CC BY
14
U- FUNCTION IN APPLICATIONS
Igor Ushakov
Sun Diego, California e-mail: [email protected]
The Method of Universal Generating Functions (U-functions) was introduced in [Ushakov, 1986]. Since then the method has been developed, in first order, by my friends and colleagues -Gregory Levitin and Anatoly Lisnianski. They actively and successfully apply the method of U-function to optimal resources allocation, to multi-state system analysis and other problems. Frankly, now I feel like a hen sat on duck eggs and then wanders how hatched chicks fearlessly swim so far from shore. ©
I decided to remind you a Russian folk proverb: "new is well forgotten old". What is U-function? It is, first of all, generalization of a classical Generation Function (GF) permitting perform more general transforms. From technical side, this method represents a modification of the Kettelle's Algorithm conveniently arranged for calculations with the use of computer.
U-FUNCTION
One can represent any discrete distribution of random variable (r.v.) xk as a set of pairs:
Si ={( xf, pf), (xf, pk2)),..} (1)
This distribution can be represented in the form of polynomial (common generating function)
< (-) = I Pi1) ^) . (2)
1S ]<nk
where nk, in principle, can be infinite.
The polynomial representation allows one to obtain the distribution of sums of random variables, using the convolution procedure.
If instead of the distribution of x(+x2 one wish to obtain the distribution of an arbitrary function fx(, x2), for any combination of realizations x1 = X((1) and x2 = X2(), the realization of the function fx(, x2) takes the value of f (X1(1), X2(1)) with probability p1(1) • p21). Having the discrete distributions of two random variables x1+x2 in the form S1 ={(X1(1),p1(1)),...,(X1(n1),p(1"1))} and
s 2 = {( x 21
, p21)),..., (X 2n), p(2"2))} one can obtain the discrete distribution S of the function
f(x1, x2) as a "Descartes composition" of two sets S1 and S2 (here we use the term "interaction" because the operation reminds Descartes product but does not coincide with it) .
Sf = {(f(X(1), Xf); p® • pf),...,(f (X1(n1), X2n2)); p® • p2"2)),
(f (X(2), Xf); p(2) • pf),...,(f (X((2), X2n2)); p(2) • p2n2)),
... (3)
(f (X((n(),X2()); p(n1) • pf ),...,(f (X((n(),X2n2)); p1(^) • p$)
This distribution can be conveniently obtained using the "composition procedure" over functions <p((z) and <(z) :
f \ f ( 7 ) - — ,(<■ )„ *2' )
f
(z)= ® (<pi(z) ,((z)) = Z pi(l) zXl ® Z p2) z
v1-i<n( y
f
V 1<Z<n2 y
= Z ZPi( 1) ■ p2° zf(X1 ■X2(4)
1< 1<n2 1<Z<n2
The composition operator ® possesses commutative property, i.e.
f
® (((z), ((z) )= ® (((z), ((z) ) (5)
ff
and associative property, i.e.
® ((l(z) ® ((2(z) ),(3(z)) ) == ® ((((z), ® (((l(z), ((3(z))) ® ( ((z), ® ((l(z),(2(z) )). f f f f f f
(6)
if the function f possesses these properties. In the most applications this is the case, though some exceptions exist (see, for example, [ Levitin, 2005]).
USING U-FUNCTION FOR SOLVING THE OPTIMAL REDUNDANCY PROBLEMS
Let us consider a series system consisting of n subsystems. Each of subsystem k contains at least one unit with probability of failure-free operation (PFFO) pk and cost ck . For increasing the system reliability, one can introduce redundancy in each subsystem. Denote the number of redundant units of subsystem k as xk. All the possible PFFOs and costs of any subsystem k for different levels of redundancy can be represented by the set of triplets
Sk = {(Pk (Xk), Ck (Xk), Xk ),1 < Xk <»} where Ck (xk) is the total cost of xk redundant units (usually, a linear function); and Pk (xk) PFFO or
availability coefficient) of the subsystem when it contains xk redundant units. It is well-known that for loaded redundancy (the so-called "hot standby") of group including one main and xk identical redundant units takes the form
Pk (xk) = 1 - (1 - Pk)xk+1; and for an unloaded redundant (the so-called "cold standby") units takes the form
Pk (xk) = Z j- exp(-V);
0<j<xk J-
The set of triplets Sk can be represented in the GF form as
( (z) = Z Pk () zC") ^ . (7)
1<Xk
Now consider a general procedure of optimal redundancy with the use of U-function. First take units 1 and 2 and arrange the Descartes interaction procedure between sets S1 and S2. To distinguish interaction procedure from common product of two generating function, let us introduce symbol ®.
f ^ y) = ®fa(z,y), %(z,y)) = ® ZP(X()zCl(ll)y!ll!, ZP2(x2)zC2(l2)yix2i
-, U
^ l<X2
®(C1(X1),C2(X2) ) ®(*1,*2 )
= Z P(Xl) • P2(x2) z ^—y ^ = Z P,( Xi) . P2(x2) Z^™ y{X-2! (8)
l<X(<œ 1<Xi <<»
1<X2 l<X2
= Z Pu(Xu)zCi,2(Xl,2)yX1,2
l<X, 2 <«>
i, 2
From (8) clear that composition operator + means summation corresponding components and
operator ® means a collection of corresponding components.
Thus the obtained U-function (,2(z, y) represents the set of related PFFO, costs and numbers of redundant units for different configurations of the series connection of subsystems 1 and 2. The group of subsystems 1 and 2 now can be treated as an equivalent "aggregated" subsystem. Notice that solving optimal redundancy problem, one has to make some kind of "sifting" of function (,2(z, y). One has to order all terms of the final expression in (8) by increasing values of C1,2 and exclude all terms of (,2(z, y) that have value of P1,2 equal or smaller than previous terms. (If two terms have the same values C12 and P1,2 one leaves an arbitrary single of them.). In the remaining terms, all X1,2 are numbered by natural numbers in accordance with their order by increasing cost. This procedure is equivalent to deleting dominated terms
At the next step one obtains the UGF for the group of three subsystems 1, 2 and 3 applying the same convolution operator over U-function (,2(z) representing the aggregated subsystem and ((z) representing the subsystem 3:
f \ (1,2,3 (z, y) = ®(u( z, y), (3(z, y))=® Z ^1,2(^1,2) * zCl,2№,2) y №,2\ Z P.( X3) zCs(l3) y{
+ ,U
l< X3 j
= Z P,,2(XU) • P,( X3) Z?(Ci,2(Xl2),C3(X3))yU(Xl,2,X3 )= Z PU(XU) • P,( X3) Z^^W y^^
,2 (X 1,2) • P 3(x3," ' "'U
l< Xi 2<œ 1< Xi 2<œ
l < X3 l < X3
= Z Pl,2,3(Xi,2,3)zCi,2,3(Xi,2,3)y
l< Xi 2 3 <œ
(9)
This procedure continues until necessary final UGF representing the entire system is generated. Instead of further abstract presentation of the procedure, let us turn to a simple illustrative numerical example.
Consider a series system of two units. Let unit-l and unit-2 are characterized by strings
Si = {M i(l), m 2l),..., M«!
and
s 2 = {M 12), m 22),..., m^!
Each multiplet M is a set of parameters Mj) = {a[kj \...,a(N)! where N is the number of
parameters in each multiplet.
"Interaction" of these two strings is an analogue of the Cartezian product whose memberts fill the cells of the following table:
, l< X( 2 <w
Ushakov I. - U-FUNKTION IN APPLICATIONS , „ 0RT&A # 03 (26)
(Vol.7) 2012, September
Table 1.
M ((() M 2() M«
M 12) M 1(1) ® M12) M2() ® M((2) M^ ® M((2)
Ml 22) m 1(1) ® m 22) M2() ® m 22) M^ ® m 22)
• • • • • •
M^ M((() ® M^ m2() ® m\(22) M^ ® M^
Interaction of multiplets consists of iteractions of their similar parameters, for instance,
Mf) ®M(h) = {«} Ca^),«) Ca^),...,«) ®«Nh})}
f1 fl fN
Operator ®, as well as each operator ® , in most natural practical cases possesses the
fs
commutativity property, i.e.
and the associativity property, i.e.
(3)
® ( a , b )= ® (b , a ) , (4)
f f
® (a, b, c) = ® (a ® ( b , c) )= ® ((a ® b ), c ). (5)
f f f f f
U-FUNCTION IN GENERAL CASE
Of course, operator ® depends on the physical nature of parameter as and the type of
fs
structure, i.e. series or parallel.
Table 2.
Type of parameter Type of structure Result of interaction
A) a is unit's PFFO series a%> ® «Ah> = «A} x«A}
parallel «A} ® «A} = 1 - (1 -«2 >) x (1 «})
B) a is number of units in parallel series aB} > ® «Bh} = «>; B^h>)
parallel > ® a£> = «>; B^h>)
C) a is unit's cost (weight) series «2 > ® «(h} = a%} h}
parallel ) ® «A}=«a) h}
D) a is unit's ohmic resistance series > ® «(h}=«a > h}
parallel a%) ® a(h> = [«))-1 + (a(h>)-1
Type of parameter Type of structure Result of interaction
E) a is unit' s capacitance series a%> ® a(h> = [(a A] >)-1 + «})-1 ]"
parallel a(]} ®«A} = a%> + «Ah}
F) a is pipeline unit's capacitance series aA] ) ® «A} = min «\aA}}
parallel «A]} ®«A} = a%> +a(Ah>
G) a is unit's random time to failure series aA]) ® «Ah> = min «J\aAh>}
parallel «A} ® «Ah} = max«] >«h}}
In the problem of optimal redundancy, one deals with triplet of type "Probability-Cost-Number of units'' for each redundant group: M j = {a1 ^ ,a(^ ,a3^}. If there is a system of n series subsystems
(single elements or redundant groups) , one has to use a procedure almost completely coincided with the procedure of compiling the dominating sequences at the Kettelle's algorithm.
In accordance with the description given above, the block diagram of the using U-functions, for example, for four subsystems can be presented as follows (see Figures 1 and l).
Figure 1. Block-diagram of the sequential procedure of solving.
Figure 2. Block-diagram of the dichotomy procedure of solving. CONCLUSION
The method of U-function is one of the methods of directed enumerating. It showed its effectiveness for solving a number of practical problems concerning with optimal resources allocation and analysis of multi-state systems and system consisting of multi-state elements.
REFERENCES (in chronological order)
(1986) Ushakov, I.A. Universal Generating Function (in Russian). Engineering Cybernetics, No. 5.
(1987) Ushakov, I.A. Universal generating function. Soviet Journal Computer Systems Science, No.3.
(1987) Ushakov, I.A. Optimal standby problem and a universal generating function. Soviet Journal Computer and System Science, No.4.
(1987) Ushakov, I.A. Solution of multi-criteria discrete optimization problems using a universal generating function. Soviet Journal of Computer and System Sciences , No. 5.
(1988) Ushakov, I.A. Solving of optimal redundancy problem by means of a generalized generating function. Elektronische Informationsverarbeitung und Kybernetik, No.4-5
(1995) Gnedenko, B.V., and Ushakov I.A. Probabilistic Reliability Engineering. John Wiley & Sons.
(1998) Levitin, G., Lisnianski, A., Ben Haim, H., and Elmakis, D. Redundancy optimization for series-parallel multi-state systems, IEEE Transactions on Reliability, , No. 2.
(1999) Levitin, G., and Lisnianski, A. Importance and sensitivity analysis of multi-state systems using universal generating functions method, Reliability Engineering & System Safety, No. 65.
(2000) Ushakov, I.A. The method of generating sequences. European Journal of Operational Research, No. 2.
(2000) Levitin, G., and Lisnianski, A. Optimal Replacement Scheduling in Multi-state Seriesparallel Systems. Quality and Reliability Engineering International, , No. 16.
(2001) Levitin, G. Redundancy optimization for multi-state system with fixed resource-requirements and unreliable sources. IEEE Transactions on Reliability, No. 50.
(2001) Levitin, G., and Lisnianski, A. A new approach to solving problems of multi-state system reliability optimization, Quality and Reliability Engineering International, No. 47.
(2002) Levitin, G. Optimal allocation of multi-state elements in linear consecutively-connected systems with delays. International Journal of Reliability Quality and Safety Engineering, No. 9.
(2003) Levitin, G. Optimal allocation of multi-state elements in linear consecutively-connected systems. IEEE Transactions on Reliability, No. 2.
(2005) Levitin, G. The Universal Generating Function in Reliability Analysis and Optimization. Springer.
