STEADY STATE ANALYSIS OF QUEUEING SYSTEM WITH RANDOM VACATION SUBJECT TO SUPPLEMENTARY VARIABLE METHOD

STEADY STATE ANALYSIS OF QUEUEING SYSTEM WITH RANDOM VACATION SUBJECT TO SUPPLEMENTARY VARIABLE METHOD G. SEENIRAJ, R. JAYARAMAN, D. MOGANRAJ Department of Mathematics, Periyar University, Salem-636011, Tamil Nadu, India Department of Mathematics, Periyar University Constituent College of Arts and Science, Pappireddipatti, Dharmapuri 636905, Tamil Nadu, India Department of Science and Humanities, Er Perumal Manimekalai College of Engineering, Hosur635117, Tamil Nadu, India


INTRODUCTION
Queueing hypothesis is the numerical search of investment up lines or line. The suggestion empowers numerical assessment of a only some of related forms, including poignant base at the line, holding up in the line and being served by the server at the front of the line. The assumption allows the induction and computation of a few presentation measures including the normal property up time in the line or the framework, the ordinary number of clients and the possibility of experiencing the framework in certain states, for example, vacant, full, having an available server or trusting that a specific time will be served. In this situation the circumstance where a server is inaccessible for essential clients in periodic interims of time is known as repair. The excursions may speak to server's taking a shot at some advantageous employments, performing server upkeep review and fixes, that intrude on the client service. Allowing servers to take repair makes the line models increasingly practical and adaptable in considering genuine circumstances. In the event that it isn't in task individuals regularly go for some other stimulation. In such cases, the landing rate will be low. This reasonable circumstance precisely fits into the present model. Gupta and Sikdar (2004) [6] have analysed a single server finite-buffer bulk-service queueing model with particular vacation in which the interarrival and service times are considered to be exponentially and arbitrarily distributed, respectively. Madan and Al-Rawwash (2005) [8] have analysed a single server queue with batch arrivals and general service time distribution. Baba (2012) [2] used a batch arrival M/G/1 queue with multiple working vacation and obtained probability generating function and stochastic rotting structure of the system and some performance indices, mean system length and the mean waiting time. Vijaya Laxmi et al (2013) [11] Analysed a finite barrier renewal input single working vacation queue with state dependent vacations. They also presented an efficient computation algorithm and computed the stationary queue length for the above model along with different performance measures.
Sree Parimala and Palaniammal (2014) [12] studied bulk service queueing model with server's single and delayed vacation. For this model, the steady state solutions and the system characteristics are derived and analysed. Ibe (2015) [7] has considered a single server vacation queueing system with server time out and derived terminology for the mean waiting time and studied N-policy scheme. Balamani (2014) [3] has studied a two stage batch arrival queue with compulsory server vacation and second optional repair and has derived the steady state solutions also computed the mean queue length and the mean waiting time. Ebenesar Anna Bagyam et al (2015) [5] have considered immensity arrival multi-stage retrial queue with Bernoulli vacation,  [10] derived, A study on transient solution of single server queueing system with repair process by using generating functions of the system under the system down.
Using the above ideas, the consistent state line size dispersion at a subjective time is acquired.
Execution estimates like the normal line duration, expected length of occupied and inactive periods are inferred. The holding up point in time in the line is additionally got. A cost model is additionally inferred.

MODEL DESCRIPTION
We are used to develop the queue size distribution for following notation. Let X denote the customers arrival in random, Customer arrival when the server is busy is on Poisson arrival rate and be the when the server is on repair Poisson arrival rate, be the probability that m customers arrive in a batch and G_(z) be its probability generating function. Let ( ) and

CONCLUSION
We determined some significant framework attributes through the key likelihood producing capacity and also derived the expected waiting time of the customers and the function of the line length distribution. The model is critical since progressively broad circumstances in reasonable applications are considered in the model.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.