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5
RECURRENT SEQUENCE OF PARALLEL-SERIAL CONNECTIONS
G. Tsitsiashvili*, M. Osipova** •
* IAM FEB RAS, Vladivostok, Russia, ** FEFU, Vladivostok, Russia * guram@iam.dvo.ru, ** mao1975@list.ru
ABSTRACT
In this paper a sequence of parallel-serial connections is considered. In this sequence next connection is obtained by parallel or serial linking of new arc to obtained connection. Distributions of random numbers of connectivity components are analyzed. These distributions are considered intensively now. Central limit theorem is proved for these distributions and parameters (mean and variance) of normal limit distribution are calculated.
1. INTRODUCTION
In the reliability theory parallel-serial connections play important role [1] - [6] etc. These connections are widely used in electrotechnics, in computer networks etc. A specific of these connections is a possibility to calculate their reliability by algorithms with linear complexity by a number of arcs.
Last years large interest is called to characteristics of networks sparseness. It means that powers of nodes (a number of incident arcs) is bounded by some positive number (see [7] and large bibliography in this article). Stochastic modeling and statistical processing of internet type networks data showed that nodes powers have distribution with heavy tails [8]. Last circumstance makes actual to consider parallel-serial connections which are free of this lack.
Last time a distribution of numbers of connectivity components in different random networks are analyzed intensively now [9] - [11]. In this paper numbers of connectivity components in recurrent sequence of connections obtained by parallel or serial linking of new arc is considered. For this sequence central limit theorem is proved and parameters of limit normal distribution are calculated.
A problem to calculate a mean and mainly a variance of limit normal distribution in this model is technically sufficiently complicated. In this paper it is based on central limit theorem for discrete Markov chains [12] and on a construction of special and sufficiently fast algorithm of such calculations.
2. MODEL DESCRIPTION
Consider the sequence <An, n> 1, of ports defined recursively by a sequential or parallel connection of new arc bn to the port <An. Denote a type of connection by || or ^ accordingly. Suppose that random variable a>n characterizes a type of the arc bn connection to the port <An and put
n^ = P(u>n = P(u>n = ||) = 1 — n_>, 0 < n^ < 1.
Here random variable pn characterizes a state of the arc bn:
P((3n = 1) = P(bn in working state) = p, P(fin = 0) = 1 — p = q, 0 < p < 1.
The sequences of random variables n > 1}, n > 1} are independent and each of them consists of independent and identically distributed random variables.
The port <An with randomly working arcs is characterized by random vector (an,qn) there an is an indicator of a connectivity between initial and final nodes of parallel-sequential connection <An and is a number of connectivity components in <An. Introduce auxiliary random variables
an+1 = «n A ^ rjn+1 = + 1 - ^ (1)
then
(an+i.^n+i) = K^n =^)(an+i,*7n+i) + K^n = ||)(an+l,^n+1), (3)
where 1(C) is an indicator of an event C.
3. LIMIT THEOREM FOR MARKOV CHAIN CHARACTERIZING CONNECTIVITY OF PARALLEL-SERIAL CONNECTIONS
Denote An+1 = — then the sequence Xk = (ak, Afc), k > 1, is Markov chain with the states set X = {(i,j), i = 0,1, j = —1,0,1} as follows
(an+1,An+1) = l(un ^Xa^1 — pn) +l(un = ||)(an V£n,— pn + an0J.
From the equalities (1) - (3) and the conditions 0 < p < 1,0 < n^ < 1 we see that Markov chain Xk,k > 1, states are interconnected. Consequently from the central limit theorem for discrete Markov chains with finite states set [12, chapters V,VI} there are normally distributed random vector N(0,B) with the dimension six and with zero mean and with covariance matrix B and real numbers A(x), x £ X, which do not depend on initial state Xx so that for any real t(x), x£l,
P £ X) > (t(x), x £ X)) ^ P(N(0,B) > (t(x), x £ X)), n ^ (4)
Here Nn(x) = I(Xk = x) and the inequalities are defined componentwise. Introduce auxiliary numbers a(x), x £ X:
a(i, 0) = 0, a(i, 1) = 1, a(i, —1) =—1, ¿ = 0,1.
From the formula (4) it is simple to obtain that there is normally distributed random variable N(0, S) with zero mean and with the covariance B > 0 so that for any real t
P a(x)(Nn(x) — nA(x)) > t) ^ P(W(0,B) > t), n ^ (5)
Using obvious equality £x£x a(x)Nn(x) = Afc = n > 1, rewrite the formula (5) as follows
P > t) ^ P(N(0, B)>t),n^™,A = !xeX a(x)A(x). (6)
Remark 1. A calculation of the vector (A(x), x G X)) and especially of covariance matrix B in the formula (4) is sufficiently complicated procedure [12, chapters V, VI}. So to define the mean A and the covariance B we use following limit formulas
A = lim B = lim ^ (7)
which are corollaries of the formula (6) with special initial distribution of Xx.
4. CALCULATION OF LIMIT NORMAL DISTRIBUTION PARAMETERS
Choose random vector (at, Ax) = (at, which does not depend on random sequences n > 1}, n > 1} and satisfies the equalities
P((a i,%) = (1,1)) = P = P((«i,%) = (0,2)) = Q = 1 - P (8)
with P(an = 1) = P, P(an = 0) = Q. Random sequence an, n > 1, is stationary Markov chain. Theorem 1. The equalities
A = Qn^q, (9)
B = n^qQ(1-n^qQ + 2PQ) > 0 (10)
are true.
Proof. To define the constants A, B from (7) we construct recurrent algorithm. Denote
Mn = MVn, An = M(nn \an = 1), Bn = M(nn \an = 0), Mn = + BnQ, (11)
M; = M^, ^ = M(ti\an = 1), s; = M(V*\an = 0), M; = ^p + B^Q (12)
where
^ = 1, Si = 2, = 1, s; = 4. Using the formulas (1) - (3), (11) obtain for n > 1:
_ AnPn^p + 4wPrcMp + (Bn - 1)QrcMp +
^n+l = p ,
_ BnQn^p + (4W + 1)Pn^q + (gw + +
= Q ,
Mn+1 = ^ + - + Pn^q + Qn^q = Mn + Qn^q = Mt + nQn^q, Mt = 1 + Q.
Then from (7) we obtain the equality (9). And
^n+i - = - Bj! - (2n^q + rcNp), n > 1, A = + rc^p < 1.
(13)
so
1 _
+ (2n^q + ^\\p) 1 - A
= - 1 - Q, An+1P + Bn+1Q = Mn+1
consequently
An+i = Mn+1 + Q[AnQ -1-Q], Bn+1 = Mn+1 - P[AnQ -1-Q],n>1. (14) Begin now a calculation of M'n+1. Using the formulas (1) - (3), (12) obtain for n > 1:
_ A'nPn^p + A'nPnnp + (B!n - 2Bn + 1)Qnnp + A'nPn\\q
An+1 = p ,
B!nQn^p + (A'n + 2An + 1)Pn\q + (B!n + 2Bn + 1)Qn^q + B^Qnnq
K+i =M'n + 2AnQn{lp + 2BnQ(n^q - n^p) + n^q(1 + P). So from (14) we obtain
n-l n-l
K+i =M[ + 2QnllP ^ Ak+1 + 2Q(n^q - nl]P) ^ Bk+1 + nn^q(1 + P) =
k=0 k=0
n-l
Z, 1-An
Mk+1 - 2nQP(1 + Q)n]]P + nn^q(1 + P) + 2n^qP2Q-^—j =
k=0
= M[ + 2Qn^q(n(1 + Q)+ n^qQn(n - 1)/2) - 2nQP(1 + Q)n]]P + +nn^q(1 +P) + 2P2Q2(1-An), M[ = 1 + 3Q.
Consequently
DVn+i = - = nn^q[1 + P- Q2n^q - 2P2(1 + 0)] + 2P2Q2(1 - An) + QP.
Then from (7), (13) we have
B = + P- Q2n^q - 2P2(1 + Q)) = n^qQ(1 - n^qQ + 2PQ) > 0.
Theorem is proved.
Remark 2. From Remark 1 is possible to replace the condition (8) by more natural suggestion
P{(^i,Vi) = (1,1)) = P, i,Vi) = (0,2)) = q
so that the equalities (6), (9), (10) are true also.
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