Bounding availability of repairable computer systems

Abstract

Markov models are widely used for the analysis of availability of computer/communication systems. Realistic models often involve state space cardinalities that are so large that it is impractical to generate the transition rate matrix let alone solve for availability measures. Various state space reduction methods have been developed, particularly for transient analysis. In this paper we present an approximation technique for determining steady state availability. Of particular interest is that the method also provides bounds on the error. Examples are given to illustrate the method.

Bounding steady-state availability models with group repair and phase type repair distributions

“…An approach which has been developed in the last few years is the use of bounding algorithms which require the generation of only a portion of the state space [2,3,5,6,8,9,10,12,17,18]. Those algorithms perform well when, as in the case of availability models, the probability mass is concentrated in a small portion of the state space.…”

“…Those algorithms perform well when, as in the case of availability models, the probability mass is concentrated in a small portion of the state space. The first of such algorithms was developed by Muntz et al [12] using results from Courtois and Semal [5,6] concerning bounds for conditional steady-state distributions in subsets of Markov chains. Let N be the number of components of the system.…”

“…Let N be the number of components of the system. Denoting by C k , 0 ≤ k ≤ N , the subset of states with exactly k failed components, by G the subset of generated states, and by U the subset of non-generated states, the basic algorithm proposed in [12] takes G = ∪ 0≤k≤K C k and bounds the behavior in U using a submodel with states c k , K < k ≤ N associated to the subsets C k . This basic algorithm requires the solution of |C K | linear systems of size |G| + N − K, which is typically very costly.…”

“…This basic algorithm requires the solution of |C K | linear systems of size |G| + N − K, which is typically very costly. In order to reduce the computational cost of the algorithm a state cloning technique is developed in [12] which introduces some looseness in the bounds but reduces the number of linear system to be solved to |C F |, where 0 ≤ F < K. Lui and Muntz [8] have proposed a refinement of the algorithm for the particular case F = 0 including a reuse technique which, at the price of an additional looseness in the bounds, avoids a complete reapplication of the algorithm each time K is incremented in the search for the desired accuracy. The additional looseness has been reduced in another paper from the same authors [9].…”

Towards optimal IT availability planning

“…Therefore, guaranteeing the availability (defined as: ensuring that authorised users have access to information and associated assets when required [46]) of business-supporting IT systems has become important for these organisations [60,89,109,105]. IT outages are a cause of competitive liability, chipping away at a company financial performance and reputation.…”