This paper describes an unreliable server batch arrival retrial queue with two types ofrepair and second optional service. The server provides preliminary first essentialservice (FES) to the primary arriving customers or customers from retrial group. Onsuccessful completion of FES, the customer may opt for second optional service (SOS) withprobability α. The server is subject to active break downs. The customerunder FES (or SOS) during the failure decides, with probability q, tojoin the orbit(impatientcustomer) and, with complementary probabilityp, to remain in the server for repair in order to conclude hisremaining service (patientcustomer). Both service and repair times areassumed to have general distribution. It is considered that the repair time of serverduring the presence of patient customer and the repair time of the server while thecustomer (impatientcustomer) joining the orbit due to failure, aredifferent. For this queueing system, the orbit and system size distributions are obtained.Reliability of the proposed model is analysed. Some particular cases are also discussed.Other performance measures are also obtained. The effects of several parameters on thesystem are analysed numerically.

Web servers have to be protected against overload since overload can lead to a server breakdown, which in turn causes high response times and low throughput. In this paper, a stochastic model for breakdowns of server systems due to overload is proposed and an admission control policy which protects Web servers by controlling the amount and rate of work entering the system is studied. Requests from the clients arrive at the server following a nonhomogeneous Poisson process and each requested job takes a random time to be completed. It is assumed that the breakdown rate of the server depends on the number of jobs which are currently being performed by the server. Based on the proposed model, the reliability function and the breakdown rate function of the server system are derived. Furthermore, the long-run expected number of jobs completed per unit time is derived as the efficiency measure, and the optimal admission control policy which maximizes the efficiency will be discussed.

It is shown that random variables X exist, not exponentially or geometrically distributed, such that

P{X – b ≧ x | X ≧ b} = P{X ≧ x}

for all x > 0 and infinitely many different values of b. A class of distributions having the given property is exhibited. We call them ALM distributions, since they almost have the lack-of-memory property. For a given subclass of these distributions some phenomena relating to service by an unreliable server are discussed.