with finite mean t is a conditional probability of having n, n 0 busy servers in the model at moment t, if at starting time t = 0 the model is empty.

Transcription

1 INFINITE-SERVER M G QUEUEING MODELS WITH CATASTROPHES K Kerobya The fe-erver qeeg model M G, BM G, BM G wh homogeeo ad o-homogeeo arrval of comer ad caarophe are codered The probably geerag fco (PGF) of jo drbo of mber of by erver ad dffere ype erved comer, a well a he Laplace-Selje Traform (LST) of drbo of by perod ad drbo of by cycle for he model are fod Keyword: qee model, by perod, caarophe, by cycle Irodco The fe-erver model M G wh Poo arrval of comer ad geeral ervce me drbo [- 3] ha bee wdely ed dffere feld, ch a mahemacal ecology, bology, face, phyc, relably heory, dral egeerg, raffc egeerg, raporao, ec, de o her followg valable propere: he aoary drbo of he model eve o he form of ervce me drbo; he deparre proce of erved comer follow Poo drbo [4] The fe-erver model M G oe of he poplar qeg model There are mber of geeralzao of h model for dffere arrval procee: Bach Poo Procee, Marov Arrval Procee (MAP), Phae Type Procee (PH), Geeral (G) or Reewal Procee (RN) [4-7] Trae drbo of he mber of by erver M M model ad rae probable of correpodg Marov proce were derved by Rorda [5] I [6], Sevayaov preeed h famo heorem of evy of aoary drbo of M G model He proved ha aoary drbo of h model eve (varace) o he hape of ervce-me drbo I [7], Bee carred o he rae probable of M G model By g probablc mehod ad embedded Marov cha mehod, Taac [8] derved he probably geerag fco (PGF) of he mber of by erver for M G ad G M model Ug dffereal eqao ad Laplace raformao (LT) [9], Shabhag derved he jo drbo of he mber of by erver, he mber of erved comer ad oal occpao me of erver a mome for he model BM G wh bach arrval of comer I [], Brow ad Ro geeralzed hee rel by coderg BM G model wh o-homogeeo Poo arrval of comer ad amed ha boh he bach ze ad ervce me drbo mgh deped o he arrval me By g probablc mehod, hey derved PGF of he mber of by erver ad he mber of erved comer I [], Klmov codered may fe-erver model ad fod her ma performace dce, eg rae probable of Marov proce for he M G model ad he egral eqao for PGF of qee of he model G G Qee ze drbo of he model G G ad BG G wh geeral drbo of arrval ad ervce procee have bee codered by L [2] By perod drbo for M G ad G G model have bee derved [3] Model of PH G qee wh Phae Type arrval have bee codered by Ne ad Ramawam [4] The Bac Syem of Dffereal Eqao for qee ze ad marx expoeal olo are fod The dffereal eqao for he PGF of qee ze ad mber of erved comer of he model BMAP G wh homogeeo ad o-homogeeo Bach Marov Arrval of comer have bee derved by Breer [5], Kerobya [6] I [7] Tog fod he PGF of mber of by erver for he model M G wh ype of Poo arrval of comer Th rel wa geeralzed by Mayama ad Tae [8] for MAP G model whe -ype of comer arrve accordg o Marov Arrval Proce By Nazarov, Moeev ad all [9-2] have bee codered feerver qeeg model wh geeral ervce me drbo ad dffere arrval procee HM G, MMP M, MAP G, ad HIGI G der codo of hgh ey raffc By g arrval proce pecal hg procedre, Kolmogorov dffereal eqao, ad aympoc mehod hey defed fr ad ecod order approxmao for PGF of qee ze However, h mehod cao be appled o he model wh bach arrval procee

2 For may applcao mpora o evalae he ferece of evromeal parameer o her performace meare For example, QoS of moble ewor ha hgh correlao wh exeral ad eral evromeal parameer Moreover, he relably of cable ha hgh ferece o hroghp of chael ad lo of reqe wred ewor [22] To evalae he QoS of yem whe he ferece of evrome ha daro or caarophe characer, for example lo of commcao, falre of all chael or erver of he yem, aaeo deah of all pece of poplao, collape of facal orgazao or race compay, vr aac of comper erver ec, he qeeg model wh caarophe are ed Compreheve revew o qeeg model wh caarophe ad egave gal ca be fod Aralejo [22], Do [23], Bocharov [24] Mo of he arcle are dedcaed o M M N, G M ad M G qeeg model wh caarophe Ife-erver qeg model M M wh caarophe have bee codered by Chao [25], Boh [26], Ecoomo ad Fao [27], D Crecezo ad all [28] The aoary ad rae drbo of qee ze were obaed by g Marov Procee, Reewal Procee, dal procee, ad embedded procee The model of M G ad BM G wh homogeeo ad o-homogeeo arrval of caarophe are carred o Kerobya [29] The bac dffereal eqao, her olo ad LT of by perod ad by cycle of model are fod The qeeg model MMAP G wh homogeeo ad o-homogeeo Mared MAP arrval of dffere ype of comer ad caarophe codered by Kerobya [3] The bac yem of dffereal eqao for PGF of he model, her olo, he PGF of jo drbo of qee ze ad he vecor of mber of already olved comer are fod The qeeg model MMAP G wh homogeeo Mared MAP arrval of dffere ype of comer em-marov evrome, wh reorce vecor of comer ad caarophe codered by Kerobya [3] The bac yem of dffereal eqao for PGF of he model, her olo, he PGF of jo drbo of accmlaed reorce, qee ze, vecor of oal erved reorce are fod The geeral mehod of modelg of qeeg model radom evrome ad caarophe codered I he pree paper, we coder he model M G, BM G ad BM G wh aoary ad rae arrval of comer ad caarophe The bac dffereal eqao of he model ad her aoary ad rae olo are fod For hee model he jo drbo of qee ze ad mber of erved comer ad her mome are fod The drbo of by perod ad by cycle of model wh aoary arrval of comer ad caarophe have bee vegaed 2 o Model decrpo Le coder a fe-erver model M G wh caarophe where he comer ad caarophe arrve o he model accordg o a Poo drbo wh parameer λ ad ν, repecvely The ervce me of he comer are depede decally drbed (d) radom varable (rv) whch have a geeral drbo fco (DF) B( ) = P( ) wh fe mea b vale The model ha fe mber of erver, ad he ervce of arrvg comer ar mmedaely If a caarophe occr whe he model by, he all comer he model are mmedaely lo If a caarophe occr whe he model empy, he dappear who ay coeqece Le he rv N( ), be he mber of comer he model a mome, N ( ) = {,,2,} Sppoe ha a arg me = he model empy, N () = Le rodce probable P ( ) = P{ N( ) = N() = },, where P() a codoal probably of havg, by erver he model a mome, f a arg me = he model empy Theorem The codoal probable of he model () P afy followg Kolmogorov dffereal eqao

3 d P ( ) = [ ( B ( T )) + v ] P ( ) + v d d P ( ) = [ ( B ( T )) + v ] P ( ) + ( B ( T )) P ( ), d wh al codo P () =, P () =, Proof Le coder poble chage of ochac proce N () drg he me erval ( +, ) Arrvg a mome comer ll be he yem a mome wh probably S ( ) = B( T ) ad compleed ervce ad lef he model before mome T wh probably S( ) = B( T ) By he adard femal mehod Tjm [3] for probable P () we derve T d P ( ) = S ( ) P ( ) + v P ( ) d = d P ( ) = [ S ( ) + v ] P ( ) + S ( ) P ( ), d () wh al codo P () =, P () =, The probable P () afy followg codo: P ( ),,, From codo (2) we derve Afer bo (3) o () for probable we ge P () = (2) = P ( ) = P ( ) (3) = d P ( ) = [ S ( ) + v ] P ( ) + v d d P ( ) = [ S ( ) + v ] P ( ) + S ( ) P ( ), d (4) wh al codo P () =, P () =, 3 o Model Aaly Trae Probable of he Sae P () To olve he yem of dffereal eqao (4), we e probably geerag fco (PGF) [3,, 35] Le P( z, ) be a PGF for mber of by erver he model a mome, P( z, ) = z P ( ), z = Theorem2 The geerag fco P( z, ) afe he followg bac dffereal eqao d P ( z, ) = [ S ( )( z ) + v ] P ( z, ) + v, z (5) d

4 wh al codo Pz (,) = The olo for P( z, ) or gve by [ ( ( ))( ) ] B x z + v [ ( B( x))( z) + v] P( z, ) = e + v e d, z (6) [ ( ( ))( ) ] B x z + v [ ( B( x))( z) + v] (, ) P z = e + v e d The fr h order mome m () afe m () of mber of by erver ca be fod from (5) I well ow [3] ha m ( ) = P( z, ) z z = For m () from (5) we ge he followg dffereal eqao d m ( ) + vm ( ) = S ( ), d d m 2( ) + vm 2( ) = 2 S ( ) m ( ), d (7) d m ( ) + vm ( ) = S ( ) m ( ) d Wh al codo: m() =, m2() =,, m () = The olo of (7) have form v( x) ( ) ( ), m = S e (8) v( x) ( ) = ( ) ( ) m S m x e x x2 v vx = m ( )! e S ( x ) S ( x ) S ( x ) e Parclarly, for fr wo mome of mber of by erver from (8) we fd v( x) m ( ) S( x) e =, (9) The probable P () of he model ca be fod by x2 2 v vx 2( ) 2 ( 2) ( ) 2 m e S x S x e = ()

5 Theorem3The rae probable of he model d P ( ) = P( z, ) z= dz P () are gve by ( ) v () [ ( ( ) ( )) + v( )] ( ) = + [ ( ) ( )]!! P e v e d () where ( ) = ( B( x)) ( B( x)) b = b, = b Le coder ome performace meare for he model M D wh caarophe f x b, Bx ( ) = f x b ( ) ( + v) [ + v]( )] [ ( )] e + v e d f b,!! P () = b ( b) b v [ ( b ) + v( )] [ ( b )] e + v e d f b,!! v e, f b, m () = v, f b, ( e ) e f b m2 () = v v, f b v 2 v 2 2,, I may applcao, for example moble comper ewor ad elecommcao ewor, comer ad caarophe arrval have o-aoary are [2, 6, 34] Th fac lead o e he o-aoary Poo procee o model he arrval of comer ad caarophe To defe he PGF P( z, ), a well a he rae probable P (), fr ad ecod mome of a mber of by erver m() ad m () 2 a mome he model for o-aoary arrval of caarophe, (6), (9), () ad () we have o chage v o v () [ ( ( ))( ) ( )] B x z + v x [ ( B( x))( z) + v( x)] P( z, ) = e + v( x) e d, z (2) v( ) d x m ( ) = ( B( x)) e (3) 2 v( ) d x m ( ) = 2 ( B( x)) m ( x) e

6 v = ( ) ( ) v x [ ( ( ) ( )) + v( x) ] () ( ) = + ( ) [ ( ) ( )]!! P e v e d (4) Whe, we have qeg model M G wh fe mber of chael ad who caarophe From (6), (9), ad (), we derve he rel ha are well ow qeg heory [3, 8] (, ) = [ ( B( x))( z)], P z e z ( B( x)) P! e ( B( x)) (, ) =, m ( ) = ( B( x)), m2 ( ) = ( B( x)) 2 I may applcao, o defe he mber of chael eceary for ramo of a gve amo of formao he drbo of he mber of erved comer o me erval ad mome ca be ed Le rv M () be he mber of comer erved drg me erval [, ) Defe he jo drbo of rv M () ad N () erved o [, ) Le Pm() be a probably ha here are m erval: P ( ) = P( N( ) = m, M ( ) = ),, m m [, ) by erver a mome, ad comer are Le P( z, y, ) be a jo PGF of he mber of by erver a mome [, ) erval ad mber of erved comer m = m= m P( y, z, ) = z y P ( ), z, y Theorem 4 The geerag fco of ochac proce ( N( ), M ( )) afy he followg bac dffereal eqao (,, ) = [ + ( ) ( ( ))] (,, ) + B( x)( z),, P y z v zb y B P y z ve z y (5) wh al codo P( y, z,) =, z, y Proof Le coder rao of ochac proce ( N( ), M ( )) drg he me erval ( +, ) By he adard mehod Tjm [3] we ca wre he correpodg Kolmogorov dffereal eqao: d P ( ) = ( + v ) P ( ) + v P ( ) d = d P ( ) = ( + v ) P ( ) + S ( ) P ( ) + v P ( ), d m m=

7 d P m( ) = ( + v ) P m( ) + S ( ) P m ( ), m d d P m ( ) = ( + v ) P m( ) + S ( ) P m ( ) S ( ) P m ( ),, m + d (6) wh al codo P () =, P () =,, m m The from (6) for PGF P( y, z, ) we ge he followg dffereal eqao P( y, z, ) = [ + v zs( ) ys ( )] P( y, z, ) + vp(, z, ), z, y (7) wh al codo P( y, z,) =, z, y From he followg dffereal eqao, we fd he PGF P(, z, ) P(, z, ) = S( )( z) P(, z, ), z, (8) wh al codo P( z,,) =, z P(, z, ) Noe ha a PGF of he mber of erved comer drg me erval [, ) (8), doe o relae o he proce of caarophe ad, a follow from The olo for dffereal eqao (8) wh correpodg al codo ha a form of P(, z, ) e = B( x)( z), z (9) Theorem 5 The olo P( y, z, ) of bac dffereal eqao (5) gve by { ( [ ( ( )) ( )]) } y B x + zb x + v [ ( B( x))( y) + v] P( y, z, ) = e + ve d, z, y (2) For he model wh o-aoary arrval of caarophe he PGF P( y, z, ) gve by { ( [ ( ( )) ( )]) ( )} y B x + zb x + v x [ ( B( x))( y) + v( x)] P( z, y, ) = e + v( ) e d, z, y (2) Corollary From (5) ad (9), follow ha PGF P( y, z, ) ca be preeed a facor form P( y, z, ) = P(, z, ) P( y,, ) Here, P( y,, ) a PGF of he mber of by erver, ad P(, z, ) a PGF of a mber of comer erved drg [, ) erval I ca be how ha qeeg model M G wh caarophe he oly oe whch allow h decompoo From (2) whe v =, for P( y, z, ) we receved well ow rel for he model M G who caarophe Maveyev ad Uhaov [35]:

8 (,, ) = [ zb( x) y( B( x))],, P y z e y z Remar The rel ca be geeralzed for he model BM() G wh bach arrval of comer Sppoe he bache of comer arrve accordg o o-homogeeo Poo proce wh parameer [8,] The mber of comer he bach depede, ad decally drbed (d) rv wh a drbo q ( ) = P(, = r), r =,2, ad wh PGF r r= () r Q( z, ) = z q ( ) The ervce me of comer are d rv wh geeral drbo fco B () ad fe mea b Le coder he model wh o-aoary Poo arrval of caarophe wh parameer Le N () a mber of by erver a mome v () P( y, z, ) r be he PGF of ochac proce ( N( ), M ( )), where ad M () a mber of erved comer erval [, ) By g adard femal argme we ca defe he PGF of jo drbo of rv Theorem 6 The PGF P( y, z, ) of he model B()M G gve by { ( )[ ( ( ( )) ( ))] ( )} x Q y B x + zb x + v x { ( x)[ Q( y( B( x)) + zb( x))] + v( x)} P( y, z, ) = e + ve d, z, y (22) Ug (22) we ca defe may performace merc of he model: he probable Pm() of havg m by erver a mome ad erved comer erval, he mome of by erver ad erved erval [, ) comer, ad probably of dle ae of he model P () I parclar, for he model wh v( ) = v, ( ) = for probably P() we ge [, ) N () ad M () [ ( ( ( ))) ] Q B x + v [ ( Q( B( x))) + v] P ( ) = e + v e d (23) Remar 2 Le coder he qeg model M G wh ype of comer ad caarophe whch geeralze he Tog [7] rel The arrval of comer ad caarophe follow he Poo drbo wh parameer, =,2,, ad v Servce me of -h ype of comer d rv wh geeral drbo B ( ), =,2,,, ad fe fr mome Sppoe ha a arrval epoch, he caarophe deroy all he b comer he model f he model by Le rv N ( ), a mber of ype comer he model a mome, N ( ) = {,,2,}, =,2,, ad rv M () a mber of erved ype comer erval [, ) A al me =, he model empy, N () =, M () =, =,2,, Le rodce followg oao ad defo: z = ( z, z2,, z ), y = ( y, y2,, y ), N( ) = ( N( ), N2( ),, N ( )), 2 = (,,, ), M( ) = ( M( ), M2( ),, M ( )), m = ( m, m2,, m ), Pm( ) = P{ N( ) =, M( ) = m N() =, M() = },, m, =,2,,, where Pm () a codoal probably of he eve: he model here are comer of he fr ype, 2 comer of he ecod ype,, comer of he -h ype a mome, ad m comer of he fr ype, m 2 comer of he ecod ype,, m comer of he -h ype are erved erval [, ), f a al mome = he model wa empy ad here are o ay erved comer

9 Deoe by he jo PGF of mber of by erver comer erved erval [, ) M () P( z, y, ) N () he model a mome ad mber of m P( z, y, ) = z z z y y y P ( ), z, y = = m= m= 2 m m2 2 2 Theorem 6 The PGF P( z, y, ) of he model M G gve by [( ( )) ( ) ] B x z + B x y + v [( B ( x)) z + B ( x) y ] + v = = P( z, y, ) = e + v e d (24) Proof By g adard femal argme for PGF of ochac proce ( bac dffereal eqao m N () P( y, z, ) = [ + v zs( ) ys ( )] P( y, z, ) + vp(, z, ), z, y wh al codo P( y, z,) =, z, y The olo for dffereal eqao (8) wh correpodg al codo ha a form of, M () ) we ca wre he P(, z, ) e B( x)( z) =, z From (24) we ca defe he probably P (), he mome of he mber of each ype comer he m model a mome, mber of each ype comer erved erval [, ), ad probably of dle ae of he model P() I parclar, for P() we fd ( ( )) B x + v ( B ( x)) = = P ( ) = e + v e d (25) Remar 3 Now le coder he qeg model BM G wh bach arrval of dffere ype of comer ad caarophe The arrval of caarophe ad comer follow a Poo drbo wh parameer v ad ( ), =,2,,, =,,2,, where he mber of comer he bach, ( ) = P( = ) The mber of -h ype comer he bache d rv wh a drbo qr = P( = r), r =,2, ad r wh PGF Q ( z ) = z q r r= Servce me of -h ype comer d rv wh geeral drbo B ( ), =, 2,,, ad fe fr mome b Sppoe ha a arrval epoch, he caarophe deroy all he comer he model Le rv N ( ), a mber of -h ype comer he model a mome, N( ) = {,, 2,}, =, 2,, A al me =, he model empy, N () =, =,2,, For h model he PGF of jo drbo of rv N () ad () M yz = 2 2 P(,, ) P( y, y,, y, z, z,, z, ) ca be fod

10 P e ve d { [ ( ( ( )) ( ))] } Q y B x + z B x + v { [ Q ( y ( B ( x)) + zb ( x))] + v} = = ( yz,, ) = + From PGF mome P( yz,, ) ca be defed he PGF of mber of erved comer ad mber of by erver a P e ve d { [ ( ( ( )) ( ))] } Q y B x + B x + v { [ Q ( y ( B ( x)) + B ( x))] + v} = = ( y,, ) = + P e ve d { [ (( ( )) ( ))] } Q B x + z B x + v { [ Q (( B ( x)) + zb ( x))] + v} = = (, z, ) = + For h model, we ca defe alo o-aoary arrval of dffere ype of comer ad caarophe The correpodg PGF of jo drbo of mber of erved comer ad mber of by erver a mome ha form P e v e d { ( )[ ( ( ( )) ( ))] ( )} x Q y B x + z B x + v x { ( x)[ Q ( y ( B ( x)) + zb ( x))] + v( x)} = = ( yz,, ) = + ( ) Le ow ppoe ha arrvg bache ca coa dffere ype of comer The model who caarophe have bee codered by mber of ahor, eg Fao [32], Tag [7], Cho ad Par [33] Le radom varable be he mber of h ype comer he bach The vecor x= ( x, x2,, x ) deoed x he bach ze ad ha a PGF 2 Q( x ) = Q( x, x,, x ) = q( x =, x =,, x = ) z z z Where q( x =, x2 = 2,, x = ) a probably ha arrvg bach coa comer of fr ype, comer of ecod ype, comer of ype The ervce me of comer are d rv ad for ype of comer have drbo B ( x ) Comer of ype arrvg mome ll be he yem a mome wh probably B ( ) ad q he model afer compleg he ervce before mome wh probably B ( ) A al mome he model empy Theorem 7 The PGF of he model gve by 2 [ Q( w( x, ))] d 2 ( z, ) = ( ( ) =, 2( ) = 2,, ( ) = ) 2 = (26) 2 P P N N N z z z e P ( z, ) = P( N ( ) =, N ( ) =,, N ( ) = ) z z z = e Q( w( x, )) d To olve h model we wll e mehod of collecve mar Le mar - color each ype comer arrvg bach depedely from oher comer he bach ad he yem wh red color by

11 probably z or ble color by probably z The Q( x) ca be erpreed a probably of eve arrvg bach doe o coa ble comer w ( z, ) = B ( ) + ( B ( )) z - a probably of eve ype comer arrvg a he mome ll he yem a mome ad ha red color, w( x, ) = ( w ( z, ), w ( z, ),, w ( z, )) The he probably of eve arrvg a he mome 2 2 bach x= (, 2,, ) whch comer ll are he yem a mome comer = = doe o coa ble color [ B ( ) + ( B ( )) z ] = w ( z, ) Probably of eve arrvg a mome bach whch comer ll are he yem a mome 2 2 = doe o coa ble color comer R( x,, ) = q(,,, ) [ B ( ) + ( B ( )) z ] = Q( w( x, )) The bache arrve accordg o Poo proce wh parameer Le each arrvg bach mar wh probably Q( w( x, )) ad mar wh probably Q( w( x, )) The, a how by Rechel [34], he plg procee are ohomogeeo Poo procee wh parameer Q( w( x, )) d ad repecvely Hece, he probably of eve o ble comer he yem a mome [ Q( w( x, ))] d P( z, ) = P( N ( ) =, N ( ) =,, N ( ) = ) z z z = e he probably of eve o red comer he yem a mome [ Q( w( x, ))] d, P ( z, ) = P( N ( ) =, N ( ) =,, N ( ) = ) z z z = e Q( w( x, )) d Where P( z, ) a PGF of he mber of comer ha have bee erved before me, ad P(, ) 2 z a PGF of he mber of comer ha erved he yem a he mome The correpodg PGF for he model wh caarophe ca be fod by followg argme [3] Le oe ha he dyamc of he model wh caarophe bewee wo coecve caarophe he ame a he dyamc of he model who caarophe Afer he caarophe he model jmp o he dle ae ad coe dyamc from ha ae a he model who caarophe If he caarophe occr accordg o Poo proce wh parameer collecve mar mehod v he he PGF of he model wh caarophe P ˆ( z, ) we defe by g P ˆ( z, ) he probably ha o ble comer he model wh caarophe a mome Ideed, h eve ca happeed f caarophe do o occr [,) ad o ble comer arrve (he probably of h eve P( z, ) e v ), or a caarophe occr a he mome x, x [, ), (he probably v ), he model jmp o dle ae ad drg he me x caarophe do o occr ad o ble comer arrve (he probably of h eve v( x) P( z, x) e v ) By g he oal probably rle for PGF P ˆ(, ) v v( x) z we ge Pˆ( z, ) = P ( z, ) e + v P ( z, x ) e (27)

12 Le P ˆ( z, ) he Laplace Traformao (LT) of PGF P ˆ( z, ) ˆ P( z, ) = e Pˆ ( z, ) d The from (9) we fd v v Pˆ( z, ) = P ( z, + v ) + P ( z, + v ) = P ( z, + v )( + ) (28) If h rel wre he form P ˆ( z, ) = P( z, + v)( + v) (29) he accordg o Kerobya [3] ad Maveyev [35] ha mple probablc erpreao Le ppoe ha depedely of he model ca occr eve A accordg o Poo proce wh parameer The he lef ad rgh de of (2) ca be erpreed a probable of followg eve: lef de; he fr eve A occrred whe he model wh caarophe are o ay ble comer, (wh probably ˆ P( z, ) = e Pˆ ( z, ) d ) rgh de; he fr eve of oal flow of caarophe ad eve A occrred whe he model who caarophe are o ay ble comer (wh probably ( + v) ( ) (, ) (, ) + v P z + v = e P z d ) Th model geeralze he rel of Fao [32], Tag [7], Cho ad Par [33] The mome of he mber of dffere ype of comer ca be fod by he adard way, ee for example Fao [32] 4 o Model Aaly: By Perod To defe he DF of by perod of he model wh caarophe, we hall e he mehod of collecve mar [, 7, 35, 36] Le be he probably ha he model empy a mome A al me = he model empy, () P() P =, ad P () a Laplace Traform (LT) of a fco P ( ) = e P ( ) d, Le rv be he legh of by perod of he model By perod beg whe he fr comer arrve a he empy model ad ed whe he model free of comer The rv be he legh of by cycle ad defed a he m of by perod ad dle perod of he model For fe-erver M G model a rv wh expoeal drbo ad parameer The Laplace-Selje Traform (LST) of by perod ˆ( ) ad by cycle ˆ( ) defe P() ˆ( ) = e d ( ), ˆ ( ) = e d( )

13 Accordg o [35] for he cla of coervave qeg model wh Poo arrval of comer, he probably ha he model empy, he drbo of by perod ad he by cycle ca be defed from he followg eqao Where P ˆ ( ) = + ( ) P( ), + + (3) ˆ ( ) = ˆ ( ) + P P e d P e v e d [ ( ( )) ] B x + v [ ( B( x)) + v] ( ) = ( ), ( ) = + (3) Noe, ha he coderg fe-erver model wh caarophe coervave Therefore, olvg (3) for by perod ad by cycle drbo, we derve ˆ( ) = + = + ( ), P ( ) P ( ) + ˆ( ) = P() (32) If dle ae probably of he model M G who caarophe oe by p () The from (3) we fd () p = e ( B( x)) [ ( B( x)) v] + ( B( x)) + v( ) P () = e e d + v e e d ( v+ ) v( ) ( ) ( ) (33) = e p d + v e p e d v = p ( v + )( + ) Th rel ca be wre he form P = ( v + ) p( v + ) ad erpreed by mlar o (29) way I parclar, whe v =, from (3) we derve he well ow rel for clacal model M G who caarophe Sadje [37]: ˆ ˆ( ) = + [ ], ( ) = [ ( B( x)) ] + [ ( B( x)) + ] e d e d + Le coder he by perod ˆ( ) ad by cycle ˆ( ) drbo for he model M D wh caarophe

14 f x b, Bx ( ) = f x b The for by perod ˆ( ) ad by cycle ˆ( ) we fod ( + + v) b v + ( + ) e P ( ) =, ( + + v) b ( + + v) e ( + vb ) v+ e lm P ( ) =, v + (34) ( + + v) b ve ˆ( ) = + +, ( v) b + ( v) e ( + + v) b ( + )( + + v) e ˆ( ) = ( + + v) b + ( v + ) e e = + ve ( + vb ), ( + vb ) ( + vb ) 2[ ( + b( + v) e ] 2 = ( + vb ) 2 [ v+ e ] (35) (36) If [38]: v =, from (9) we derve he well ow rel for clacal model M D who caarophe Nazarov + ˆ( ) =, ( b ) + + e 2 ( + ) ˆ( ) = ( + b ) + e The by perod ad by cycle of he model BM G ad M G ca be fod by g he dle ae probable of correpodg model (23) ad (25) For he LST of by perod of he model BM G ad M G we ge ˆ BM G ( ) = + [ ] [ ( Q( B( x))) v ] + + [ ( Q( B( x))) + v] e + v e d d ˆ M G ( ) = + [ ] ( B ( x)) v + + ( B ( x)) = = e + v e d d, (37) (38) The rel of h paper ca be geeralzed followg dreco by coderg model wh: Bach Marov Arrval Proce, Mared Marov Arrval Proce or more geeral arrval procee of comer ad caarophe; aoary ad rae arrval procee of comer ad caarophe; ervce me drbo of comer deped o her ype, me mome of her arrval ad a addoal parameer vecor; he model ca be codered ome radom evrome

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16 3 Kerobya K, Kerobya R Trae aaly of fe-erver qee MMAPr() Gr wh Mared MAP arrval ad daer// Proc of 7-h I Worg Cof HET-NET 23 Performace & Secry Modellg ad Evalao of Cooperave Heerogeeo Newor, -3, Nov 23, Illey, We Yorhre, Eglad, UK-P- 3 Kerobya K, Eaoa K, Kerobya R A fe-erver qeeg model MMAP G wh mared MAP arrval, em-marov radom evrome ad bjec o caarophe/proc of 8-h I Cof Comper Scece ad IT, Sraov, 28, Ra, P Fao D The Ife Server Qee wh Arrval Geeraed by a No-Homogeeo Compod Poo Proce The Joral of he Operaoal Reearch Socey P Cho BD, Par KK The M M qee wh heerogeeo comer a bach // J Appl Prob P Rechel F Sochac Procee Scece, Egeerg ad Face Chapma&Hall, Maveyev VF, Uhaov VG Qeg Syem-M: MSU, p 36 Reberg, JT O he e of Collecve mar qeeg heory, Proc Symp Cog h UNC, chapel Hll, 965 -P Sadje W The by perod of he qeeg yem M G // J Appl Prob P Nazarov A, Trepgov A Qeeg Theory Tom Sae Uvery NTL, p