Reliability indices of a Geo/G/1/1 Erlang loss system with active breakdowns under Bernoulli schedule

Abstract

This paper deals with a discrete-time Erlang loss system, in which breakdowns occur randomly at any instant while the server is serving the customers. As soon as breakdown occurs, the server is sent to repair directly, the customer being served before server breakdown decides, with probability 1 - q, to depart the system (impatient customer) and, with complementary probability q(0 ≤ q ≤ 1), to wait for the server to complete his remaining service (patient customer). Additionally, Bernoulli vacation schedule is introduced into this model: after completion of each repair without interrupted customer waiting there or after completion of service, the server either goes for a vacation with probability θ(0 ≤ θ ≤ 1) or may wait for serving the next customer with complementary probability 1 - θ. Firstly, we obtain the z-transforms of the probabilities of server state by using a new type discrete supplementary variable technique. Secondly, we present some performance measures of this model, such as the steady-state availability, failure frequency of the system, mean time to the first failure and the probabilities when the server is idle, busy, down or on vacation. Finally, some numerical examples show the influence of the parameters on some crucial performance characteristics of the system.