defined as 0, when server is free and on working vacation 1, when server is free and in regular service 2, when server is occupied with first essential service at time t ϐ(t) = 3, when server is busy with second optional service at time t 4, when server is busy and on working vacation 5, when server is waiting for repair 6, when server is undergoing repair Retrial Process -When server is busy or breakdown or under repair/delayed repair state, the customer joins a virtual pond known as orbit. From the orbit, the customers keep trying for service until they get it. We assume that only the customer at the head of the queue is permitted to access the server. Further, let the inter-retrial time follows a general random variable S with distribution S(t) and its Laplace Stieltjes Transform (LST) S*(ϑ).
Service Process -Service is assumed to be multi-optional. The service time for first essential service (FES) obeys a general distribution with random variable B0 and distribution function B0(t). Also, the service time for second optional service (SOS) obeys a general distribution with random variable Bi and distribution function Bi(t) (1≤i≤k). The corresponding LST for FES and SOS are B0 * (ϑ) and Bi * (ϑ), 1≤i≤k respectively. Also, we have taken mi (0≤i≤k) as the probability that the customer opts for i th service.
Working vacation process -If the system is free of customers the server leaves for a working vacation. In such type of vacation, the server does not terminate the service completely rather it serves with a comparatively lower service rate. This period obeys an exponential distribution with parameter η. On completion of a working vacation period if server observes that the system is vacant then either it remains idle for a new customer with probability p or it continues its vacation with probability q(=1-p). But if customers are there at the end of a working vacation then vacation interruption may take place wherein the server immediately returns to normal busy mode. While being on working vacation, the server obeys a general distribution with random variable Bv and distribution function Bv(t). The corresponding LST is Bv * (ϑ) and bw and bw 2 are the first and second moments respectively.
Repair and Delayed Repair Process -The entrance of a negative customer during the service of a positive customer leads to a server breakdown. The arrival rate of such customers is θ. After a breakdown the server needs to get repaired. It may be possible that repairman takes time to initiate the repair. This is known as delayed repair time. Let the repair time obeys a general distribution with random variable G0 for FES and Gi for SOS. The corresponding distribution function is G0(t) and Gi(t). So the LST will be G0 * (ϑ) and Gi * (ϑ). On similar pattern the delayed repair time obeys a general distribution H0(t) and Hi(t) with random variable H0 and Hi. Thus, the LST for delayed repair function becomes H0 * (ϑ) and Hi * (ϑ), where 1≤i≤k. The r th moment for delayed repair at FES is h0 r and for SOS it is hi r . Similarly, the r th moment for repair state at FES is g0 r and for SOS it is gi r , 1≤i≤k Bernoulli Feedback -We have also considered the concept of Bernoulli Feedback. If the customer is dissatisfied by the service due to some issue, then it may join the queue again for availing the same service. Let the probability that a customer goes for feedback be s and will exit the system once it is served be r, where s=1-r.
The hazard rates for retrial, busy (with FES and SOS), working vacation, repair and delayed repair state of the server respectively are as follows: Let us denote the elapsed retrial time, elapsed service time (for FES and SOS), elapsed working vacation time, elapsed repair time (during FES and SOS) and elapsed delayed repair time (during FES and SOS)

Practical Application
This study is quite applicable in telecommunication sector, for instance, we consider a communication system which is built for making restaurant reservations. Let us assume that a restaurant takes its bookings and offer various other services via telephone system, where a customer can call and book a table for itself (positive customer). This telephone system is staffed with a manager (single server) who answers all the phone calls. After booking a table (completion of first essential service) the customer may hang up the phone call (exit the system) or it may enquire about some party bookings, book tickets for an upcoming musical concert, etc. (optional services). There may be a miscommunication due to the lack of clarity in network or any other such issues and a person has to call again to confirm his bookings (Bernoulli feedback). Whenever the manager is busy in supervising other parts of the restaurant then he is unable to answer the calls (on vacation mode) and in such situation the assistant manager (substitute server) serves usually but comparatively at a slower pace (working vacation). During this phase, once phone call is completed (at the service completion) if there are calls in the system then the manager returns immediately (i.e. vacation interruption occurs). However, if there are no calls once the manager has finished his secondary job (vacation is over) then manager continues to handle other issues of the restaurant (vacation mode). Chances are there that the line is busy when a customer makes a call, then customer calls back after a certain period of time (retrial). There is a possibility that during a phone call, signal may not be appropriate or low network coverage or virus attack (negative customers) due to which customer loses his service. The telephone line then has to get repaired and the repairman may take time to initiate the repair process (delayed repair). Once the signal repair is completed, the communication system works as good as new. There is also a chance that enormous number of calls arrives on a weekend then either manager works more efficiently or the line may get jam (state dependent condition). This scenario completely matches with our model in consideration. Thus, all the results and analysis presented in our work will be very profitable for the system manager. Numerical illustration for the discussed example is given in section 6. This model is also applicable in the field of packet switching network, Simple Mail Transfer Protocol (SMTP), call centres, etc.

Performance Measures
For completely analysing a queueing system we must know how well it can perform. This can be done by obtaining the performance indices of our interest. A queueing system with no waiting line or queue length, is the most ideal one, however it is not practical to form such model. Therefore, it is important to predict average number of customers and their delay time which has a great effect on whether a customer will join a queue or not. Also, the probability of the server in idle state, busy state or any other different state is another essential performance measure. If all these fruitful results are available, then it is beneficial for the system designers and for the arriving customers as well. On one hand, the system managers/designers can make the system better by predicting the service time, repair time, etc. whereas on the other hand, customers are at an ease if they are aware of the average waiting time in the queue. This particular section is devoted to all the performance indices calculated. Very firstly, we have provided the marginal generating functions of different states of the server.

Lemma 1
The marginal probability generating functions for retrial period, busy period with FES and SOS, working vacation period, delayed repair period under FES and SOS and repair period under FES and SOS are respectively given by

Queueing Measures
In this subsection, we aim to find the long run probabilities of the state of the server as well as the queue size and waiting time of the customer which is a great source of information for making a model more sufficient and economically friendly. The results are presented in the following two theorems as follows:

Theorem 1
Chances that server is idle during retrial time P(PR), server is occupied with FES P(π0), with SOS P(πi), server is on working vacation P(Qv), server is under delayed repair while providing FES P(D0), while providing SOS P(Di) and server is under repair while providing FES P(R0) and while providing SOS P(Ri) (for 1≤i≤k) are respectively given by The average queue length Lq i.e. the total number of customers waiting in the orbit and the exact expected waiting time is respectively given by All the above symbols used are given in appendix A.
Proof: The PGF of the total customers present in the orbit is given by Solving the above equation by using L' Hospital's rule we obtain our desired results.

Reliability Measures
In this system, the server is unreliable so reliability measures provide the necessary information that is required to improve the model. We have obtained the availability measure (AV) which is the probability that the server is either rendering service to a positive customer or it is in idle period. We have also derived the expression for the failure frequency (FF) of the server.

Lemma 2
The steady state availability of the server (AV) is

Maximum Entropy Approach
Maximum entropy approach was originated by Jaynes (1957). 'Entropy' basically refers to the variation of a system or probabilistic distribution. Many a times in probabilistic analysis it happens that we are unable to evaluate an appropriate distribution. In queueing systems, the information available for the queue size, steady state probabilities, etc. can be considered as constraints. However, there is a large variety of distributions satisfying them. In this method, we obtain the best distribution that suits them using the available information. The probability distribution that maximizes the entropy function is the optimal one.

Construction of Entropy Function
Let P1(n), P2(n), P2i(n), P3(n), P4(n), P4i(n), P5(n) and P5i(n) (1≤i≤k), n being the total number of customers there in the system, be the probabilities that server is in retrial period, busy period, working vacation period, delayed repair and repair period respectively.
Using the work done by El-Affendi and Kouvatsos (1983), we express the entropy function Y as We will find the steady state probabilities using the following set of constraints: where, ) P (n), P (n), P (n), P (n), P (n), P (n), P (n), J(P   The maximum entropy solutions for approximate values of the steady state probabilities, subject to constraints are given by Proof: Partially differentiating Lagrange's function J with respect to P1(n), P2(n), P2i(n), P3(n), P4(n), P4i(n), P5(n), and P5i(n) respectively (1≤i≤k) and then equating the result to '0' and further manipulating the equations, we get the desired results.

Theorem 4
The approximate expected waiting time of the customers present in the queue (via maximum entropy approach) is Proof: Consider a particular customer, say M who has n customers preceding him in the queue. At this instant server may be idle, busy, under working vacation, repair or delayed repair. Therefore, five cases arise depending upon the state of the server. An analysis of each case is given below in detail (a). Idle state -This is the most convenient case for the customer. The server is free to serve. So, the batch of incoming customers is served instantly. This implies that service time of the customers preceding the customer M will be the approximate expected waiting time.
. Busy state -In this case server is occupied, so the batch of arriving customers joins the orbit. Firstly, n customers who are ahead in the queue get the service then those who are ahead of the arbitrary customer M will receive the service. So the approximate expected waiting time becomes (c). Working vacation state -The customer M waits for the fulfilment of working vacation period. When the server returns to normal busy state, those customers which are present in the orbit are served first. After that the customers before M get their service.
(d). Delayed repair state -Now the waiting time includes the time taken for the start-up of repair plus the time taken for the completion of repair. Once the server comes back to serve the customers, the ones there in the orbit are served first then those who are in front of M in the queue.
(e). Repair state -It is quite similar to the above case. Only the start-up time is not included.
By adding the waiting time of all the five cases discussed above, we obtain the required result.

Particle Swarm Optimization
With a rapid growth of telecommunication and computer networking an appropriate analysis of queueing model is highly required to provide suitable service and reduce congestion. This is possible if maximum number of people reach to a cost effective system. Therefore, systems including cost optimization are very much helpful and recommended. Recently,  and  have performed a cost analysis for a queueing model with vacation and unreliable server. We have made an attempt to solve the financial restrictions in networking systems using the technique of particle swarm optimization (PSO). This method is capable of finding the best suitable solution from a very large space of solutions. Another advantage of using PSO is that International Journal of Mathematical, Engineering and Management Sciences Vol. 6, No. 2, 541-563, 2021 https://doi.org/10.33889/IJMEMS.2021.6.2.033 554 the objective function is not required to be differentiable unlike many other optimization techniques. There are numerous applications of this method. Some of them include the fields of signal processing, telecommunications, designing and management, etc. In this method we first initialize a given set of population comprising of numerous candidates/particles. Then these candidates are made to move in the search space with respect to the given constraints and objective function over the candidate's location and velocity. The fitness value of each particle is calculated to further evaluate personal best (pbest) and global best (gbest) values. If any particle's pbest value is better than gbest then that becomes the new gbest value. This process continues till the number of iterations is completed.
PSO method is applied to the above defined cost elements using the MATLAB software for k=3 optional services. To understand the concept better, let us describe an algorithm for this technique. For that purpose assume that    n :  is the cost function which is to be minimized. Firstly,  takes an individual solution as an argument and generates an output which becomes the objective function value of that individual solution. We are interested in finding a solution i for which for each j in the search-space.
Let the number of individuals in the community be M and every individual has a location pm and velocity velm. Let posm be the best location of candidate m and P the universal best location (best among the entire swarm). Also, lo and up are the lower and upper bounds of the search space respectively. The algorithm is then given by We have calculated the total cost of the system by varying some of the parameters and observe that the minimum cost obtained is $2.5862 for λ1 = 7.1 and (θ * , η * ) = (0.008, 0.58).

Sensitivity Analysis
Numerical examples are very much mandatory so that we can predict the outcome of parameters considered in our model on the performance metrics. Performing numerical analysis of any model predicts its behaviour in real life scenario.

Numerical Simulation of Practical Example
In this particular section we have illustrated the example considered in Section 2.1, i.e. how a communication system of a reservation counter of a restaurant behaves in different situations by performing a numerical experiment. We assume that customers book the table via phone call with rate λo=0.28/min. The manager answers the phone call with rate βo=19/min. Let us consider that server provides three optional services such as party lawn booking, home delivery and pick and drop service with rate β1=25/min, β2=15/min and β3=20/min. The probability that customer opts for these services is mi=0.0055 for i=1, 2, 3 and arrival rate is λ1=7.2/min, λ2=5.5/min and λ3=3/min respectively. Whenever there is a miscommunication then customer calls again for feedback with probability s=0.9. There is also a possibility that line is busy and customer has to make the phone call again with rate α=0.009/min. If the line has to get repaired due to this then the delayed repair and repair parameters are respectively ξo=12, ξi=8 and ζo=14 and ζi=14 for i=1, 2, 3. When the manager is off duty then assistant manager (substitute server) takes the charge with rate βv=10/min. At the end of his vacation when there is no phone call then the manager returns to provide service with probability p=0.001. The results obtained are P(PR)=0.1596, P(πi)=0.0014 (0≤i≤3), P(Qv)=0.023 and Lq=1.002.

Effect of System Parameters on Long Run Probabilities and Average Queue Size
We have also explored some other numerical results. These results convey how different variables can have repercussions on system performance measures. MATLAB software is used to illustrate the following results:     In Figures 1 and 4, we see a decrease in average queue length with an increase in η and β0. While in rest of the two figures we notice that it is directly proportional to the values of λ1 and α.

Comparison between Exact Waiting Time and Expected Waiting Time
In the next two tables we show a relative study comprising the exact expected waiting time (   From Tables 4 and 5, we can observe that the waiting time is directly proportional to θ and βo. However, it is completely opposite for the rest of the parameters. Also, APE is comparatively less in Case 1 than in Case 2. From the numerical analysis discussed above we can arrive at a conclusion that the results obtained are similar to practical situations that are observed.

Conclusion and Future Possibility
This investigation involves a bulk retrial G-queue under state dependent, voluntary services, Bernoulli feedback along with delay in repair of the repairman. Performance indices like the long run probabilities and average queue size of the system are discussed which can manage and reduce the queue size. The main motive of our investigation is that we present a relative study corresponding to the exact and approximate expected waiting time for the developed model so that the system planners and decision makers can reduce the waiting time of the customers in the queue which in turn improves the grade of service of their respective systems. This proposed model is applicable in industrial queues, management and production industry, communication networks, supermarkets, etc. Basically, in all these huge sectors it is next to impossible to build a model in which the server never fails or never deactivates. The concept of repairing of the server and the delay in repair is also very relative in today's world of high speed technology. Thus, this The remaining symbols of Theorem 2 are given below 3 2 *'' *''' * 2 1 2 2 1 2 3 2 3 *'' 2 *' 2 *' 2 1 2 2 1 2 2 1 2 2 3 2 3 3 2 1 2 3 1 1 2 2 1 1 2 1 (1 ( ))(2 ( )) (1 (1 ( ) 1 ( ))[ 6 ( )( ( ) ( )) -3 ( )( ( ) ( )) 6 ( )( )];