Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times

This paper investigates the transient behavior of a \begin{document}$ M/M/1 $\end{document} queueing model with N-policy, system disaster, repair, preventive maintenance, balking, re-service, closedown and setup times. The server stays dormant (off state) until N customers accumulate in the queue and then starts an exhaustive service (on state). After the service, each customer may either leave the system or get immediate re-service. When the system becomes empty, the server resumes closedown work and then undergoes preventive maintenance. After that, it comes to the idle state and waits N accumulate for service. When the \begin{document}$ N^{th} $\end{document} one enters the queue, the server commences the setup work and then starts the service. Meanwhile, the system suffers disastrous breakdown during busy period. It forced the system to the failure state and all the customers get eliminated. After that, the server gets repaired and moves to the idle state. The customers may either join the queue or balk when the size of the system is less than N. The probabilities of the proposed model are derived by the method of generating function for the transient case. Some system performance indices and numerical simulations are also presented.


1.
Introduction. Queueing models with disaster and repair have been studied by many researchers in the past few decades as they possess wide applications in modeling many practical situations related to computer networks, communication systems, etc (refer [3], [4], [7], [8], [13], [14], [15], [19], [20], [25]). Closedown the system when it becomes empty and setup the system before starting the service, play a key role in various real life situations as they support economically to minimize the expenses of an organization. The preventive maintenance of the server is essential as it extends the life of the server. Only few works are investigated in literature related to closedown, preventive maintenance and setup times (refer [2], [10], [16]).
considered a Markovian queueing model where the server undergoes closedown and then preventive maintenance whenever the system becomes empty. They derived the transient solutions and some system measures such as asymptotic behavior of various system state probabilities, average system size, average workload, etc. Arumuganathan and Jeyakumar [2] considered a batch arrival bulk service N-policy vacation queueing model where the server closedown the system when it becomes empty and setup the system after the completion of vacation to resume the service.
Haight [6] introduced the balking behavior of customers in queueing models. Kumar et al. [11] derived the time-dependent system size probabilities of the M/M/1 queueing model. Jain [8] investigated a batch arrival queueing model with unreliable server and balking. She obtained the probability generating function of the queue size at an arbitrary epoch. Vijayalaxmi and Jyothsna [24] studied a renewal input finite buffer multiple working vacation queueing model with balking.
Takacs [23] introduced the queue with feedback customers where each customer either immediately joins the queue for another service or leaves the system, after the service completion. He obtained the steady state queue size distribution and distribution function of sojourn time of a customer in the system. Choi and Kim [5] analyzed a two phases vacation queueing model with feedback customers where the first phase is batch service followed by a single service in the second phase. They obtained the steady state system size probabilities of that model.
The significance of the proposed research work is in the following points: Closedown and setup periods are introduced in a Markovian queueing model and the transient system size probabilities are obtained in terms of modified Bessel function of first kind using the method of generating function. The main contribution of this research work is to derive the most significant transient system size probabilities of an M/M/1 queueing model with the presence of closedown, setup periods, preventive maintenance, balking, re-service, N-policy, system disaster and repair. In the literature, so far no work is carried out to derive the time-dependent system size probabilities of the proposed model. This gives us the motivation to carry out this research work.
The remaining sections are as follows. The M/M/1 queueing model with Npolicy, system disaster, repair, preventive maintenance, balking, re-service, closedown and setup times is described and the transient probabilities are derived in section 2. The system performance measures such as expectation, variance, probabilities of closedown, maintenance, failure, setup and empty state are also obtained in section 3. Numerical simulations of the proposed model are presented in section 4. The conclusion and future scope are provided in section 5.
2. Model description. We consider a single server Markovian queueing model with N-policy, system disaster, repair, preventive maintenance, balking, re-service, closedown and setup times. Figure 1 gives the state transition diagram of the present model. The assumptions to derive the system size probabilities under transient state for the model corresponding to the Figure 1 are described as follows: • When the server is busy in on state (idle in off state), customers join the queue at the rate of λ 1 (λ 0 ) which is exponentially distributed. It is assumed that the arrival rate during the off state is less than that of the on state of the server. That is, λ 0 < λ 1 .
• The server stays dormant until N customers accumulate in the queue and then starts an exhaustive service which is exponentially distributed with the rate µ. • After the service, each customer may either move out from the system with probability η or get immediate re-service with probability 1 − η. • When the system becomes empty, the server resumes closedown work exponentially at the rate δ and then undergoes preventive maintenance exponentially with parameter ξ. • After that, it comes to the idle state and waits N accumulate for service.
When the Nth one enters the queue, the server commences the setup work exponentially at the rate γ and then starts the service. • Meanwhile, the system suffers disastrous breakdown during busy period which is distributed exponentially with parameter θ. • It forced the system to the failure state and all the customers get eliminated.
After that, the server gets repaired exponentially at the rate ω and moves to the idle state. • The customers may either join the queue with probability σ or balk the queue with probability 1−σ when the size of the system is less than N and the server is idle in off state. • The service of customers is based on first-come first-served.
• Assume that inter-arrival, service, failure, repair, closedown, setup and maintenance times are all independent.
2.1. Practical applications of the proposed model. In the process of Plasma Arc Welding, the welding of metal pieces (job) takes place one by one. It will be performed if there are sufficient quantity of work pieces (jobs) available for welding. In order to meet the operating cost, it will be put in waiting for the arrival of N (threshold value) work pieces (jobs). During this waiting period, as there is no service available to the arriving jobs, some of the arriving jobs decide to leave the work place without waiting in the queue (balking). After the arrival of N th job, the tungsten electrodes are connected and the position of nozzle is corrected in order to start the welding process. Then the welding process begins and completes service for all the work pieces exhaustively. After the completion of service, immediate re-service can be given to the work piece if it is required. If there is no work piece waiting for welding after welding the last work piece, all the connections of welding torch are removed and then the electrodes as well as nozzle are cleaned. During the welding process, the torch utilized for welding may be breakdown. Then immediately all the work pieces (jobs) are removed from the work place and the torch will be sent for repair. After the repair, the welding torch becomes ready to start the welding process and waits for the arrival of N work pieces. This situation is modeled mathematically in this research work. Let {Ω(τ ), τ ≥ 0} and ζ(τ ) are size of the system and the server state respectively at time τ . Let ζ(τ ) = 0, off state of the server, 1, on state of the server at time τ .
2.2. Transient probabilities. The transient system size probabilities are derived for the proposed model in this section.
Theorem 2.1. The expressions for the probabilities π 1,j (t), for j ≥ 1 are derived from (3), (4) and (5) as where I j (τ ) is the modified Bessel function of the first kind of order j.
4. Numerical illustration. The numerical simulation is carried over for the model under consideration using MATLAB software in this section. Probability values π 1,1 (τ)   All the probability curves, except π 0,0 (τ ) increase initially and become steady state in the long run. Figure 4 and 5 depict that the average and variance of number of jobs in the system increase when the rate σ of joining the queue increases. It is natural in every practical situation that the customers get accumulated when their joining rate increase. Figure 6 and show that the expected value and variance of customers in the system go down with the increment of leaving rate of the system. This happens as more customers leave the system while η increases.

5.
Conclusion and future scope. The analysis of a single server Markovian queueing model with N-policy, system disaster, repair, preventive maintenance, balking, re-service, closedown and setup times is carried over. Using the method of generating function, the probabilities of number present in the system are computed for the transient state. Numerical illustrations motivate us to visualize the influence of various system parameters. The future extension of this model may be a multiserver queueing model with closedown, setup times and maintenance.