N O N H O M O G E N E O U S S T O C H A S T I C P R O C E S S E S C O N N E C T E D T O P O I S S O N P R O C E S S

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1 S C I E N T I F I C J O U R N A L O F P O L I S H N A V A L A C A D E M Y Z E S Z Y T Y N A U K O W E A K A D E M I I M A R Y N A R K I W O J E N N E J 218 (LIX) 2 (213) DOI: /sjpna N O N H O M O G E N E O U S S T O C H A S T I C P R O C E S S E S C O N N E C T E D T O P O I S S O N P R O C E S S Franciszek Grabski Polish Naval Academy, Faculy of Mechanical and Elecrical Engineering, Śmidowicza 69 Sr., Gdynia, Poland; f.grabski@amw.gdynia.pl ABSTRACT Some generalizaions of he Poisson process and heir properies are presened in he paper. The nonhomogeneous Poisson process allows o consruc a probabilisic model describing he differen kinds of accidens number. The nonhomogeneous compound Poisson process enables o describe mahemaically he various ypes of accidens consequences. Theoreical resuls give possibiliy o anicipae he accidens number and heir consequences. Key words: nonhomogeneous Poisson process, nonhomogeneous compound Poisson process, safey characerisics. Research aricle 218 Franciszek Grabski This is an open access aricle licensed under he Creaive Commons Aribuion-NonCommercial-NoDerivaives 4. license (hp://creaivecommons.org/licenses/by-nc-nd/4./) 5

2 Franciszek Grabski INTRODUCTION In 1837 Simeon-Denis Poisson derived his disribuion o approximae he Binomial Disribuion when a parameer p, deermining he probabiliy of success in a single experimen, is small. Applicaion of his disribuion was no found; when von Borkiewisch (1898) calculaed from he daa of he Prussian army he number of soldiers who died during he 2 consecuive years because of he kick by a horse. A random variable, say X, denoing he number of solders killed accidenally by he horse kick per year, urned ou o have Poisson disribuion p(k) = P(X = k) = (Λ)k e Λ, k S = N = {,1,2, } wih parameer Λ =.61 [ 1 ]. We calculae several probabiliies according o his year disribuion. P(X = ) = , P(X = 1) = , P(X = 2) =.119, P(X = 3) =.2555, P(X = 4) =.3135, P(X 4) = , P(X > 4) =.425. I can be noiced, ha he probabiliy of no soldiers killed accidenally by he horse kick per year is over 5% and he probabiliy ha more hen 4 solders was killed during he year is.425. A nonhomogeneous Poisson process and nonhomogeneous compound Poisson process are generalizaions of he Poisson process. The nonhomogeneous Poisson process allow o consruc a probabilisic model describing he numbers of differen ypes of accidens The nonhomogeneous compound Poisson process gives possibiliy o describe mahemaically differen kinds of he accidens consequences. Theoreical resuls presened in [1, 3 7] enable o anicipae he accidens number and heir consequences. HOMOGENUOUS POISSON PROCESS A random process {X(): } is said o be process wih independen incremens if for all 1,, n such ha < 1 < 2 < < n he random variables X(), X( 1 ) X(),, X( n ) X( n 1 ) are muually independen. If he incremens 6 Scienific Journal of PNA Zeszyy Naukowe AMW

3 Nonhomogeneous sochasic processes conneced o Poisson process X(s) X() and X(s + h) X( + h) for all, s, h >, s > have he idenical probabiliy disribuions hen {X(): } is called a process wih he saionary independen incremens (SII). I is proved ha for he SII processes such ha X() = an expecaion and a variance are where E[X()] = m 1, V[X()] = σ 1 2, (1) m 1 = E[X(1)] and σ 1 2 = V[X(1)]. (2) An example of a SII random process is a Poisson process. Definiion 1 A sochasic process {X(); } aking values on S = N = {,1,2, }, wih he righ coninuous and piecewise consan rajecories is said o be a Poisson process wih parameer λ > if: 1. X() =. 2. {X(): } is he process wih he saionary independen increamens. 3. For all >, h. P(X( + h) X() = k) = (λ h)k For = we ge a firs order disribuion of he Poisson process: p k (h) = P(X(h) = k) = k! (λ h) e λ, k N. (3) e λ h, k N. (4) For h = 1 we obain he Poisson disribuion wih parameer λ. Hence E[X(1)] = λ and V[X(1)] = λ. Therefore, from (1) and (2), we obain he expecaion and he variance of he Poisson process: E[X()] = λ, V[X()] = λ,. (5) For a fixed his formula deermines he Poisson disribuion wih parameer Λ = λ : p(k) = P(X = k) = Λk e Λ, k N. (6) NONHOMOGENEOUS POISSON PROCESS We will begin wih a reminder of he concep of nonhomogeneous Poisson s process. 2 (213) 218 7

4 Franciszek Grabski Le τ = θ =, τ n = θ 1 + θ θ n, n N, (7) where θ 1, θ 2,, θ n are posiive and independen random variables. Le τ = lim n τ n = sup{τ n : n N }. (8) A sochasic process {N(): } defined by he formula N() = sup{n N : τ n } (9) is called a couning process corresponding o a random sequence {τ n : n N }. Le {N(): } be a sochasic process aking values on S = N = {,1,2, }, value of which represens he number of evens in a ime inerval [, ]. Definiion 2 A couning process {N(): } is said o be nonhomogeneous Poisson process (NPP) wih an inensiy funcion λ( ),, if 1. P(N() = ) = 1. (1) 2. The process {N(): } is he sochasic process wih independen incremens, he righ coninuous and piecewise consan rajecories. 3. P(N( + h) N() = k) = +h ( λ(x)dx) k e +h λ(x)dx. (11) From his definiion i follows ha he one dimensional disribuion of NPP is given by he rule P(N() = k) = ( λ(x)dx) k e λ(x)dx, k =,1,2, (12) The expecaion and he variance of NPP are he funcions given by The corresponding sandard deviaion is Λ() = E[N()] = λ(x)dx ; (13) V() = V[N()] = λ(x)dx,. (14) D() = V[N()] = λ(x)dx,. (15) 8 Scienific Journal of PNA Zeszyy Naukowe AMW

5 Nonhomogeneous sochasic processes conneced o Poisson process The expeced value of he incremen N( + h) N() is Δ(; h) = E(N( + h) N()) = The corresponding o i sandard deviaion is +h λ(x)dx. (16) σ(; h) = +h λ(x)dx. (17) An nonhomogeneous Poisson process wih λ( ) = λ, for each, is a regular Poisson process. For he Poisson process wih parameer λ he random variables θ 1, θ 2,, θ n, n = 2,3, are muually independen and exponenially disribued wih he idenical parameer λ. The Poisson process is a couning process which is generaed by he random sequence {τ n : n N }, where τ n = θ 1 + θ θ n, n N. The incremens of an nonhomogeneous Poisson process are independen, bu no necessarily saionary. A nonhomogeneous Poisson process is a Markov process. COMPOUND POISSON PROCESS Le {N(): } be a Poisson proces wih inensiy λ > and X 1, X 2, be sequence of independen and idenically disribued (i.i.d.) random variables independen of {N(): }. A sochasic process is called a compound Poisson process (CPP). X() = X 1 + X X N(), (18) The probabiliy discree disribuion funcion of {N(): } a k is p(k; ) = P(N() = k) = We quoe a well-known resul [1, 3]. (λ )k e λ, k =,1,2, If E(X 2 1 ) <, hen 1. E[X()] = λ E(X 1 ). (19) 2. V[X()] = λ E(X 2 1 ). (2) The conceps and facs can be generalized. 2 (213) 218 9

6 Franciszek Grabski Definiion 3 Le {N(): } be a nonhomogeneous Poisson process (NPP) wih an inensiy funcion λ( ), such ha λ( ) for, and X 1, X 2, is a sequence of he independen and idenically disribued (i.i.d.) random variables independen of {N(): }. A sochasic process {X(): } deermines by he formula X() = X 1 + X X N(), (21) is said o be a nonhomogeneous compound Poisson process (NCPP). Proposiion 1 If {N(): } is a nonhomogeneous Poisson process (NPP) wih an inensiy funcion λ( ), such ha λ( ) for, hen cumulaive disribuion funcion (CDF) of he nonhomogeneous compound Poisson process is given by he rule where G(x, ) = I [,) (x)e Λ( ) (k) + k=1 p(k; )F X (x), (22) F X (k) (x) denoes he k-fold convoluion of CDF of he random variables Xi, i = 1,2, and p(k; ) = (Λ( ))k e Λ( ),, k =,1, ; (23) Λ( ) = E[N()] = λ(x)dx is discree probabiliy disribuion of NPP. (24) P r o o f: Using oal probabiliy low we obain cumulaive disribuion funcion (CDF) of NCPP. G(x, ) = P(X() x) =P(X 1 + X X N() x) = = k= P(X X N() x N() = k) P(N() = k) = (k) p(k; )F X (x) = I [,) (x)e Λ( ) + (k) k=1 p(k; )F X (x). This proposiion is generalizaion of resulas presened in [1]. k= = Corollary 1 If he random variables, i=1,2, are absoluely coninuous wih densiy funcion f X ( ), hen he densiy of NCPP is given by he rule (k) g(x, ) = k=1 p(k; )f X (x), >, (25) where f X (k) (x) denoes k-fold convoluion of he densiy funcion fx (x). 1 Scienific Journal of PNA Zeszyy Naukowe AMW

7 Nonhomogeneous sochasic processes conneced o Poisson process Example 1 Le he random variables X i, i = 1,2, have normal disrbuion N(m, σ). I means ha a probabililiy densiy funcion of X i = X is f X (x) = 1 (x m) 2 2πσ e 2 σ 2, σ >, m (, ). x (, ). (26) The sum X 1 + X X k has normal disribuion N(km, kσ). Hence i s densiy is k-fold convoluion of he densiy funcion f X (x) given by (25): f X (k) (x) = Therefore he densiy of NCPP given by (18) akes he form g(x, ) = 1 2π σ (Λ( )) k k (x km) 1 2 2π kσ e 2 k σ 2 (27) e Λ( ) e (x km)2 2 k σ k=1 2, x, >. (28) Fig. 1. The densiy funcions of CPP for λ =.46, m = 1.2, δ = 3.2, and = 5, = 8 Corollary 2 If he random variables X i, i=1,2, are discree disribued wih probabiliy funcion p X (x) = P(X = x), x S hen he discree probabiliy disribuion of NCPP is given by he rule (k) g(x, ) = k=1 p(k; )p X (x), >, (29) where p X (k) (x) denoes k-fold convoluion of he discree disribuion funcion p X (x). 2 (213)

8 Franciszek Grabski Example 2 Assume ha random variables X i, i = 1,2, have a Poisson disribuion wih parameer μ > : p X (x) = μx x! e μ, x =,1,2, k-fold convoluion of his discree disribuion funcions is Then he rule (18) akes he form g(x, ) = p (k) X (x) = (kμ)x e kμ, x =,1,2, x! (Λ( )) k Λ( ) e (kμ)x k=1 x! e kμ, x =,1,2,, >. (3) Assuming μ =.64 we compu probabilies (3). The resuls are shown in able 1. Tab. 1. The values of he funcion (3) x g(x,4),13358,2632,262122,181373, x g(x,4),42511,159536,525922,155298,41687 Fig. 2 shows a discree probabiliy disribuion of a nonhomogeneous compound Poisson process under assumpions ha random variables X i, i=1,2, have Poisson disribuion wih parameer μ =.64 and = 4, Λ( 4) = 32,48.,3,25,2,15,1, Fig. 2. A discree probabiliy disribuion of NCPP corresponding o ab Scienific Journal of PNA Zeszyy Naukowe AMW

9 Nonhomogeneous sochasic processes conneced o Poisson process Proposiion 2 Le {X(): } be a nonhomogeneous compound Poisson process (NCPP). If E(X 1 2 ) <, hen 1. E[()] = Λ( )E(X 1 ). (31) 2. V[X()] = Λ( ) E(X 1 2 ). (32) P r o o f: Applying he propery of condiional expecaion we have E[X()] = E[E(X() N())] Using a formula E[E(X() N())] = E (E(X 1 + X X N() ) N()) = = E(X 1 + X X N() N() = n)p(n() = n) = n= = E(X 1 + X X n ) P(N() = n) n= = n=. = E(X 1 ) n P(N() = n) = E(X 1 )E(N()) we ge V[X()] = E[V(X() N())] + V[E(X() N())] = V(X 1 + X X N() N() = n)p(n() = n) n= = V(X 1 + X X n ) P(N() = n) = n= = n= V(X 1 ) n P(N() = n) = V(X 1 )E(N()) = V(X 1 )Λ( ); V[E(X() N())]= = V (E(X 1 + X X N() ) N()) = V(E(X 1 )N()) = (E(X 1 )) 2 V(N()) = Therefore = (E(X 1 )) 2 Λ(). V[X()] = V(X 1 )Λ( ) + (E(X 1 )) 2 = Λ()[E(X 1 2 ) (E(X 1 )) 2 + (E(X 1 )) 2 ] = = Λ( ) E(X 1 2 ). Corollary 3 Le {X( + h) X(): } be an increamen of compound nonhomogeneous Poisson process (CNPP). 2 (213)

10 Franciszek Grabski If E(X 1 2 ) <, hen E[X( + h) X()] = Δ(; h) E(X 1 ); (33) where D[X( + h) X()] = Δ(; h) E(X 1 2 ), (34) Δ(; h) = +h λ(x)dx. (35) CONCLUSIONS The resuls presened in he aricle are generalizaions of heorems known in he lieraure. These generalizaions concern incremens of he nonhomogeneous Poisson process and incremens of he nonhomogeneous compound Poisson process. The resuls allow o anicipae he number of differen kinds of accidens and heir consequences. The nex paper will discuss applicaion hese heoreical resuls in modelling marine accidens [2, 5, 8, 9]. REFERENCES [1] Andrzejczak K., Sochasic modeling of he repairable sysem, J. of KONBiN, 215, No. 3(35). [2] Annual repor on Shipping accidens in he Balic Sea 213, HELCOM 214. [3] Di Crescenzo A., Marinucci B., Zacks S., Compound Poisson process wih Poisson subordinaor, Journal of Applied Probabiliy, 215, Vol. 52, No. 2, pp [4] Fisz M., Probabiliy and Mahemaical Saisics [in Polish], PWN, Warsaw [5] Grabski F., Nonhomogeneous Poisson process applicaion o modelling accidens number a Balic waers and pors, Journal of Polish Safey and Reliabiliy Associaion, 217, Vol. 8, No. 1, pp [6] Grabski F., Semi-Markov models o reliabiliy and operaion [in Polish], IBS PAN Warsaw 22. [7] Grabski F., Semi-Markov Processes: Applicaions in Sysems Reliabiliy and Mainenance, Elsevier, Amserdam Boson Heidelberg London NewYork Oxford Paris San Diego San Francisco Sydney 215. [8] Herdzik J., Zdarzenia wypadkowe na morzu i ich główne przyczyny, Auobusy, 216, No. 1, pp [Accidens evens a sea and heir main causes available in Polish]. [9] Limnios N., Oprisan G., Semi-Markov Processes and Reliabiliy, Springer-Birkhauser, Boson Scienific Journal of PNA Zeszyy Naukowe AMW

11 Nonhomogeneous sochasic processes conneced o Poisson process [1] Shipping in he Balic Sea Pas, presen and fuure developmen relevan for Mariime Spaial Planning, Projec Repor I, Balic LINes 216. [11] Shiryayev A. N., Probabiliy, Springer-Verlag Berlin Heidelberg, New York Tokyo N I E J E D N O R O D N E P R O C E S Y S T O C H A S T Y C Z N E Z W I Ą Z A N E Z P R O C E S E M P O I S S O N A STRESZCZENIE W arykule przedsawiono wybrane uogólnienia procesu Poissona i ich własności. Skupiono się na dwóch uogólnieniach niejednorodnym procesie Poissona i niejednorodnym złożonym procesie Poissona. Niejednorodny proces Poissona pozwala na skonsruowanie modelu probabilisycznego opisującego liczbę różnych rodzajów wypadków. Niejednorodny złożony proces Poissona pozwala maemaycznie opisywać konsekwencje ych wypadków. Przedsawione u wyniki eoreyczne dają możliwość przewidywania liczby wypadków i ich konsekwencji. Słowa kluczowe: niejednorodny proces Poissona, niejednorodny złożony proces Poissona, charakerysyki bezpieczeńswa. Aricle hisory Received: Reviewed: Revised: Acceped: (213)