Inaccuracy of Availability Metrics Estimated by the Serial-Parallel Model János Szigeti, Tibor Cinkler High-Speed Networks Laboratory Department of Telecommunications and Media Informatics Budapest University of Technology and Economics Magyar tudósok krt 2, H-1117 Budapest, Hungary Email: {szigeti,cinkler}@tmit.bme.hu Abstract—Provisioning QoS network connections with the desired availability relies on estimating the availability of the connections in advance. The redundancy of the protection – which denotes how many failures can be survived – gives a good hint on the availability, thus lower bound estimation can be carried out. If we want to achieve a more accurate approximation, we have to use other methods. The Serial-Parallel availability modeling and calculation method, based on the availability metrics of the components of the connection, offers a fast estimation, with the complexity of O(n) in case of n components. However, the result of this estimation can be inaccurate since the model does not take into account the overlapping of components, i.e., when a component is member of more different series. In this paper we analyze the inaccuracy of the Serial-Parallel method. We prove that the estimated availability is always less than the exact one, define an upper bound onto the inaccuracy of the estimated unavailability and show where does this inac- curacy converge by increasing the availability of the network components. Index Terms—availability, conditional probability, protection. I. I NTRODUCTION Measuring and estimating the availability of network de- vices and connections is important since the provisioned network connections have to fulfill predefined availability requirements. The availability of a network device can be improved by extending its mean time to failure (MTTF) or by shortening its mean time to repair (MTTR) attribute. Assuming these attributes are predefined, generally, high con- nection availability can be achieved by setting up redundant backup paths along the working path, which are protecting parts (links or segments) of it or the whole path (end-to-end protection). However, the more complex the protection is, the more resources the connection requires [1], [2], and the more difficult is to derive its availability. The availability of connection using dedicated protection can be estimated accurately, since the backup paths do not interfere. However, if the backup resources are shared, the The work described in this paper was carried out with the support of the BONE-project (“Building the Future Optical Network in Europe”), a Network of Excellence funded by the European Commission through the 7th ICT- Framework Programme and the CELTIC TIGER II project. availability estimation may become a complex problem. Ex- haustive availability evaluation has to enumerate each possible network failure state variation, which can be carried out only for small systems. With stratified sampling we can eliminate the state space while achieving still a good approximation [4]. In [5], [6], [7] routing algorithms for shared protection with guaranteed availability are proposed without evaluating the exact connection availability. The p-cycle protection scheme [9] provides a special way resource sharing, and in [8] we already have presented a fast evaluation method of the accurate availability for p-cycle- protected connections. Among the numerical results of [8] we compared the accurate availability to the availability estimated with the well-known Serial-Parallel (S-P) method 1 [10] and the experienced behaviour of the inaccuracy lead us to study the S-P heuristic more deeply in link-protected connections. The rest of the paper is organized as follows. In the next section we define the scope of the work and introduce the notation for our availability model. In Sect. III we examine S-P inaccuracy with single link overlaps, next, in Sect. IV we extend these single overlaps to multiple link overlaps and deduce lower, upper bounds and limit value of the error of S-P method. Illustrative examples are studied in Sect. V, and finally, Sect. VI summarizes the results of the work. II. NOTATION AND SCOPE OF THE WORK The network consists of atomic components which may fail. The whole set of these components is denoted by E, while for denoting a single component we use e (e ∈ E) which is used usually with an index or other special markings (e.g., e i , e ∗ , etc.) In our availability model we assume that each network component e has two states (S(e)): it may be either operational (up, S(e)=1) or in failure (down, S(e)=0) state, and the state of each component is independent from the state of any other component (∀e ′ ,e ′′ ∈ E,e ′ = e ′′ : S(e ′ )| S(e ′′ ) = S(e ′ )). The availability of a component is a 1 The principle behind the Serial-Parallel method is that if dual-state (up/down) components are connected serially, the series is up only if all the components are up; whereas a block of parallelly switched components requires only a single component to be in up state. 978-1-4244-7283-3/10/$26.00 c  2010 IEEE