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Piotr Chołda, Andrzej Jajszczyk To show how reliability measures are calculated and how to compare different recovery procedures, we chose only a few typical protection schemes. We were also paying attention to simplicity and transparency of the relevant formulas. We can note that there are also other reliability measures, e.g., the reliability polynomial for the whole network, down time ratio or average traffic loss. However, the availability is the most common measure and can be used both in very simple as well as very sophisticated (e.g., the most recent [2]) analyses. It is very important to note that many different types of availability are defined. Formally, the so called instantaneous availability is defined in terms of the ITU-T Recommendation [3] as the probability that an item is in an up state at a given instant of time. However, for our calculations we will use the concept of asymptotic (steady-state) availability, which means the limit, if such exists, of the instantaneous availability when the time tends to infinity. During calculations, it is assumed that the failures of the elements are statistically-independent. The availability defined in such a way could be understood as an average ratio of time in which an item or a connection works properly, to the whole time. 2. AVAILABILITY OF THE CONNECTION WITHOUT PROTECTION An unprotected connection consists only of a working path. The recovery (alternative) path is not used. The connection is interrupted when any of its elements fails. Thus, its availability equals simply an availability of a path on which this connection is established. As a path we consider a set of #L links and #N nodes through which a signal from an input node (source) to an output node (sink) is transmitted. From the reliability standpoint, this is a serial structure, i.e., the structure remaining in the operation state only if all of its elements are in the up state. Therefore, the availability of a path ( A ) is determined as [4]: A = # L ∏ ALi N j A P × # N ∏ A P Li N j i= 1 j= 1 where is the availability of link i; A is the availability of node j. 3. DEDICATED PROTECTION The simplest protection scheme is called dedicated protection or 1+1 protection. From the reliability point of view it is a parallel-serial structure with hot standby, because traffic is carried both through the working and the protection paths all the time. The parallel structure means that only if all N paths fail, the connection is down. Availability for the parallel structure is calculated according to the following formula [5]: (1)
Availability Assessment of Resilient Networks N ∏ i= 1 A(parallel item1, …, itemN) = 1 − P (itemi = unav) = 1 − ( 1 − A ) (2) where P(event) is the probability of an event; ‘unav’ means that an item is unavailable; Ai is the availability of item i. Therefore, the availability of the whole connection using the 1+1 protection ( A1+ 1 ) will be calculated as follows (see Fig. 1): ASo ASp ( ( AP )( AP ) ASw A 1 2 Si 1 1 1 A1+ 1 = − − − (3) where , A are the availabilities of the source and sink node, respectively; ASo Si APi P2 is the availability of the first path, and of the second path; , A are the availabilities of a splitter and switch, respectively. N ∏ i= 1 A ASp Sw Using the example of a Unidirectional Path-Switched Ring (UPSR), a particular case of the 1+1 protection, we present how practically calculate the availability. The availability of connections established in some of ring topologies is relatively simple to calculate; some effort has already been made [6]. In UPSR each pair of nodes is connected by two separate paths. Traffic from the source node is transmitted simultaneously in both directions: clockwise on the working path and counterclockwise on the protection path. The working path is located on a fiber which is different from the fiber carrying the protection path. In the case of any working path element failure, the sink node starts to receive the signal from the protection path [7]. Figure 1. Dedicated protection: reliability block diagram If we assume that availabilities of all nodes are equal and each span is of an identical length (which means that also availabilities of all spans are equal) the availability of a connection between two nodes in UPSR ( A ) is given by the UPSR following formula (cf. the reliability block diagram shown in Fig. 2): A A 1− 1− A 1− A A ( ( )( ) ) = UPSR = N WP PP N k k −1 n n−1 ⎛ ⎛ ⎞⎛ ⎞⎞ = ⎜ − ⎜1− ∏ A S × ∏ AN ⎟⎜1− ∏ A × ∏ ⎟ ⎜ S ⎟ ⎟ = ⎝ ⎝ i= 1 i= 1 ⎠⎝ i= k + 1 i= k + 1 ⎠⎠ 2 A N 1 AN 2 = A k k −1 1− 1− A A n−k n−k −1 1− A A n−k n−k + 1 k k + 1 = A A + A A − A N n n ( ( )( ) A S N S N S N S N S i N (4)
Page 1: 12th GI/ITG CONFERENCE ON MEASURING
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