U- FUNCTION IN APPLICATIONS

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1 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September U- FUNTION IN ALIATIONS Igor Ushaov Sun Diego aliornia igusha@gmail.com The ethod o Universal Generating Functions U-unctions was introduced in [Ushaov 986]. Since then the method has been developed in irst order b m riends and colleagues Gregor Levitin and Anatol Lisniansi. The activel and successull appl the method o U- unction to optimal resources allocation to multi-state sstem analsis and other problems. Franl now I eel lie a hen sat on duc eggs and then wanders how hatched chics earlessl swim so ar rom shore. I decided to remind ou a Russian ol proverb: new is well orgotten old. What is U- unction? It is irst o all generaliation o a classical Generation Function GF permitting perorm more general transorms. From technical side this method represents a modiication o the Kettelle s Algorithm convenientl arranged or calculations with the use o computer. U-FUNTION One can represent an discrete distribution o random variable r.v. as a set o pairs: S p p... This distribution can be represented in the orm o polnomial common generating unction p. n where n in principle can be ininite. The polnomial representation allows one to obtain the distribution o sums o random variables using the convolution procedure. I instead o the distribution o + one wish to obtain the distribution o an arbitrar unction or an combination o realiations and the realiation o the unction taes the value o with probabilit p n o two random variables + in the orm... n p p ` n... n S p p ` one can obtain the discrete distribution S o the unction p. Having the discrete distributions S and as a Descartes composition o two sets S and S here we use the term interaction because the operation reminds Descartes product but does not coincide with it. S... n ; p ; p n ; p p p... p n n n ; p n n ; p ; p p n n p p n n n This distribution can be convenientl obtained using the composition procedure over unctions and : 78

2 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September = = The composition operator and associative propert i.e. p n i n i p i i i = p p. 4 n i n possesses commutative propert i.e. = == 5. 6 i the unction possesses these properties. In the most applications this is the case though some eceptions eist see or eample [ Levitin 005]. USING U-FUNTION FOR SOLVING THE OTIAL REDUNDANY ROBLES Let us consider a series sstem consisting o n subsstems. Each o subsstem contains at least one unit with probabilit o ailure-ree operation FFO p and cost c. For increasing the sstem reliabilit one can introduce redundanc in each subsstem. Denote the number o redundant units o subsstem as. All the possible FFOs and costs o an subsstem or dierent levels o redundanc can be represented b the set o triplets S where is the total cost o redundant units usuall a linear unction; and FFO or availabilit coeicient o the subsstem when it contains redundant units. It is well-nown that or loaded redundanc the so-called hot standb o group including one main and identical redundant units taes the orm p ; and or an unloaded redundant the so-called cold standb units taes the orm 0 t! ep t; The set o triplets S can be represented in the GF orm as h. 7 Now consider a general procedure o optimal redundanc with the use o U-unction. First tae units and and arrange the Descartes interaction procedure between sets S and S. To distinguish interaction procedure rom common product o two generating unction let us introduce smbol. 79

3 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September From 8 clear that composition operator means summation corresponding components and operator means a collection o corresponding components. Thus the obtained U-unction represents the set o related FFO costs and numbers o redundant units or dierent conigurations o the series connection o subsstems and. The group o subsstems and now can be treated as an equivalent "aggregated" subsstem. Notice that solving optimal redundanc problem one has to mae some ind o siting o unction. One has to order all terms o the inal epression in 8 b increasing values o and eclude all terms o that have value o equal or smaller than previous terms. I two terms have the same values and one leaves an arbitrar single o them.. In the remaining terms all are numbered b natural numbers in accordance with their order b increasing cost. This procedure is equivalent to deleting dominated terms At the net step one obtains the UGF or the group o three subsstems and appling the same convolution operator over U-unction representing the aggregated subsstem and representing the subsstem :.. 9 This procedure continues until necessar inal UGF representing the entire sstem is generated. Instead o urther abstract presentation o the procedure let us turn to a simple illustrative numerical eample. onsider a series sstem o two units. Let unit- and unit- are characteried b strings... n S and... n S Each multiplet is a set o parameters... N where N is the number o parameters in each multiplet. Interaction o these two strings is an analogue o the arteian product whose memberts ill the cells o the ollowing table:

4 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September Table. n n n n n Interaction o multiplets consists o iteractions o their similar parameters or instance n n n h i h i h i... N N h Ni Operator as well as each operator in most natural practical cases possesses the commutativit propert i.e. and the associativit propert i.e. s a b = a b c = a b a 4 b c = a b c. 5 U-FUNTION IN GENERAL ASE O course operator structure i.e. series or parallel. depends on the phsical nature o parameter s s and the tpe o Tpe o parameter Tpe o structure Result o interaction A α is unit s FFO B α is number o units in parallel α is unit s cost weight D α is unit s ohmic resistance series h h parallel h h h h series B Bi B ; BRi parallel h h ; B series h h parallel h h series h h B parallel Bi h B Ri h Table. 8

5 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September Tpe o parameter Tpe o structure Result o interaction E α is unit s capacitance F α is pipeline unit s capacitance G α is unit s random time to ailure series h parallel h h series h min h parallel h h series h min h parallel h ma h h In the problem o optimal redundanc one deals with triplet o tpe robabilit-ost-number o units or each redundant group:. I there is a sstem o n series subsstems single elements or redundant groups one has to use a procedure almost completel coincided with the procedure o compiling the dominating sequences at the Kettelle s algorithm. In accordance with the description given above the bloc diagram o the using U-unctions or eample or our subsstems can be presented as ollows see Figures and. Figure. Bloc-diagram o the sequential procedure o solving. 8

6 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September ONLUSION Figure. Bloc-diagram o the dichotom procedure o solving. The method o U-unction is one o the methods o directed enumerating. It showed its eectiveness or solving a number o practical problems concerning with optimal resources allocation and analsis o multi-state sstems and sstem consisting o multi-state elements. REFERENES in chronological order 986 Ushaov I.A. Universal Generating Function in Russian. Engineering bernetics No Ushaov I.A. Universal generating unction. Soviet Journal omputer Sstems Science No Ushaov I.A. Optimal standb problem and a universal generating unction. Soviet Journal omputer and Sstem Science No Ushaov I.A. Solution o multi-criteria discrete optimiation problems using a universal generating unction. Soviet Journal o omputer and Sstem Sciences No Ushaov I.A. Solving o optimal redundanc problem b means o a generalied generating unction. Eletronische Inormationsverarbeitung und Kberneti No Gnedeno B.V. and Ushaov I.A. robabilistic Reliabilit Engineering. John Wile & Sons. 998 Levitin G. Lisniansi A. Ben Haim H. and Elmais D. Redundanc optimiation or series-parallel multi-state sstems IEEE Transactions on Reliabilit No Levitin G. and Lisniansi A. Importance and sensitivit analsis o multi-state sstems using universal generating unctions method Reliabilit Engineering & Sstem Saet No Ushaov I.A. The method o generating sequences. European Journal o Operational Research No Levitin G. and Lisniansi A. Optimal Replacement Scheduling in ulti-state Seriesparallel Sstems. Qualit and Reliabilit Engineering International No Levitin G. Redundanc optimiation or multi-state sstem with ied resourcerequirements and unreliable sources. IEEE Transactions on Reliabilit No

7 Ushaov I. U-FUNKTION IN ALIATIONS RT&A # 0 6 Vol.7 0 September 00 Levitin G. and Lisniansi A. A new approach to solving problems o multi-state sstem reliabilit optimiation Qualit and Reliabilit Engineering International No Levitin G. Optimal allocation o multi-state elements in linear consecutivel-connected sstems with delas. International Journal o Reliabilit Qualit and Saet Engineering No Levitin G. Optimal allocation o multi-state elements in linear consecutivel-connected sstems. IEEE Transactions on Reliabilit No Levitin G. The Universal Generating Function in Reliabilit Analsis and Optimiation. Springer. 84