Evaluation of hazard distribution in network structures

Transcription

1 Rsk Analyss V: Smulaton and azard Mtgaton 363 Evaluaton of hazard dstrbuton n network structures J. Auguts,, R. Krkstolats,, V. Matuzas & E. Uspuras Lthuanan Energy nsttute, Lthuana Vytautas Magnus Unversty, Lthuana Abstract The subject of ths work s the analyss of hazard dstrbuton n the network systems. n the age when transport flows n the ralway and road systems, nformaton flows n the nternet channels, etc. the ntensty and hazard dstrbuton n the network systems becomes a more relevant and mportant ssue. n ths paper varous theoretcal hazard type dstrbutons n the network systems are analysed. A mathematcal model of hazard dstrbuton and accumulaton n the nodes of the network was created. Theoretcal results are llustrated by smulaton of hazard fuel transportaton on Lthuanan roads. t s shown that usng ths methodology t s possble to construct an algorthm, whch enables one to operate and decrease fuel transportaton hazard and rsk. Keywords: hazard, dstrbuton, rsk assessment, network systems. ntroducton azard n rsk analyss s defned as a feature or characterstc of materal, technologcal process, nformaton, human actvtes or other phenomena that specfes a potental possblty of endangerng human lfe, health, nature, buldngs, eupment, etc. Some llustratons of hazards could be posonous chemcal substances kept n stock, open source of radaton, a car wth brakes that are out of order, a lock that s bult on a rver that flows through a cty, etc. n ths work we wll use numercal values for the hazard, such as the uantty of posonous materals, the uantty or flow of hazardous nformaton, etc. callng these characterstcs hazard level or, n some cases, smply hazard. do:.495/rsk634

2 364 Rsk Analyss V: Smulaton and azard Mtgaton Development and expanson of varous networks and network structures creates favourable condtons for the spread of hazard through network channels. One of the most vsble examples of such phenomena s the spread of computer vruses n the nternet network. t seems that a smlar stuaton s developng n the networks of moble connecton as well. t s also obvous that together wth the mprovement of means of transportaton and the ncrease n uantty and sze of loads, the assessment of hazard dstrbuton becomes more promnent n the systems of transportaton. n the context of ths work, hazard s understood as the amount of hazardous materals, the flow of hazardous nformaton, dsease concentraton, etc. azard can be dstrbuted through the channels of varous networks and concentrated n the nodes of the networks. azard transmsson through the channels that connect network nodes can take place n many ways: for example, hazard can be transmtted to a sngle or to several nodes, as an undvded value or dvded nto parts. Each node can also have certan protecton or mmunty aganst hazard whch blocks ts transmsson or dmnshes t. n all the cases, for hazard dstrbuton n network nodes, teratve euatons were made and theorems that descrbe the suffcent convergence condtons for these processes were proven. For the calculaton of the margnal dstrbutons, systems of lnear euatons were made. To llustrate the theoretcal fndngs, software was created and numercal examples were gven. A hazard assessment model was used to evaluate the rsk of fuel transportaton on Lthuanan roads. Margnal dstrbuton of the nfntve source of the hazard n the network nodes wth mmunty n the majorty of real network systems, hazard n ts whole or n part s elmnated durng certan tmes n a number of cycles. For example, n the computer networks, antvrus systems that destroy the majorty of vruses are mplemented, human mmune system destroys dsease vruses, customs does not allow free movement of prohbted goods, etc. Of course, f the source of the hazard n such systems was a pont source (sngle), the hazard would be entrely destroyed or mnmzed to the secure level by the system durng certan perod of tme or after a certan number of cycles. n the real systems, however, sources of hazard are often nexhaustble,.e. nfntve and, at the begnnng of every cycle, the hazard can renew. Despte the effort of software developers to fght computer vruses, there are stll many hackers that create them and the number of new vruses s not decreasng. Advanced technologes that are used nowadays do not decrease the amount of fake documents and money n the networks of banks and supermarkets. n ths work we wll analyse only the margnal hazards n dfferent network systems wth the nfntve hazard sources and node mmuntes. The mmunty of the network node s consdered the number ( ), by whch the hazard that occurs n ths node s multpled. Thus, when =,

3 Rsk Analyss V: Smulaton and azard Mtgaton 365 hazard s fully transmtted (such networks have been analysed so far), and when =, all of the hazard that makes way to the node s destroyed. Let network flow matrx wll be marked as follows [ j ] Q =, where, j =,,,. Let us assume that the frst node of the network s the nfntve source of hazard that ncreases hazard by the value durng each new cycle. Frst we shall make an assumpton that a margnal hazard dstrbuton exsts under such condtons,.e. n ( n) lm =, where =,,,. Then, f at the end of each cycle the hazard that s located n the node s multpled by the mmunty <, and the node n whch the source of hazard s located does not have mmunty (let s say t s the frst one), n ths case hazard n the frst node wll not change, f t s reduced by the value after each cycle,.e. ( ) = 3 n the second node, the ncomng hazard multpled by the mmunty has to be eual to the outgong hazard. Thus, ( ) = ( + + ) The case n analogous wth the other nodes: ( ) ( ), j,3,, j j j j j j = = Jonng those euatons we get a system of lnear euatons: = ; = ; =. () The man matrx of ths system s:

4 366 Rsk Analyss V: Smulaton and azard Mtgaton ~ Q = 3 3 3, T constants vector B = [ ], and unknowns vector T = [ ]. Thus, ths system can be wrtten n the form of matrx ~ Q = B. () To nvestgate the exstence of the margnal dstrbuton, hazard dstrbuton wll be expressed n such teratve process: ( n). (3) ( n + ) = ( n) Q As there s the mmunty, we shall ntroduce a dagonal matrx = ow we can express the system of euatons ths way: [ ( n + ), ( n + ),, ( n + ) ] = [ ( n), ( n),, ( n) ] Q [,,,] + Q = We mark n) = [ ( n) ( n) ] T ( n), ( ) = [ ] T obtan that ( ( n ) = ( n) Q + ( ) and + (4)

5 Rsk Analyss V: Smulaton and azard Mtgaton 367 n the paper the convergence condtons of the process (4) were not fully set; however, for ths process, general condton that s necessary and suffcent for the convergence of teratve processes apples: for the process (4) to converge, t s necessary and suffcent that all the absolute values of the Q matrx are less than [7]. 3 azard assessment of ol products transportaton n Lthuanan roads As an example of the margnal dstrbuton of the nfntve source of hazard, we wll analyse the hazard of the fuel transportaton n Lthuanan roads n two way: analytcal calculaton usng () euaton system and computer smuloaton usng (4) teratve process. Fuel transportaton from the Mažekų nafta refnery termnal to the bggest cty of Lthuana Vlnus was nvestgated. All possble and most probable routes are presented n Fg.. Fgure : Lthuanan roads graph. For the analyss, an orented graph wth nodes and 7 lnks was chosen. n ths case, Lthuanan towns consttuted a set of nodes and the network of Lthuanan roads a set of lnks. The mmuntes of all the nodes, except for Vlnus, were eual to, as we were nterested n hazard dstrbuton from the ntal node Mažeka to the fnal node Vlnus. The source of hazard (Mažeka) durng each cycle s supplemented wth a relatve hazard unt, whch s eual to

6 368 Rsk Analyss V: Smulaton and azard Mtgaton ( = ). Snce the hazard s generated durng each cycle, n some nodes total hazard can exceed. Usng () euaton system the followng soluton s obtaned: Table : azard dstrbuton n Lthuanan roads (analytcal calculatons). ode name o. azard level ode name o. azard level Mažeka..733 Panevėžys..7 Seda Kėdana Retavas Cnkškės Kryžkalns Kaunas Skaudvlė Jonava Šaula Ukmergė Kelmė Kupšks Radvlšks Utena Tytuvėna Molėta..796 Rasena Elektrėna..873 Šeduva Vlnus..83 As we see from the table, the hghest hazard, when transportng fuel by the route Mažeka Vlnus, s n Kaunas and n the source of hazard (Mažeka). The reason why Kaunas has the hghest level of hazard s that t s a relatve centre of the network of Lthuanan roads [6], []. The hazard was also estmated usng euaton (4). Modellng results are gven n Table. The results that were presented are settled after teratons. As we can see, results are almost the same n both cases. t means that for large systems we can use modellng, because analytcal soluton s becomng complcated. Table : azard dstrbuton n Lthuanan roads (computer smulaton). ode name o. azard level ode name o. azard level Mažeka..695 Panevėžys..735 Seda..366 Kėdana Retavas Cnkškės Kryžkalns Kaunas Skaudvlė Jonava Šaula Ukmergė Kelmė Kupšks 8..6 Radvlšks Utena Tytuvėna Molėta..55 Rasena..865 Elektrėna..855 Šeduva Vlnus.

7 Rsk Analyss V: Smulaton and azard Mtgaton Conclusons The man am of the paper s to present the developed hazard dstrbuton mathematcal model and ts analyss. ere was analysed the mechansm of hazard propagaton n network systems n the case when hazard arse durng each cycle. The most mportant cases are hazard propagaton n Markov chans and network systems, where peaks have an mmunty or resstance to the hazard characterstc. From the materal presented n the paper we can see that margnal hazard dstrbuton network system can be determned as lnear euaton system. n the paper llustraton of the results applcablty s presented. Based on the margnal hazard dstrbuton mathematcal model the hazard caused by fuel transportaton by fuel trucks was estmated n the fragment of Lthuanan roadway network. References [] Lefevre C, Pcard P (993) An Epdemc Model wth Fatal Rsk, Mathematcal Boscences, ew York, 7:7-45 p. [] Purdy G (993) Rsk analyss of the transportaton of dangerous goods by road and ral, Journal of azardous Materals, 33, p [3] Gnedenko B, Pavlov V, Ushakov, Chakravarty A (999) Statstcal Relablty Engneerng, ew York, Y: John Wley & Sons. [4] Klenrock L and Gal R. (996) Queung Systems: Problems and Solutons, John Wley & Sons. [5] Chapra SC and Canale RP () umercal Methods for Engneers: Wth Software and Programmng Applcatons McGraw-ll Scence/Engneerng/Math Date Publshed. [6] Auguts J, Krkštolats R, Matuzas V, Ušpuras E (5) Rsk of ol products transportaton n Lthuana, Proceed. Of the nternatonal Topcal Meetng on Probablstc Safety Analyss. San Francsco, SB: X [7] Rosen K (998) Applcatons of Dscrete Mathematcs, 4th ed. ew York: McGraw-ll. [8] Roeleven D, Kok M, Stpdonk L, de Vres WA nlan (995) Waterway transport: Modellng the probablty of accdents. Safety Scence. Volume 9, 9- p. [9] Adams J (995) Rsk, London: UCL Press Ltd. [] Volkovas V, Krkštolats R, Matuzas V, Doroševas V, Augutenė E. (5) Rsk Analyss of Petrol Transportaton. Envronmental research, engneerng and management. Kaunas, o.(3), P