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AN ASYMPTOTIC ANALYSIS OF A RELIABILITY OF INTERNET TYPE
NETWORKS
Tsitsiashvili G.Sh., Losev A.S. •
Institute for Applied Mathematics, Far Eastern Branch of RAS 690041, Vladivostok, Radio str. 7, guram@iam.dvo.ru, alexax@bk.ru
Introduction
In this paper a problem of a construction of accuracy and asymptotic formulas for a reliability of internet type networks is solved. Analogously to [1] such network is defined as a tree where each node is connected directly with a circle scheme on a lower level with n>0 nodes. A construction of accuracy and asymptotic formulas for probabilities of an existence of working ways between each pair of nodes of the internet type network is based on a recursive definition of these networks and on asymptotic formulas for a reliability of a random port. This asymptotic formula represents the port reliability as a sum of probabilities of a work for all ways between initial and final nodes of this port. An estimate of a relative error and a complexity of these asymptotic calculations for a radial-circle scheme are shown.
1. An asymptotic formula for a reliability calculation of a port and its accuracy
\JUP , denote Pr* P(Up).
An asymptotic formula for a reliability of the general type port with low reliable arcs.
Consider the no oriented graph r with the final nodes set U, the arcs set W, the fixed initial and final nodes u, v and the set of the acyclic ways {Rj,...,Rn}between u, v. Suppose that the probability pw of the arc w eW work depends on the parameter h > 0: pw = pw (h) and pw (h) ^ 0, h ^ 0. Denote P(Up) - the probability of the event U that all arcs wf,...of the way Rp work. Then
f. n ^
the reliability of the port r is Pr = P p
v p=i ) p=1
Remark that for p ^ q the arcs sets {w e Rp}, {w e Rq} can not satisfy the inclusion
{w e Rp} c {w e Rq}. In an opposite case there is the node u* in which the ways Rp, Rq diverge by
the arcs (u*, up), (u*, uq). But as the arc (u*, up) e{w e Rq} so there is a circle in the way Rq. This
statement contradicts with a suggestion that the way Rq. is acyclic. As the inclusion
{w e Rp} c {w e Rq} is not true for p ^ q so the way Rp contains the arc w g Rq and consequently
P(UpUq) = o(P(Up)), h ^ 0, p ^ q. An induction by n gives the inequality
Pr* - ZP(UpUq) *Pr^ Pr*. (1)
1<p<q<n
But
Z P(U pUq) < n max pw (h) Pr*
weW
1<p<q<n
and consequently from the formula (1) we obtain
Pr ~ P*. (2)
Denote by A = | Pr* /Pr -1| the relative error of the asymptotic formula (2). It is obvious that
A(h) < n maxpw(h) = 0(h) ^ 0,h ^ 0. (3)
weW
Assume that (p(h) ^ 0, h ^ 0 then for the replacement of h by (p(h) the upper bound 0(h) of the relative error is to be replaced by 0(p(h)) = o(0(h)).
Radial-circle scheme. Consider the radial-circle scheme represented on the fig. 1. This scheme has the center 0 connected with the nodes 1,.. ,,n arranged on the circle.
n n — 1
Fig.1 Radial-circle scheme
Each acyclic way from the node i, 1 < i < n, on the circle (the circle node) to the center 0 of this scheme consists of a peace along the circle and a transition to the center 0. A way from the circle node i to the circle node j, 1 < i ^ j < n, has a peace from the node i along the circle, a transition to the center 0, a transition to the circle and a peace along the circle to the node j.
Define the connection matrix P =|| Pij j^-=0 of the radial-circle scheme in which Pij is the
probability that there is a working way between the nodes i, j of this scheme. Represent the results of the matrix P calculation with n=6 and
p01 = 0.0471595 p02 = 0.0469944
p03 = 0.0287418 p04 = 0.0499121
p05 = 0.0135117 p06 = 0.00822811
p12 = 0.0490761 p23 = 0.0340865
p34 = 0.0442866 p45 = 0.0004677
p56 = 0.00818179 p16 = 0.0173955
Here the matrix P is calculated by the Monte-Carlo method with 1000000 realizations during 14 hours. Denote by P* =|| P* H^-=0 the connection matrix with elements calculated by the asymptotic formula (2). The matrix P* have been calculated during one minute that is
approximately 1000 times faster. As P^ = PJiand Pii = 1 we show only the elements Pv, P* with
1 < i < j < n.
- 0.0496627
0.050371 0.0523323
0.0326335
0.00549431
0.0378183
0.0512658 0.00262581 0.00408933 0.0458115
0.0136101
0.000817883
0.000753723
0.000532299
0.00123165
0.00920024 0.0178084 0.00131303 0.00112028 0.00129662 0.00912706
P=
0,049758
0,049859 0,052157
0,03268
0,005359
0,037725
0,051263 0,002637 0,004073 0,045997
0,013703 0,000844 0,000743 0,000555 0,001253,
0,009279 0,017839 0,001327 0,001108 0,001301 0,009229
The matrix of the relative errors A =|| A ||
ij Il0<i<j<6
satisfies the equality:
- 0.00191948
0.0101643 0.00335063
0.00142467
0.0246274
0.00246654
0.0000537835 0.0042632 0.00399357 0.00404871
0.00682727
0.0319321
0.0142263
0.0426474
0.0173307
0.00856111
0.00171756
0.0106371
0.0109635
0.00337921
0.0111695
Remark. Analogously it is possible to obtain asymptotic formulas for a general type network or a radial circle scheme with high reliable arcs. But in this case it is necessary to replace a work probability by a failure probability and a way by a cross section.
2. Recursively defined networks
A calculation of the connection matrix in recursively defined networks. Suppose that D* is the set of networks r with no intersected sets of arcs. Define recursively the networks class D, D* c D by the condition
r = {UW e D , r = {U2,W2} e D*, W n W2 = m (4)
U n U2 = {z}, (z is a single node) ^ r u r2 e D.
Analogously to [2] in this paper we calculate{Pr, u,v eU, u ^ v}, not its single element. These calculations are based on the recursive formulas: if r ' e D, r' e D*, U'n U" = {z}, then
P, ,
Pr -, u, v e U',
Pr", u,veU " , (5)
; r
Pr 'Pr", u eU',veU '' .
In the last equality the quantity Pr, characterizes the connection between the nodes u, z and the quantity Pr - the connection between the nodes z, v. The number of arithmetical operations n(Pr) necessary to calculate {Pr, u,v eU, u ^ v}, by the recursive formulas (5) is characterized by the following statement.
Theorem. Suppose that Tj,..., r is the sequence of networks with the no intersected sets of arcs. If D* consists of sequences of independent probability copies of rj,..., r,, then for each re D the inequa,ities
m-a s z n(pr) s +Z Z n(Pr,) (6)
^ u,veU,u^v ^ i=1 u,veUi, u^v
are true with l (r) the number of nodes in the graph r. From the inequalities (6) obtain that
2 Z n(Pr)
i- u,veU, u ^v 1
lim -= 1.
i (H^» l (r)(l (r) -1)
So asymptotically when l (r) ^» to calculate a connection probability for a single pair of nodes it is necessary a single arithmetical operation.
Proof. Suppose that the inequality (6) is true for r' then from the recursive formulas (5) and the equality l (r' u r'') = l (r') +1 (T") -1 we have
Sn<pr,,>^Z Zn(Pr,)+l^p-l)+«E2)<l(p-l) +
u,veU 'uU', u^v i =1 u,veUi, u^v
+ (TO - 1)(l(r2) - 1) = Z Z n( Pr. ) + l (r1 ur2)(l 2r1 ur2) - 1).
i=1 u,veU:, u^v
A calculation of the transition matrices in the internet type networks. Analogously to [1] define the class of the internet type networks as the recursively defined class of networks D with
the set of originating schemes D* which consists of radial-circle schemes and in the formula (4) the
node z is the center of the radial-circle scheme r2.
2 3
Fig.2. The internet type network
So if we have the transition matrix for the radial-circle schemes it is possible to calculate the transition matrix of the internet type network by the formula (5). This algorithm is significantly faster than general type algorithm from [1]. It contains fast algorithm to calculate the transition matrix in the radial-circle scheme and practically optimal algorithm to calculate the transition matrix for the internet type networks.
REFERENCES
1. Ball M. O., Colbourn C. J., and Provan J. S. Network Reliability. In Network Models. Handbook of Operations Research and Management Science. 1995. Elsevier. Amsterdam. Vol. 7. P. 673-762.
2. Floid R.W, Steinberg L. An adaptive algorithm for spatial greyscale.// SID 75 Digest. 1975. Pp. 36-37.
