Indexes of a telecommunication network - Reliability ... - IEEE Xplore

57

IEEE TRANSACTIONS ON RELIABILITY, VOL. 37, NO. 1,1988 APRIL

Performance Indexes of a Telecommunication Network Ali M. Rushdi, Senior Member IEEE King Abdul Aziz University, Jeddah

.

This paper interprets the two new performance indexes in terms of certain mean capacities in the network. The Aggarwal index [lo] is the mean normalized s-t capacity of the network, while the Trstensk9-Bowron index [ll] is simply Key Words - Positive and complementary indexes, Mean an overall counterpart, viz, it is the mean normalized overall normalized capacity, Pseudo-switching function, Map method, capacity of the network. Both are positive (direct) indexes, but other complementary (inverse) indexes can be defined. Reduction rule, Generalized Max-Flow Min-Cut Theorem The algebraic properties of the capacity function are inRcodrr Aids vestigated, and it is observed that once the capacity function Purpose: Tutorial is put in a sum-of-products (s-o-p) form, it becomes readily Special math needed for explanations: Probability convertible to its mean value. Based on this observation, Special math needed to use results: Same some of the manual techniques used in conventional Results useful to: Reliability and telecommunication analysts reliability analysis are adapted here for the computation of the new performance indexes. This approach stresses both Abstract - Two recently proposed performance indexes for the similarities and differences between the two applicatelecommunication networks are shown to be the s-t and overall tions. The first of these techniques is a map procedure that versions of the same measure, vlz, the mean normalized network results in simple symbolic expressions for the performance capacity. The network capacity is a pseudo-switching function of indexes. However, this technique can only be applied the branch successes, and hence its mean value is readily obmanually to small or moderate networks. A second techtainable from its sum-of-products expression. Three manual nique applies network transformations or reduction rules in techniques of conventional reliability analysis are adapted for the computation of the new performance indexes, viz, a map pro- such a way that the network capacity function is preserved. cedure, reduction rules, and a generalized cutset procedure. Four These rules are straightforward to apply in the case of seriestutorial examples illustrate these techniques and demonstrate parallel subnetworks. The case of bridging branches is, their computational advantages over the stateenumeration however, more difficult and is handled by a function expansion or network decomposition. A third technique technique. generalizes the “Max-Flow Min-Cut Theorem” [6-91 for network states X other than the ideal state (X = 1). This technique is very fast when the network minimal cutsets I. INTRODUCTION [12], and possibly its minimal paths [12] are known. None A telecommunication network is usually modeled by a of the aforementioned techniques has been computerized. stochastic graph G = ( V , E ) (where Vand E are the sets of Nevertheless, they offer computational advantages for vertices (nodes) and edges (branches) of G) on which a set manual use, and the concepts of the last two techniques lay K E V is distinguished [l]. A standard index of network the foundations for the development of computerized performance is that of network reliability which is simply a algorithms that can handle complex networks. measure of probabilistic connectivity since it equals the probability that certain connections (directed or un11. ASSUMPTIONS, NOTATION, directed) exist in G among the nodes in K [l-51. Two im& NOMENCLATURE portant special cases are those of: i) source-to-terminal (s-t) reliability for which IK( = 2 and one of the nodes in K is designated as a source, and 2) overall reliability (all- Assumptions terminal reliability) for which K = V. A second standard 1 . The telecommunication network is modeled by a index of network performance is that of network s-t linear graph whose nodes (vertices) are perfectly reliable capacity which equals the maximum flow that can be and of unlimited capacities. passed from a source node t o a terminal node so that no 2. Each network branch (edge or link) can have only branch capacity is violated, and under the assumption that two states, good or failed. The branch failures are all branches are working [6-91. Traditionally, each of the statistically independent. performance indexes of reliability and capacity is used in3. Each network branch is assigned specific values for dependently of the other one; whenever one is considered its reliability and capacity. The branch capacity is the upthe other is disregarded. Recently, Aggarwal [lo] and per bound on the branch flow in either direction. Trstensk9 & Bowron [l 11 have proposed two new performance indexes for telecommunication networks that in- Notation tegrate the reliability and capacity criteria. These two performance indexes have been computed through computa- n number of branches (links) in the logic diagram of tionally uneconomical stateenumeration procedures. the network.

0018-9529/88/O4OO-OO57$01.0001988 IEEE


indicator variables for successful and unsuccessful operation of branch i. These are switching random variables that take only values belonging to the bivalent discrete domain Bz = (0,l};Xi = 1 and X i = Oifiisgood,andXi = OandXi = I i f i i s failed. indicator variables for successful and unsuccessful operation of the network. Successful operation can be equivalent to connectivity, or to the satisfaction of a certain flow requirement [lo, 12, 131. reliability and unreliability of branch i: pi = Pr{Xi = l}; q i = Pr{Xi = l } = 1 -pi. Bothpi and qitake real values on the closed interval [0, 11. network reliability and unreliability; R = Pr{S = l} =E{S},F= 1 - R , O < R , F d l . capacity or number of channels of branch i; ci 0. n-dimensional vectors of branch successes, reliabilities and capacities: X = ( X J Z ... X , f ; p = @1pz ... pn)T; c = (CICZ ... C?JT. a superscript, implies transpose. state k of the network, denoted by a particular value of the n-dimensional vector X, k = 0, 1 , 2, ..., 2" - 1. capacity (effectiveness) of interconnection from node i to node j in state X. This is the maximum flow from i to j that does not violate branch capacities; Cij(X)2 0. Since X is a binary random vector, C,(X) is a discrete random variable of a probability mass function of no more than 2" distinct values. maximum capacity (effectiveness) of interconnection from node i to node j: Cum- = Cij(l) = capacity of interconnection from i to j in the ideal case when all branches are good. This maximum capacity can sometimes be achieved at certain states xk other than 1. source, terminal node. positive and complementary performance indexes of the network; L = 1 - M, 0 d L,M < 1 . The subscripts st and o may be added to either M o r L. In the high performance limit (M 1, L 0), L is more useful than M when expressed in the same floating-point number system. normalized s-t capacity (effectiveness) of the network in state X; Wst(X) = Cst(X)/Cst(l). normalized overall capacity (effectiveness) of the network in state X; *(X) = aij Wij(X); aij =

- -

Cij(l)/

c Cij(1).

assignment of the two functional values (0 or 1) for all possible 2"values of X [14, p 581. Pseudo-switching (-Boolean) function C(X): A mapping R where R is the field of real numbers, ie B; C(X)is an assignment of a real number for each of the possible 2" values of X [15, p 211. multiaffine function: A function of several variables which is a first-degree polynomial in each of its variables. Examples of multiaffine functions include: 1. Certain algebraic functions such as system reliability/unreliability as a function of component reliability/ unreliability [ 12, 161, system availability/unavailability as a function of component availability/unavailability [16, 171, and the mean capacity as a function of branch reliabilities. 2. Pseudo-switching functions [15, pp 21-22] such as network s-t or overall capacity as a function of branch successes.

-

111. PERFORMANCE INDEXES

Aggarwal [lo] has proposed an s-t performance index which can be expressed in the present notation as:

The sum in (1) is taken over values of k that correspond to success states. It can as well be taken over all values of k since Ws&) = 0 for failure states. Then the states {X = xk}are exhaustive and disjoint, and hence (1) can be rewritten in the form:

which means that s-t performance is measured by the mean normalized s-t capacity, ie, by the mean value of the s-t capacity normalized by its maximum. On the other hand, Trstensky & Bowron [l 11 have introduced an overall performance index, which can be expressed in the present notation as:

i*j

=

c Cij(X)/ c Cij(1).

i#j

i#j

CiJ@1 13, CiJ@IO,) the function Cij(X)when X,is set to 1 or 0. Meanings of Cij(X I l,, l,,,), ..., etc. follow

similarly.

(4)

Substituting (4) into (3), then interchanging the orders of the summation and expectation operators results in the following alternative expression for M,

Nomenclature

-

Bz Switching (Boolean) function S(X): A mapping B; where Bz = (0,l}, ie, S(X) is any one particular

+j

=

C aijMij. i#j

~

59

RUSHDI: PERFORMANCE INDEXES OF A TELECOMMUNICATIONNETWORK

Equation ( 5 ) means that overall performance is measured by the mean normalized overall capacity, ie, it is measured by the sum of the mean values of s-t capacities when summed over all node pairs divided by the corresponding summation of their maximum values. As a result, the overall performance index Mo is simply a particular weighted average of the s-t performance indexes Mij. Both the indexes M,, and Mo are positive or direct indexes; the higher M , the better the performance. The definition of M i s analogous to that of traditional direct indexes such as reliability and availability. Complementary indexes L whose definition parallels that of unreliability or unavailability can also be defined, such that if L is lower, the performance is better, namely:

Lst = 1 - Mst = E{ 1 - Wst(X>},

(6)

Lo = 1 - Mo = E{l - *(X)}.

(7)

Trstensk9 & Bowron [l 11 have characterised the scatter of q(X) around Mo by the variance:

can be &rived from it directly by replacing the arguments X , and X , by their means p r and qr respectively, viz,

-

Cij(X)(s-o-p) {*, xr} 9

.-.{pr2 qr} +E{C,} (p) (s-o-p).

(1 1) Eq. (1 1) results immediately from the fact that the mean of a sum is the sum of means, and the assumption that the X;s are statistically independent. Not only the capacity Cij(X) but also the capacity squared C$(X)is a pseudo-switching function. Therefore, C$(X)can also be put in s-o-p form, so that it becomes readily convertible into its mean:

C$(X)(s-0-p)

~

{*r

3

zr}

{Poqr), E{ C:.} @) (s-0-p)

(12) On account of (11) and (12), the problem of finding the mean of the capacity and its variance reduces to that of expressing the capacity itself and its square in s-o-p forms.

V. A MAP PROCEDURE var{*(X)} =

ck(*(Xk)

- Mo)’ Pr{X

= xk}

= E{P(X)} - Mb.

(8)

Eq. (8) is a correction of (4) in [ll]. The scatter of (1 q(X))around Lo is characterised by Var{ 1 - q(X)}which around is given also by (8). Similarly, the scatter of W,,(X) M,, is:

vu{wst(X)}= c k ( W s t ( X k ) = E{ W%W}

- Mst)’ Pr{X

=

x k }

- M;t,

(9)

while the scatter of (1 - W,,(X)) around L,, is characterised by Var{ 1 - Wst(X)} = Var{ Wst(X)}. IV. CAPACITY AND ITS MEAN The source-to-terminal capacity as a function of component successes Cij(X)is a real-valued function of binary arguments, and hence it is a pseudo-switching function that obeys the algebraic decomposition formula: Cij(X) = Cij(X 103 + [Cij(X 113

- Ci,QlIOl)]X,,

P = 1,2, ...,n.

(10)

Eq. (10) can be easily proved by perfect induction over all possible values of X, namely, { X IOr} and {X 1 If}. It means that Cij(X)is a multiaffine function. Hence C,@) can always be written in a sum-of-products (s-o-p) form. Furthermore, Cij(X)is completely specified by the 2” coefcorresponding to the 2“ values X k that its ficients Ci&) argument X takes. Consequently, C,@) can be conveniently expressed in the form of a truth table or a Karnaugh map of real entries. If the random function Cij(X)is written in s-o-p form, then its mean value

The pseudo-switching function Cij(X) can be specified by a modified Karnaugh map. The map variables are the elements of X and the map entries are the real numbers c,&) which represent the s-t capacity for states x k , and hence are not necessarily 1’s and 0’s. These numbers can be obtained individually or collectively via any of the procedures in sections VI and VII. To express Cij(X)in an almost minimal s-o-p form, it is necessary to cover the nonzero entries of the map by the smallest possible number of map loops. Each of these loops should be the largest that combines 2’ {i = 0, 1, 2, ..., n} adjacent cells of the map containing as a minimum a certain (so far uncovered) value. The contribution of such a loop to the s-o-p expression of C,(X) equals its covered value multiplied by the usual loop term. To allow for the choice of larger loops, a cell entry may be partitioned into several values to be covered by several loops. Such a partition is usually possible for integer-valued entries in maps describing small-size networks. Once a portion of an entry is covered, that entry is replaced by its uncovered portion. In particular, if an entry is totally covered, then it is replaced by a zero. The procedure terminates when all entries in the map become 0’s. The above map procedure results in capacity expressions that are simpler than those obtained by the direct stateenumeration method in [lo]. It is particularly useful when the map entries belong to a small set of integral values, which is usually the case when the branch capacities are integer valued. Though the map procedure suffers the limitation that it is capable of handling only small networks (of six branches or less), it can be extended to handle moderate networks through the use of variable-entered Karnaugh maps (VEKMs) [181.

Example I This example applies the map procedure to the network discussed in [lo]. This network is shown in Figure 1 and its branch capacities are:


60

c = [lo 4 5 3 4IT. \x3xIx6

-x,

--

-xj

"4

"1

Fig. 3. A 2-Step Map Procedure to Cover the Pseudo-Switching Function c14P).

The pseudo-switching function c14@) can be converted into the switching function of success &4@) by suppressing all non-unity numerals and replacing the arithmetic operators { +, *} by their logic counterparts, viz,

"3 Fig. 1. A 5-Branch Bridge Network.

s14@) = X5(X2

u X12&3)

u X4(Xl u x l X & 3 5 > .

(15)

If all branches have the same reliabilityp and unreliability

I

i'

t-x

4

4

q, then

E{C14}@) = 4P2(1 + qp)

+ 3p2(1 + q2p) =p2(7 + qp(4 + 3q)). Forp = 0.9 and q = 0.1

Fig. 2. Modified Karnaugh Map Representing the PseudoSwitching Function CI4(X).

The numerical value of M14 agrees exactly with the corresponding value in [lo], while (14) and (16) are much simpler than their equivalents [lo; (6) & (7)l. Figure 2 shows the modified Karnaugh map representing A minimal s-o-p expression for