ASSESSMENT OF POTENTIAL TRANSPORT OF TRANSPORT COMPANY FOR RANDOM DEMAND FOR TRANSPORT SERVICES

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1 ASSESSMENT OF POTENTIAL TRANSPORT OF TRANSPORT COMPANY FOR RANDOM DEMAND FOR TRANSPORT SERVICES Gustw Konopcki* * Institute of Infortion Systes Deprtent of Cybernetics, Militry University of Technology, Wrsw, Kliskiego Str., -98, Polnd, E-il: gkonopcki@wt.edu.pl Abstrct Will be considered trnsport copny eploiting the unifor in the sense of destintion, ens of trnsport such s tnkers, with totl trnsport potentil equl to loding units (e.g. tones). The copny opertes in the trnsport rket, where dend for trnsport services is rndo. Forulte the proble of stisfying the trnsport dend in full by trnsport copny nd shll be given the foruls to clculting the probbility of such n event with the generl ssuptions bout the dend for trnsport services. Prcticlly useful foruls for estiting such probbility is given for the dend for trnsport services the described by norl sttionry stochstic process. The results re illustrted by n eple of the clcultion. Pper type: Reserch Pper Published online: 3 October 14 Vol. 4, No. 4, pp ISSN (Print) ISSN (Online) 14 Poznn University of Technology. All rights reserved. Keywords: potentil trnsport copny, odeling, stochstic process

2 84 G. Konopcki 1. INTRODUCTION Iportnt role of rod trnsport in the econoy should be seen in the fct tht, ong other odes, it stnds out bove ll: obility: you cn get it nywhere where there re no ril, ship, etc., high operbility service, involving the vilbility of reltively lrge nuber of ens of trnsport, high high vilbility: getting lower cr prices nd getting better technicl preters, tieliness nd punctulity perfornce of services. Unfortuntely. the jor disdvntges of this type of trnsport re: dependent on clitic conditions, not very eco-friendly, high rte of ccidents, reltively sll volue of individul ens of trnsport, not very low intennce costs of vehicle. In prctice, however, the dvntges of rod trnsport outweigh its disdvntges, wht is the reson for its continued presence in the trnsport rket. One of the bsic probles of the trnsport copny nged is to provide its continued presence in the rket of trnsport services, inly by providing pproprite potentil trnsport for the nticipted dend for trnsport services. The lter in this chpter will be considered proble of ssessing the bility to stisfy rndo dend for trnsport services by ens of trnsport eploited in trnsport copny t the epected tie horizon.. DESCRIPTION OF THE PROBLEM Will be considered trnsport copny, in which re eploited vehicles (otor vehicles) for the se destiny (e.g. tnks), used to eet the dend for hoogeneous trnsport services (e.g. crrige fuels). It is ssued tht the crs do not hve to be the se, i.e. not necessrily hve the se design solution nd the se pylod. The trnsport potentil trnsport copny will constitute totl cpcity of ll vehicles concerned. This potentil will be denoted by nd treted s constnt, constnt over tie, the bility of trnsport copny to provide trnsport services t ost this size. Wheres, trnsport dend generted by the trnsport rket will be tie-vrying rndo vlue. Therefore, you y eperience the following two cses the lck of fit potentils trnsport copnies to dend: dend for trnsport services y t soe finite period of tie cn be less thn the size of the potentil trnsport, dend for trnsport services y t soe finite period of tie cn be greter thn the size of the potentil trnsport.

3 Assessent of potentil trnsport of trnsport copny 85 These cses re teporry istches nd tke plce only in finite tie intervls whose lengths re dependent on the of the process chrcteristics of fortion of the dend for trnsport services. Net will be considered the second cse, give rise to - it sees - ore serious consequences for the trnsport copny: inbility to eet dend y led to reduction in the trnsport position of the copny on the rket of trnsport services, nd even to his flling out of the rket. In view of this it sees necessry to erlier understnding of the trnsport copny s to either his possibilities to function on the trnsport rket, in the cse of hving of the trnsport potentil t the level of. This knowledge y be useful to tke pre-eptive ction, such s the purchse of dditionl ens of trnsport, or chnge of the profile provision of trnsport services. 3. FORMULATION AND SOLUTION OF THE PROBLEM In the following discussion, it is ssued tht rndo fctors influencing the dend for trnsport services, while cusing rndo chnges its vlue, which will be described by continuous-tie stochstic process X(t). It is ssued tht process X(t) is sttionry, ergodic nd differentible in the en-squre sense (Gichn & Skorochod, 1968). Further issue will be considered for stochstic process X(t) is designted the probbility of eceeded by this process fied vlue trnsport potentil equl to. In ddition, the epected vlue of durtion of such eceeded, i.e. the durtion of eceeded by this process of fied vlue trnsport potentil equl to will be clculte. An eeplry ipleenttion of stochstic process X(t) of the chnge dend for trnsport services s function of tie is shown in Figure 1, in which tp indictes oent of the eceeded vlue through the process. X(t) t p t Fig. 1 An eple of reliztion of the stochstic process X(t) describing the dend for trnsport services. This proble deterining of the probbility distribution of residence tie of stochstic process X(t) over defined threshold vlue is coputtionlly difficult. Fortuntely, however, in prctice, often enough to know the epected

4 86 G. Konopcki vlue of the residence tie of stochstic process over defined threshold vlue, which gretly fcilittes the nlyticl solution. The further considertions re true for ny clss of continuous-tie stochstic processes, nd finl, prcticlly useful coputtionl foruls cn be obtined reltively esily only for norl stochstic processes. First will be considered the proble of deterining the probbility P(,t) of the residence tie of stochstic process X(t) over defined threshold vlue. in the tie intervl [,t]. Consider the probbility tht in the infinitesil tie intervl dt, ieditely following oent t, the process X(t) eceeds threshold vlue. This probbility is presented in the following forul: : P{X(t) X(t dt) } Given the ssuption of continuity of the process X(t) nd tking into ccount the rte of chnge of vlue of this process V(t), for rbitrrily sll intervl dt cn be given the following forul: Given the bove equlity in the epression (1) is obtined P{ V(t) dt X(t) } Let the f(,v t) will be density function of the two-diensionl distribution of the process X(t) nd its speed V(t) t oent t. Thus, the likelihood of (3) will be equl to: P{ V(t) dt X(t) } where the liits of integrtion stisfy the following condition After trnsfortion of forul (5) with respect to tht dt is rbitrrily sll of intervl tie, is obtined (6) Due to the fct tht the probbility of eceeded threshold in the infinitesil tie intervl dt is proportionl to the length of this intervl, you cn use the density function p ( t), denoting the probbility eceeded threshold vlue by the process X(t) per unit tie. Considering this ssuption in the forul (3) is obtined: Fro foruls (6) nd (7) tht: p( t) X(t dt) X(t) V(t) dt V(t) dt X(t) f(, vdt P{ V(t) dt X(t) } P{ V(t) dt X(t) } for V(t) v t) ddv for V(t) f(, v t)vdv p( t) f(,v t)vdv (1) () (3) (4) (5) (7) (8)

5 Assessent of potentil trnsport of trnsport copny 87 Using the forul (8) cn be clculted for ech tie intervl of length T the epected vlue of residence tie of the process X(t) over the threshold vlue. Let the bove-entioned tie intervl [,T] is divided into n equl subintervls ech of length dt, i.e. [tj, tj+dt], (j = 1,,..., n). Hence, the probbility tht the process X(t) eceeds the threshold vlue in the tie intervl nuber j is equl to (9) P{X(t j ) } h( t where h( t) is density function of the distribution of the process X(t) t oent t. If dt is sufficiently sll tht it will be possible to neglect cses ultiple (t lest double) crossing in ech of these intervls the level by the process X(t), the epected vlue of the residence tie of the process bove this level in the intervl [, T] is epressed by forul (see [Konopcki & Worw, 1]): T E T h( In prcticl pplictions of interest is usully epected residence tie of the process ore thn the threshold vlue is not in ny period of tie, but the epected durtion of only one such clernce..let is rndo vrible representing the durtion of single crossing threshold vlue by the process X(t). The epected vlue of this rndo vrible is the quotient of the epected vlue E(T ) nd the epected nuber of eceednces E(N ) of the threshold vlue by the process X(t) in the tie intervl of length T. Epected vlues of E(N ) nd E() re epressed by the following foruls (detils in [Konopcki & Worw, 1, Pierievierziev, 1987]: E T N v f,v T h( t) ddt E T (1) v f(,v t) dvdt In the cse of sttionry process X(t), the density functions h( t) nd f(,v t) no longer depend on the tie nd receive, respectively, the fors of h() nd f(,v). Thus, for sttionry processes will pply the following foruls: j )d t) ddt t dvdt (1) (11) E T E T N T v E τ h() d f(,v) dv h() d v f(,v) dv (13) (14) (15)

6 88 G. Konopcki For the sttionry process X (t) cn be deterined the epected nuber of eceednces of threshold vlue in the unit of tie, which is epressed by the following forul: n E N T To obtin prcticlly useful foruls necessry to know the density functions, h() nd f(,v), which in the generl cse is difficult. Significntly siplifies the sitution in the cse of norl sttionry stochstic process X(t). Unfortuntely the describing processes the behvior of the dend for trnsport services under the ction of rndo fctors re not lwys sttionry nd dditionlly - norl. Thus, foruls obtined on the ssuption sttionrity nd norlity of the stochstic process X(t) hve definite cognitive vlue for the nlysis of the considered proble, but the results obtined with their help should be considered only s pproite. Verifiction of these results is usully on the bsis of sttistics. Despite these reservtions, however, tke ccount of the sttionrity nd norlity of the process X(t) is often the only option when is necessry 'priori to ke quntittive ssessent of the proble, i.e., under conditions where there is not still the possibility of sttisticl studies the trnsport rket. Norl sttionry stochstic process is clerly defined, when is known, nd K () - its epected vlue nd correltion function. The density function of of the process is s follows: h() where (18) For sttionry norl stochstic process its vlue nd the rte of chnge of the vlue of the fied point in tie re independent rndo vribles. Therefore, the density of two-diensionl probbility distribution f(,v) in this cse is equl to: v f(,v) dv 1 ( ep π σ ) σ σ K () (16) (17) f(,v) where the vrince σ v 1 ( ep π σ σ of process V(t) is equl to: 1 v ep π σ ) σ v v (19) σ v d K ( ) d nd the epected vlue of this process is equl to zero due to the sttionrity of the process X(t). Tking into ccount the epression (19) in (16) we obtin: σ v n ep n ep πσ σ σ ()

7 Assessent of potentil trnsport of trnsport copny 89 (1) where n is the epected nuber of eceednces through the process X(t) of its epected vlue per unit of tie. Tking into ccount the epression (1) in (15) we obtin: σ () E π ep 1 Φ σ v σ σ where the function () is the integrl function of Lplce. One of the ost iportnt prcticl questions in the ngeent of the trnsport copny is the proble of deterining the likelihood tht tie horizon of length T is not the dend for trnsport services, not once does not eceed threshold vlue, i.e. the trnsport potentil of the trnsport copny.. This is proble difficult to solve by nlyticl ethods, even in the cse of norl stochstic processes. Help in the solution of this proble is when the epected nuber of eceednces of the threshold vlue in certin period of tie is sll enough tht the net eceednces of the threshold cn be treted s independent rndo events. In this cse, it cn be ssued tht the nuber of eceednces of the threshold vlue by the process is rndo vrible of Poisson distribution nd proble of clculting the likelihood tht in ny tie intervl of length T there will be no eceeding of threshold vlue by the considered process cn be resolved in stisfctory nner. Foruls (17-) llow to clculte this likelihood for the sttionry norl processes, which is equl to: " T K,T ep ep π K σ P (3) 3. CONCLUSION In solving prcticl probles, in generl, there re the difficulties of estblishing of nlyticl forul of correltion function s function of the process X(t), which is necessry for the clcultion of n vlue, which, in turn, is essentil to clculte the likelihood (3). In this cse is ost often ccepted the upper estite of this vlue tht is equl to [Pierievierziev, 1987]: 1 n π (4) In the prcticl re often used estite of likelihood of not eceeded of the threshold vlue for norl sttionry stochstic process X(t) in the tie intervl of length T. There re: P (,T) Φ σ ( n T ep σ in ) P lower estite (5)

8 9 G. Konopcki upper estite P (,T) Φ σ ( ep n T ep σ ) P (6) Estiting (5) cn be use if T stisfies the following inequlity: T Φ σ n ( ep σ ) If it is possible to observe the process X(t), vlue n cn be estited using sttisticl ethods without hving to know the nlyticl for of the correltion function of the process. The prcticl iportnce is the deterintion of the two vlues of trnsport potentil: the upper liit, which is included in the bove clcultions perfored, lower liit - liit of vibility (e.g., econoic) to provide trnsport services by trnsport copny, nd deterintion of probbilities of eceeding ech of the (the first up nd the other down) through the process X(t). This is proble ore difficult thn considered, but possible for stisfctory solution. Eple The trnsport copny hs the potentil to trnsport equl to = 6 units (e.g. tons), while the dend for trnsport services rket is described by norl sttionry stochstic process X (t) of the epected vlue = 4 units loding nd stndrd devition = 1 units loding. (7) Fig. Eple of lower (Pin) nd upper (P) estites of likelihood of the bility to stisfy the rndo dend for trnsport services in full by the trnsport copny in the ccepted tie horizon for the sple dt: = 6, = 4, = 1.

9 Assessent of potentil trnsport of trnsport copny 91 Figure shows the chnges in the lower nd upper estites of the probbility of not eceeding of the trnsport potentil (the bility to stisfy the dend for trnsport services in full) by the dend on trnsport services s function of tie horizon T (e.g, onths). REFERENCES Gichn I. I., & Skorochod A. W., (1968), Wstęp do teorii procesów stochstycznych,. PWN, Wrszw. Feler W., (1996), Wstęp do rchunku prwdopodobieństw, PWN, Wrszw. Hggstro O., (1), Finite rkov chins nd lgorithic pplictions, Chlers University of Technology. Konopcki G., & Worw K., (1), Proble eliinowni fłszywych lrów w koputerowych systech ochrony peryferyjnej, Biuletyn Instytutu Systeów Infortycznych WAT w Wrszwie, Nr 5/1, s Krwczyk S., (1), Zrządznie procesi logistycznyi, PWE, Wrszw. Lwler G. F., (1995), Introduction to Stochstic Processes, Chpn & Hll / CRC. Mitzencher M., & Upfl E., (9), Metody probbilistyczne i obliczeni, WNT, Wrszw. Nowk M., (7), Strefy ochrony posesji, dostępny w: Norris J. R., (1977), Mrkov Chins, Cbridge Series in Sttisticl nd Probbilistic Mthetics. Ppoulis A., (197), Prwdopodobieństwo, zienne losowe i procesy stochstyczne, WNT, Wrszw. Pierievierziev E.C. (1987), Słuczjnyje procesy v pretriczeskich odelch ndiożnosti. Nukovj Duk, Kijev. Ross S.M. (1996), Stochstic processes. John Wiley & Sons, New York.. Sobczyk K., (1996), Stochstyczne równni różniczkowe, WNT, Wrszw. BIOGRAPHICAL NOTES Gustw Z. Konopcki is n ssistnt professor t the Institute for Infortion Systes Deprtent of Cybernetics of the Militry University of Technology in Wrsw. He is lso n ssistnt professor t University of Wrsw. In heded the Deprtent of Coputer Progring in the Deprtent of Cybernetics MUT. Dels with the issues of odeling of sfety nd security fcilities. Author of books on the design of infortion systes nd co-uthor of four books of theticl nlysis nd sttistics. Author of ore thn eighty reserch publictions. It dels inly with the probles of theticl odeling of processes.

10 9 G. Konopcki