On a Z-Transformation Approach to a Continuous-Time Markov Process with Nonfixed Transition Rates

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1 Ge. Mah. Noes, Vol. 24, No. 2, Ocobe 24, pp ISSN ; Copyigh ICSRS Publicaio, 24 Available fee olie a hp:// O a Z-Tasfomaio Appoach o a Coiuous-Time Maov Pocess wih Nofixed Tasiio Raes T.A. Aae, S.O. Edei 2, P.E. Oguude 3 ad O.A. Odeumibi 4,2,3,4 Depame of Mahemaics, School of Naual & Applied Scieces College of Sciece & Techology, Covea Uivesiy, Oa, Nigeia 2 soedei@yahoo.com (Received: / Acceped: ) Absac The pape peses z-asfom as a mehod of fucioal asfomaio wih espec o is heoy ad popeies i dealig wih discee sysems. We heefoe obai he absolue sae pobabiliies as a soluio of a diffeeial equaio coespodig o a give Bih-ad Deah pocess via he z-asfom, ad deduce he equivale saioay sae pobabiliies of he sysem. Keywods: Maov Pocesses, Bih-Deah Pocess, Z-Tasfom, Geeaig Fucio, Chaaceisic Diffeeial Equaio. Ioducio I mahemaical scieces, egieeig, physics ad ohe fields of applied scieces, vaious foms of asfoms such as iegal asfoms, Laplace asfom, Fouie asfom, ec ae used, depedig o he poblem-whehe discee o coiuous case. Discee sysems cao be sudied usig he Laplace o he Fouie asfom because hey ae coiuous fucios. Eve whe he coiuous Fouie asfom ca be coveed o is equivale i discee fom

2 86 T.A. Aae e al. by fis fidig he Discee Fouie Tasfom (DFT), he Fouie asfom iself does o sui he discee sysems. Such sysems ca easily be modeled usig he z-asfom []. The z-asfom coves a sequece of eal o complex umbes io a complex fequecy domai epeseaio. I ca be cosideed as a discee-ime equivale of he Laplace asfom. Z-asfoms ae o diffeece equaios wha Laplace asfoms ae o diffeeial equaios. The idea of z-asfom was fis ow o Laplace ad was lae ioduced by W. Huewicz as a coollable way of solvig liea, cosa-coefficie diffeece equaios [2] & [3]. I mahemaical lieaue, he idea coaied i z-asfom is also efeed o as a mehod of geeaig fucios as ioduced by de Moive wih egads o pobabiliy heoy [4]. Bih-ad-deah pocesses ae examples of Maov pocesses ha have bee widely sudied. They ae used i he aalysis of sysems whose saes ivolve chages i he size of some populaio- such as he sudy of populaio exicio imes i biological sysems, he evoluio of gees i livig higs [5], ad also have bee used i he chaaceizaio of ifomaio soage ad flow i compue sysems [6]. Kedall i [7] gives a complee soluio of he equaios goveig he geealized bih-ad-deah pocess, i which he bih ad deah aes ae ay specified fucios of he ime. Yechiali [8] cosides a queuig-ype bih-addeah pocess defied o a coiuous ime Maov chai wih emphasis o he seady sae egime, ad he ecommeds umeical mehods fo obaiig limiig pobabiliy sice closed-fom soluios ae difficul o obai; if hey eve exis. Kudaeli i [9] used he z-asfom o aalyze a fiie sae bih-deah Maov pocess i deivig he pefomace meics of he sysem ad hei vaiaio wih he sysem paamees. I [], he exeded he esuls i [9] o aalyze a bih ad deah pocess i which he bih ad deah asiio pobabiliies ca vay fom sae o sae. I his pape, he z-asfom is sudied ad applied o a bih-ad-deah pocess wih asiio aes, ad absolue sae pobabiliies ae heefoe obaied as soluios of he coespodig diffeeial equaio. The pape is sucued as follows: secio 2 deals wih he heoy ad cocep of z-asfom, secio 3 is o basic pocesses, ad secio 4 hadles he applicaios ad coclusio.

3 O a Z-Tasfomaio Appoach o a 87 2 Theoy ad Cocep of Z-Tasfom Defiiio 2.: Le ( ) { } { } = o = g = g g = g be a sequece of ems wih Z ad z a complex umbe such ha: {, ( 3 ), ( 2 ), ( ), ( ), ( ), ( 2 ), ( 3 ), ( 4 ), } g = g g g g g g g g, he he z - asfom of g is defied as: ( ) ( ) 3 2 { } { } ( 3) z ( 2) z ( ) z () z G z = Z g = Z g = g + g + g + g { } T = T = () z + (2) z + (3) z + (4) z + g g g g Z g( ) = g( ) z = g( ) z + g( ) z T () = = = Equaio () is efeed o as a wo-sided o a bilaeal z-asfom of g. Suppose g is defied oly fo, he () becomes: { } G( z) = Z g( ) = g( ) z (2) T = We efe o (2) as he uilaeal z -asfom of g. Defiiio 2.2: Le g( ) be he pobabiliy ha a discee adom vaiable aes he value, ad he fucio G( z ) e-wie as G (s) wih sz =, he (2) becomes G(s) = g( )s (3) = whee (3) is efeed o as he coespodig pobabiliy geeaig fucio. Defiiio 2.3: Regio of Covegece (ROC). The Regio of covegece (ROC) is he se of pois i he complex plae fo which he z-asfom summaio coveges. Evey z-asfom is defied ove a ROC. Thus; ROC = z : g( ) z < (4) = Rema: The sequece oaio { } diffeece equaios while ( ) pocessig. g g = = is used i mahemaics o sudy { } g = g is used by egiees fo sigal =

4 88 T.A. Aae e al. 2. Popeies of he z-tasfom Le h = { h} ad b = { b} be wo sequeces such ha H ( z) Z { h} B z Z { b } ad as cosas, he: ( ) = wih 2 i. Lieaiy Popey: { ± 2 } = { } ± { 2 } = Z { h } ± Z { b } Z h b Z h Z b 2 = H ( z) ± B(z) 2 ii. Scalig Popey (Chage of Scale): z Z { h( ) } = H Rema: Suppose he ROC of { h( )} is z R z < R iii. Shifig Theoem (Delay o Advace Shif): { } { } Z h( ) = z H ( z) (delay shif) ad Z h( + ) = z H ( z) (Advace shif) iv. Agume as Muliplie (Muliplicaio by ): = ad <, he he ROC of { ( )} Z h is: d Z { h( ) } = z H ( z) dz Hece, d Z { h( ) } = z H ( z), dz We ema hee ha z ( z) i -imes. d dz d dz bu a epeiive opeaio of d z dz The poofs of he saed popeies, heoem ad ohe coceps such as covoluio ad ivese asfom ae foud i [] & [4].

5 O a Z-Tasfomaio Appoach o a 89 3 Basics Pocesses This secio deals wih he ioducio of some basic pocesses ad coceps eeded i he emaiig pa of he wo. 3. Maov Pocess Defiiio 3.: A sochasic pocess { X ( ), N } is called a Maov chai if, fo all ime N ad fo all saes ( i, i, i2, i3,, i ) wih P {( )} a pobabiliy fucio; { + = + =, =, 2 = 2, 3 = 3,, = } P{ X + i+ X i} P X i X i X i X i X i X i = = = (5) Tha is, he fuue sae of he sysem depeds oly o he pese sae (ad o o he pas saes). Codiio (5) is efeed o as Maovia popey. Ay sochasic pocess wih such popey is called a Maov pocess [] & [2]. Defiiio 3.2: Coiuous-ime Maov chai. A sochasic pocess { X ( ), } wih a paamee se τ ad a discee sae space Z is called a coiuous ime Maov chai o a Maov chai i coiuous ime if fo ay, < < < < < < < ad (,,,,,, ) i i i2 i3 i i +, i Z we have ha: { ( + ) = + ( ) =, ( ) =, ( 2) = 2, ( 3) = 3,, ( ) = } = P{ X ( + ) = i+ X ( ) = i} (6) P X i X i X i X i X i X i Noe 3.: The codiioal pobabiliies; { } P ( s, ) = P X ( ) = j X ( s) = i, s <, i, j Z ae he asiio pobabiliies of ij he Maov chai. Oe example of he coiuous ime Maov chai is he Bih-Deah pocess. Hece, he followig coceps. 3.2 Bih-ad- Deah Pocesses Le X ( ) = be he populaio size i a ifiiesimal ime ieval (, + ) wih a populaio e chage X (, + ) = X ( + ) X ( ) i (, + ). We suppose λ as he pobabiliy ha a idividual gives bih i (, + ) ad µ as he pobabiliy of a deah i (, + ).

6 9 T.A. Aae e al. A bih-deah pocess has he popey ha he e chage acoss a ifiiesimal ime ieval is eihe (fo bih), (fo deah) o (fo eihe bih o deah). We heefoe mae he followig assumpios: A : { } 2 P X ( ) = = λ + = λ + A : { } P X ( ) = = µ + = µ + Theefoe, fo a case of eihe bih o deah, we have: A : { } 3 P X ( ) = = ( λ + µ ) + = ( λ + µ ). Deoe P ( ) as he pobabiliy of idividuals i a ime ieval of legh. Hece, by applyig he law of oal pobabiliy, we have: { } P ( + ) = P idividuals i ad eihe bih o deah i Thus, { idividuals i ad a bih i } + P { idividuals i ad a deah i } + P + ( ) ( λ µ ) ( ) λ ( ) µ ( ) P ( + ) = P + + P + P (7) Fo µ i = iµ ad λi = iλ, i, such ha P P{ X } () = () = =, (7) becomes: ( ) ( λ µ ) ( ) λ ( ) µ ( ) P ( + ) = P + + ( ) P + ( + ) P + + Showig ha; ( ) λ ( ) µ ( ) λ µ ( ) P ( + ) P = ( ) P + ( + ) P ( + ) P ( ) + (8) + Dividig boh sides of (8) by ad aig limi as yields: ( ) ( ) ( ) P = ( ) λ P + ( + ) µ P ( λ + µ ) P ( ), (9) + P () = maes o sese sice is absobig, hece P ( ) = µ P ( ) () Theefoe, a liea bih-ad-deah pocess saisfies he sysem of diffeeial equaios (9)-().

7 O a Z-Tasfomaio Appoach o a 9 Rema 3.: Fom (9), if µ =, he we have: ( ) λ ( ) P = ( ) P λp ( ), () Similaly, λ = i (9) gives: P = ( + ) µ P µ P ( ), (2) ( ) ( ) + Equaios () ad (2) ae efeed o as liea-bih-pocess ad liea-deahpocess especively. 4 Applicaios: The Z-Tasfom o a Bih-Deah Pocess I his secio, we coside a bih-ad-deah pocess wih asiio aes [7], [2] & [3]: λ = λ ad µ i = iµ, i =,, 2,3,4, i ad iiial disibuio: { } P () = P X () = = We heefoe subjec he sysem o a z-asfom i ode o obai he absolue sae pobabiliies, P ( ) ad he saioay sae pobabiliies, Π whee ( ) Π = lim P (3) + This is doe as follows; usig he asiio aes λi = λ ad µ i = iµ, i i (9) yields a coespodig sysem of diffeeial equaio fo : ad ( ) λ ( ) ( λ µ ) ( ) ( ) µ ( ) P = P + P + + P (4) + ( ) λ ( ) µ ( ) P = P + P (5) We ivoe he z-asfom (pobabiliy geeaig fucio) i (3) e-defied as: Φ (, s) = P ( )s (6) =

8 92 T.A. Aae e al. wih Φ (, s) = such ha: Φ(, s) = P ( )s ad = Φ(, s) ( )s = P (7) s = To subjec (4) o (6) ad (7), we muliply boh sides of (4) by summaio fom o, hece: s ad ae he P ( ) s = λ P ( ) s + ( + ) µ P + ( ) s ( λ + µ ) P ( ) s (8) = = = = Adjusig he limi (idex) of he fis wo seies (wih = m + ad = especively) i he RHS of (8) gives: Φ(, s) m Pm s P s P ( ) s P ( ) s m+ = = = = + = ( ) + ( ) λ µ λ µ Sice P v( ) = fo v ad =, we heefoe wie: s s s (, ) ( ) m ( ) s s P s P s P ( ) s Φ λ µ λ µ P ( ) s = + s m m= = = = m ( ) ( ) = λ ( s ) P s µ ( s ) P s m m= = Φ(, s) = λ ( s ) Φ(, s) µ ( s ) (fo m = = ) s Φ(, s) Φ(, s) + µ ( s ) = λ( s ) Φ(, s) s (9) Equaio (9) has a coespodig sysem of chaaceisics diffeeial equaios give below: ds = µ ( s ) (2) d ad dφ(, s) = λ( s ) Φ (, s) (2) d I ode o obai Φ (, s), (2) ad (2) will be solved simulaeously, hus, fom (2),

9 O a Z-Tasfomaio Appoach o a 93 ds s = µ d ad ds ( s ) =, such ha l( s ) µ = c (22) µ d As such (2) becomes: dφ(, s) λ = λ( s ) d = ds Φ(, s) µ λ λ l Φ(, s) s = c2, bu fo ζ =, µ µ c2 = l Φ(, s) ζ s (23) Sice ad 2 fucio saisfied by Φ (, s) such ha: c c ae abiay cosas, we assume ( ) h i a abiay coiuous Hece, ( ) h c = c (24) 2 h(l( s ) µ ) = l Φ(, s) ζ s Showig ha: l Φ (, s) = h(l s µ ) + ζ s Φ (, s) = exp h(l s µ ) + ζ s h(l s µ ) ζ s = e e (25) Applyig he iiial codiio Φ (,s) = o (25), yields: h(l s ) s = e ζ e such ha: h(l s ) = ζ s (26) Leig η = l s i (26) implies ha ( s ) = e η o s = + e η So, h( η) = η( + e η ) (27) l s h(l s µ ) = ζ ( + e e µ )

10 94 T.A. Aae e al. = ζ ( + ( s ) e µ ) µ µ = ζ ( + se e ) Showig ha: Φ (, s) = e µ µ ζ ( + se e ) + sζ = e µ µ sζ ( ζ ζ se + ζ e ) e µ µ se e s = e e e e ζ ζ ζ ζ ( e µ ζ ) ζ s( e µ ) = e e (28) Seig v = ζ ( e µ ) i (28) gives: v Φ (, s) = e e sv ( ) ( ) ( ) ( ) ( ) e v sv sv sv sv sv = ! 2! 3! 4! 5! v ( sv) = e =! Showig ha v ( v) e Φ (, s) = s (29)! = Compaig (29) wih (6) gives: µ v µ ζ ( e ) ( v) e ( ζ ( e )) e P ( ) = =!!, (3) I addiio, ζ ζ e Π = lim P ( ) = +! (3) Equaios (3) ad (3) ae he absolue sae pobabiliies ad he saioay sae pobabiliies of he sysem especively. Equaios (3) is a Poisso disibuio wih iesiy (ed) fucio λ( ) = ( ζ ( e µ )). Noe: I some cases, Φ(,s) cao be easily expaded as a powe seies i s ; hece, he absolue sae pobabiliies ca be compued by diffeeiaig Φ (,s).

11 O a Z-Tasfomaio Appoach o a Discussio of Resul Fo he pupose of discussio of esul, we sudied he goveig paamees ζ, µ, ad fo some umeical calculaios wih ζ = 9.5, µ =.2, ad >. The esuls ae show i Table ad Fig. below: Table : The pobabiliies a ime Absolue sae pobabiliies Saioay sae pobabiliies E E-5.33 Figue : Gaphical epeseaio of he pobabiliies Key: Seies - Absolue sae pobabiliies ad Seies 2- Saioay sae pobabiliies

12 96 T.A. Aae e al. 4.2 Cocludig Remas We have explicily aalyzed he effeciveess of he z-asfom ad is popeies i hadlig discee sysems. I is show ha he absolue sae pobabiliy deceases as ime iceases. The cosideed bih-ad-deah pocess is of gea impoace i queuig heoy, biological scieces, ad i he aalysis of sysems whose saes ivolve chages i populaio sizes. Acowledgeme We would lie o expess sicee has o he aoymous eviewe (s) ad efeee (s) fo hei cosucive ad valuable commes. Refeeces [] K. Ashe, Z asfom, Ieaioal Joual of Applied Mahemaics & Saisical Scieces, 2(2) (23), [2] E.R. Kaasewich, Time Sequece Aalysis i Geophysics (3d ed), Uivesiy of Albea, (98), [3] J.R. Ragazzii ad L.A. Zadeh, The aalysis of sampled-daa sysems, Tas. Am. Is. Elec. Eg., 7() (952), [4] E.I. Juy, Theoy ad Applicaio of z-tasfom Mehod, Joh Wiley & Sos P, (964). [5] G.P. Kaev, Y.I. Wolf, F.S. Beezovsaya ad E.V. Kooi, Gee family evoluio: A i-deph heoeical ad simulaio aalysis of o-liea bih-deah-iovaio models, BMC Evol. Biol., 4(24), 32. [6] L. Kleioc, Queuig Sysems Theoy (Vol. ), Joh-Wiley & Sos, New Yo, (975). [7] D.G. Kedall, O he geealized bih-ad-deah pocess, A. Mah. Saisics, 9() (948), -5. [8] U. Yechiali, A Queuig-ype bih-ad-deah pocess defied o a coiuous-ime Maov chai, Opeaio Reseach, 2(2) (973), [9] H.N. Kudaeli, The z-asfom applied o a bih-deah maov pocess, Joual of Sciece ad Techology, 28(2) (28), [] H.N. Kudaeli, The z-asfom applied o a bih-deah pocess havig vayig bih ad deah aes, Joual of Sciece ad Techology, 3() (2), [] K.A. Boovov, Elemes of Sochasic Modellig, Wold Scieific, Sigapoe, (23). [2] M. Kijima, Maov Pocesses fo Sochasic Modelig, Chapma & Hall Lodo, New Yo, (996). [3] F. Beichel, Sochasic Pocesses i Sciece, Egieeig ad Fiace, Chapma ad Hall/CRC, (26).