Unreliable Server Markovian Queue with Heterogeneous Arrival Under N-Policy

Abstract

Model Description and Notations Present investigation deals with optimal management policy for Markovian queue with server breakdowns and vacations in which arrival occurs according to the state of the server. The customers arrive in Poisson fashion to get the service. The server may breakdown N λ(i=0,1,2,3) μ CC B Po(n) P1(n) P2(n) during the service and goes for repair immediately. By applying probability generating function technique queue length distribution is obtain for different states of the server. Further we determine the probability of empty system, expected number of units in the system. Following notations and probabilities are used through out the paper for formulating the model mathematically: Threshold level of queue length when server turns on Arrival rate of customers in various status Mean service rate of the server Mean break down rate of the server Mean repair rate of the server The probability of being n customers in the system and server is on vacation. The probability of being n customers in system when server is working The probability of being n customers in the system when server is found to be broken down. P1(n) The probability of being n customers in the system when server is under repair. 2. Governing Equations Steady state equations governing the model are given as follows: λo Po(0) = μPo (1) ZoPo(n) = 2o Po(n−1), 1 ≤ n ≤ N -1 (1) (2) λo Po(n) = λo Po (n-1), n> N (3) (λ1 + μ ÷ α)P1 (1) = μP (2) + BP3 (1) (4) (î, +u+α)P} + α)P(n) = ̧uP; (n + 1) + BP3 (n) + 2, P;