Jñānābha‎, Vol. 50 (2) (2020), (21-29)

ANALYSIS OF A FINITE MARKOVIAN QUEUE WITH SERVER BREAKDOWN, SETUP TIME AND STATE DEPENDENT RATE UNDER N-POLICY STRATEGY


By

Praveen Kumar Agrawal * Anamika Jain and ** Madhu Jain***

*Department of Mathematics, GL Bajaj Group of Institutions, Mathura-281406, India

**Department of Mathematics & Statistics, Manipal University Jaipur, Jaipur, Rejasthan-303 007, India

***Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247 667 India

Email: *prkuag@gmail.com, **anamika.jain@jaipur.manipal.edu, ***madhufma@iitr.ac.in

(Received: October 03, 2019; Revised: May 28, 2020; Final Form: December 19, 2020)

 

 

 This investigation analyzes a breakdown service with setup time under N-policy. Only in working condition, the server may fail and dispatch immediately for the repair job which can be performed according to exponential distribution by a repairman. The server goes on vacation if the system does not have job to perform, i.e., the system is empty and later on switches on when the system accumulates N jobs. The arrivals at service station of the customers follow Poisson fashion with rate dependent on the server’s status which may be idle, busy or broken-down state. By using generating function method, we derive the queue size distribution. The expressions for various performance characteristics including average queue length, probabilities of long-term fraction of time for which the server is idle, busy, broken down and in repair state. The optimal value of threshold parameter N which minimizes the total average cost is determined analytically. The parameters’ sensitivity on different performance measures is examined by facilitating the numerical results.



2010 Mathematics Subject Classifications: 69K25, 90B15, 90B22.

Keywords and phrases: N-policy, Finite queue, Server breakdown, Generating function, Queue size, Set up time, State dependent rate.