Joa of Aed Mahema ad Comaoa Meha 4 3( 6-73 APPLCATON OF A Z-TRANSFORMS METHOD FOR NVESTGATON OF MARKOV G-NETWORKS Mha Maay Vo Nameo e of Mahema Ceohowa Uey of Tehoogy Cęohowa Poad Fay of Mahema ad Come See Godo Sae Uey Godo Bea m.maay@gma.om o86@gma.om Aba. The oe of h eeah ae o fd he exeed ome oe Mao qeeg ewo wh ome oe ad egae meage a ay me by he mdmeoa afomao. egao wee aed o ae whe ome fom he ewo ao bewee he ae ae deem fo o deede o ewo ae ad me. amed ha a ewo yem ae oe-e. wa oed he heoem o he exeo fo he mdmeoa -afom. A agohm wa ooed fo aao of exeed ome. aaed a exame o he PC. Keywod: oe G-ewo oe ad egae meage exeed ome -afom mehod odo Mao ewo wh ome (HM-ewo ad G-ewo ae wdey ed a mahemaa mode of ao obe ome ee ad ohe fed [-3]. Fo exame G-ewo a be ed o mode he behao of e he fomao ad eeommao yem ad ewo fo foeag exede whe oa wh e [4 5]. h ae a moa a o fd h exede he yem of he odeed ewo. Peoy wo [6] fo fdg he exeed ome of yem of HM-ewo wh may-ye meage ewo a mehod wa ooed of ee aoxmao ombed wh he mehod of ee. wa odeed a oed exoea HM-ewo wh med wag me qee qeeg meage. [7] ooed a mehod of fdg he exeed ome of yem of HM-oe qeeg ewo (QN wh a oe-e yem of QN (QS oe ad egae meage he ae whe he ome fom he ae ao ae deem ewo fo deedg o ewo ae doe o deed o me. Sh a ehqe o fd he exeed ewo ome of yem a PC fo a fe mbe of ewo ae a eaoabe CPU me.
6 M. Maay V. Nameo h ae we ode aohe mehod baed o he aao of maae -afom. f we ode o odeao mdmeoa -afom fo exeed ewo ome of yem we oba fo he eao h way a [ 8]. O he ba of hee eao we a ooe a agohm fo aag he exeed ome [9 ] fo he oe G-ewo wh ome.. QN deo. Fomao of he obem Code a oe QN G-ewo wh oe-e QS. QN S ome fom ode deede Poo fow of oe ad egae meage wh ee ad eeey. The ee me of oe meage he QS S dbed exoeay wh aamee. The ao of egae meage ae debed [-5]. Poe meage eed he QS S wh obaby deed o QS S a oe meage ad wh obaby - a egae meage ad wh obaby ( meage eae he ewo o he exea eome (QS S. The meage dg he ao fom oe o aohe QS bg yem ad ome ome of he f yem eded by h amo eeey. Code he ae whe he ome fom he ae ao ae deem fo of he ewo deedg o ewo ae. The ewo ae w be he eo... whee ( - he mbe of meage a he ( ( ( ( ( ( me mome yem S. Le ( - he oa exeed ome whh ge he yem S dg me f a he a mome he ewo a ae ad ame ha h fo dffeeabe. We ode he foowg oao: - a eo of dmeo og of eo exe fo he omoe wh mbe of whh eqa x > o ; - eo -eo; ( x - Heade fo; ( - x ome yem S a a me whe he ewo a ae. Th mea ha fo exame f he ewo a ae a mome me he he exeed ome of yem S w be ( dg he exeed ome ( ha he yem ha eeed he eo me. Le ( - ome of yem ae S whe ewo mae a ao fom ae ( ( dg ; R ( a ao fom ae ( ae ( - ome of yem f ewo mae ; ( - he ome
Aao of a -afom mehod fo egao of Mao G-ewo 63 of yem S whe he ewo mae ao fom ae ( o ( dg ; ( - ome of QS S whe he ewo hage ao fom ae ( o ( dg me. ha bee how ha he yem of dffeea eqao fo he ome ( ha he fom [4 5]: ( [ ] ( d d ( α ( [ ( ] ( [ ] whee ( [ ( ]. ( R α ( eeay o fd he exeed ewo ome of yem dg f we ow ae a he a me.
M. Maay V. Nameo 64. Aay of ome ewo by -afom mehod We ode mdmeoa -afom fo exeed ome of yem S : ( ( { /... ( <. My he yem of eqao ( o ad mmg oe a fom o. Code ome m ded o oeed yem of eqao. a be how ha ( [ ] ( [ ] ( { ( ( K (3 whee { (. K K K K K K - (4 [ ] (. (5 ( { ( [ ]. 3 (6 ( [ ] ( ( { ( [ ]. 4 (7
Aao of a -afom mehod fo egao of Mao G-ewo 65 ( ( { ( [ ]. 5 (8 ( [ ]. { 6 (9 ( { ( { ( [ ]. 7 ( ( { ( { ( [ ]. 8 ( Th g (3-( he yem of eqao ( afe he aboe afomao we go aoa. Theoem. The fo ( afe he eao ( { ( [ ] ( { ( [ ] [ ] { ( { ( [ ] ( { ( [ ] (
66 M. Maay V. Nameo { { { [ ( ( ( ] ( ( { ( { [ ( ( ] α ( [ ]. Code he a m (. Soe ha ome ( ( ( ( ( ( ( R do o deed o ewo ae.e. R ( R. Se he h ae α ( ( R [ ] (3. Le K. We ode ome oao eaed o -afom of ome of yem S : \{ \{ ( ( ( K ; ( ( ( K K ; ( ( ( K K K K ;... \{ K K ( K K K K K K... \{ K ; { ( K K. K
Aao of a -afom mehod fo egao of Mao G-ewo 67 Ad he fo agme bewee eo a o ad oy moa o K. We ee a agohm fo fdg he exeed ome g he eao (. 3. Agohm fo fdg of ome -Se: eeay o defe ( ( K \ { -Se: fd -afom (..e. -afomao of ome of yem S a mome me a he a me oaed a ae (...... To do h my he eqao (3 o. ad mmg oe a fom o. Tag o ao ha he m of (3-( a be ewe he foowg fom: ( K \ { (. (4 ( ( \ { (. (5 ( \{ ( ( (.... (6 Oboy eqa o eo f beae a. Code h m fo f he 5 ( ( ( 4 ( ( ( 3. (7
M. Maay V. Nameo 68 ad ag o ao (6 { ( ( { \ \{ 5. Ad oboy ( fo a. Smay we oba ha ( { ( ( { \ \{. (8 ( { ( ( { { ( ( { \ \ \{. (9 ( { ( ( { { ( ( \ \ \{. ( Code he exeo (3. Se he h ae ( [ ]. R R R α ( Tag o ao (4-( he yem of eqao ( afe he aboe afomao we oba fo -afom { ( \ : { ( ( (... \{ \{ \{ \ R (
Aao of a -afom mehod fo egao of Mao G-ewo 69 whee R ( defed eqao (. Fhemoe by og a yem of dffee- { ( \ a eqao ( ad exadg he fo a owe ee by we a fd he oeffe hee exao ( K \ { ( K. -Se: fd -afom. Fo h eaae a m (3-( (3 de he odo ha ; exadg he fod fo K K a owe ee fd he oeffe hee exao ( K ; og h way o -h e fdg -afom \ { K ( \{ K ( whe K K ; exao oeffe a ee owe K ge ome ( K K K fhe og o mae h e a ( -h e fd ad a -h e - -afom ( a ee owe K ; { ( whh afe ( exadg of hogh K we a fd ome ( K. 4. Exame Code a oe НМ-ewo wh egae meage og of wo QS: S ad S. The ey of he eam of oe ad egae meage ae eqa eeey: 8 3 6 4. The ey of ee meage a ewo yem ae eqa: 4. Le he ao obabe of meage be eqa eeey: 6 ad obabe ha he oe meage eed QS S e o QS S a egae meage eqa: 3. The obaby of eae meage fom he ewo o he exea eome be eqa: 7. Le he ome fom ewo ao bewee he ae be eqa: 5 R 7 R 4 3 5. Syem ( aodg o he eeed aamee of he ewo w be: ( d d 6 3 ( 4 6( 44( ( ( ( ( 4( 4( ( ( ( 9( (.7( ( 8 ( ( 4( ( ( ( 66( 6 (3
7 M. Maay V. Nameo d ( ( 4 6( 44( ( d 8 ( 4( ( ( ( ( ( ( 8( (.4( ( 4( ( 8 ( ( 9( 98 86( ( The ao ( a be ewe a foow: whee: 6 ( 6. (4 { ( { ( { 36 78 ( ( { ( { ( ( 8 ( ( { ( 4 8 { ( 3( ( ( { ( ( ( 56 6 ( ( { ( { ( { 9 ( 5 67 7 ( { 8 ( ( ( 7 4 { ( { ( ( 3 ( ( \{ ( { ( ( ( 84 39 ( ( { \{ ( ( ( ( ( (5 (6 { \{ { \{ ( ( ( ( Mae e of he agohm debed aboe o fd he exeed ome of yem S. ( \{ Deeme (. Fd -afom. Fo h we e ad my he yem of eqao (4 o ad mmg oe a fom o. Ug he m (4-( ag o ao ha \ { fom ( we oba eao fo -afom ( :.
Aao of a -afom mehod fo egao of Mao G-ewo 7 \ { ( \{ \{ 34 ( ( \{ 56 8 ( ( ( 6 \ { ( ( ( \{ \{ 86 ( 8 ( \{ 84 6 ( ( ( 39. ( ( The oo of h yem of dffeea eqao f we efy he a \{ \{ odo ( θ( ( θ ( ( V wa obaed Mahemaa ad ha he fom: \{ ( 84 34 ( 8 ( 34 4( 8 ( \{ ( e ( e ( ( 8 4 34 [( ( ( 8 ( ( 8V ( 6( ( 8V ( 6 8V ] θ( ( ( ( 84 86 84 86 8 4 86 ( ( e ( e 8 4 86 [( ( 8 ( ( ( ( 39( ( 8V 8V 8V 39 8V θ (. ( ( ( ( ] a odo θ ( θ ( afy he foowg eao \{ ( ( ( θ α * whee * α ( ohe ae \{ ( ( ( θ β * whee * β (. ohe ae
7 M. Maay V. Nameo \{ Sbg he exeo obaed fo ( (6 he Mahemaa aage fod a aaya oo of he dffeea eqao ( wh he a odo ( θ ( whee ( ( ( θ ( γ * * γ. ohe ae The exeed ome ( of yem S ae oeffe of exao of he fo ( o a dobe ee of owe ad wa oded he Mahemaa aage of mahemaa aao. * * Le V α β γ f he a odo ae: θ ( θ ( θ (. Fge hee a gah of ome S a.. 88 4 6 8 4. 88 6. 88 8. 88. 99. 99 Fg.. ome ha ( o ea [] Coo The aa moae of he e e he fa ha wh he he of he ooed mehod oe a fd he exeed ome of yem odeed ewo wh a fe mbe of ae. Refeee [] Maay M. Some e of he aay ad omao of Mao ewo wh ome ad he aao Aomao ad Remoe Coo 9 97-3 ( Ra. [] Geebe E. Pod fom qeeg ewo wh egae ad oe ome Joa of Aed Pobaby 99 8 656-663.
Aao of a -afom mehod fo egao of Mao G-ewo 73 [3] Bohao P. Vhey V. G-ewo: he deeome of he heoy of mae ewo Aomao ad Remoe Coo 3 5 46-74 ( Ra. [4] Nameo V. Maay M. Aay ewo wh oe ad egae meage a ao behao. Head of Tom Sae Uey. Maageme Come See ad foma 3 4 6-7 ( Ra. [5] Nameo V. Maay M. Aay of Mao ewo wh ome oe ad egae meage foma 4 36-44 ( Ra. [6] Koaea E. Maay M. Roo K. Fdg ome HM-ewo wh med wag me meage by ee aoxmao ombed wh he mehod of ee Head of Godo Sae Uey Yaa Kaa 3 5-3 ( Ra. [7] Nameo V. Aay of ome Mao G-ewo by mehod of ee aoxmao Head of Godo Sae UeyYaa Kaa 4 ( Ra. [8] Maay M. Thoeo O. Koaea E. Syem ad QN Newo: Aay ad Aao Godo Sae Uey Godo 88. ( Ra. [9] Maay M. Koaea E. Aay of exeed ome oe ewo g -afom Head of Godo Sae UeyYaa Kaa 8 3-9. [] Koaea E. Fdg he exeed ome he oe wo-ode HM-ewo g -afom Head of Godo Sae UeyYaa Kaa -3.
74 M. Maay V. Nameo