Joural of Applied Mathematic ad Computatioal Mechaic 6, 5(), 97- wwwamcmpczpl p-issn 99-9965 DOI: 75/jamcm6 e-issn 353-588 THE POTENTIALS METHOD FOR THE M/G//m QUEUE WITH CUSTOMER DROPPING AND HYSTERETIC STRATEGY OF THE SERVICE TIME CHANGE Yuriy Zherovyi, Bohda Kopyto Iva Frao Natioal Uiverity of Lviv Lviv, Uraie Ititute of Mathematic, Czetochowa Uiverity of Techology Czętochowa, Polad yuzherovyi@lueduua, bohdaopyto@impczpl Abtract We propoe a method for determiig the probabilitic characteritic of the M/G//m queueig ytem with the radom droppig of arrival ad ditributio of the ervice time depedig o the queue legth Two et of ervice mode, with the ervice time ditributio fuctio F ( x ) ad F ( x) repectively, are ued accordig to the twothrehold hyteretic trategy The Laplace traform for the ditributio of the umber of cutomer i the ytem durig the buy period ad for the ditributio fuctio of the legth of the buy period are foud The developed algorithm for calculatig the tatioary characteritic of the ytem i teted with the help of a imulatio model cotructed with the aitace of GPSS World tool Keyword: igle-chael queueig ytem, radom droppig of cutomer, hyteretic trategy for ervice time, potetial method, tatioary characteritic Itroductio Studie how [-3] that the radom droppig of arrival i a powerful tool for parameter cotrol of a queueig ytem The droppig ca ot oly regulate the queue legth, lo probability of cutomer, waitig time, ad queue legth variace, but alo regulate everal of thee parameter imultaeouly I order to icreae the ytem capacity, threhold trategie of the ervice iteity (ervice time) chage are ued i queueig ytem I the geeral cae, the eece of thi trategy i that the ervice time ditributio deped o the umber of cutomer i the ytem at the begiig of each cutomer ervice [4] With the help of the potetial method, we have developed a efficiet algorithm for computig the tatioary ditributio of the umber of cutomer i the ytem with threhold fuctioig trategie [4-8]
98 Y Zherovyi, B Kopyto I thi paper we coider the M/G//m queueig ytem with radom droppig of cutomer ad ditributio of the ervice time depedig o the queue legth Two et of ervice mode (the mai mode ad overload mode), with the ervice time ditributio fuctio F ( x ) ad F ( x) repectively, are ued accordig to the two-threhold hyteretic trategy The overload mode with the fuctio F ( x) tart fuctioig if at the begiig of ervice of a cutomer, the umber of cutomer i the ytem atifie the coditio > h The retur to the mai mode with the fuctio F ( x ) carried out at the begiig of ervice of the cutomer, for which h, h < h < m = where Each arrivig cutomer ca be accepted for ervice with a probability depedig o the queue legth We aig thi probability accordig to the rule: if at the time of the arrival of a cutomer cutomer are i the ytem, the the cutomer i accepted for ervice with probability β ad leave the ytem (i dicarded) with probability β Fix a threhold value h ( h< m) ad uppoe that β = for h, ad β = β (< β < ) for h m I paper [9] we alo tudied the queueig ytem with radom droppig of arrival I cotrat to thi article, i [9] the probability β doe ot chage durig the time iterval from the begiig to the completio of ervice of each cutomer Baic radom wal We coider the M/G//m ytem, where m i the maximum umber of cutomer i the queue Let λ be a parameter of the expoetial ditributio of the time iterval betwee momet of arrival of cutomer Suppoe that, if at the begiig of ervice of a cutomer the umber of cutomer i the ytem i equal to {,,, m }, the the ervice time of thi cutomer i a radom variable with ditributio fuctio F ( x) ( x ) for the mai mode ad F ( x) ( x ) for the overload mode Deote by P the coditioal probability, provided that at the iitial time the umber of cutomer i the queueig ytem i equal to {,,,, m }, ad by E (P) the coditioal expectatio (the coditioal probability) if the ytem tart to wor at the time of arrival of the firt cutomer Let η ( x) be the umber of cutomer arrivig i the ytem durig the time iterval [; x ) Let i () ( i) ( λx) ( λ) x λ i! f ( ) = f ( ) = e df ( x), f ( ) = e df ( x), i=,,, ; i () ( i) ( λx) ( λ) x λ i! f ( ) = f ( ) = e df ( x), f ( ) = e df ( x), i=,,, ;
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy 99 M = xdf ( x) <, M = xdf ( x) <, F ( x) = F ( x), F ( x) = F ( x) For Re ad {,,, h} coider the equece π i( ), qi( ) ad R ( ), defied by the relatio: i η f( ) P π ( ) = e { ( x) = i } df ( x), i {,,,, m }; π, m = η f( ) P ( ) e { ( x) m } df ( x); i P η q ( ) = e { ( x) = i} F ( x) dx, i {,,,, m };, m P η q ( ) = e { ( x) m } F ( x) dx; R ( ) = f ( ) ( ) π, () Similarly, for Re ad { h, h,, m} we et the equece π i( ), q i( ) ad R ( ): i η f( ) P π ( ) = e { ( x) = i } df ( x), i {,,,, m }; π, m = η f( ) P ( ) e { ( x) m } df ( x); i P η q ( ) = e { ( x) = i} F ( x) dx, i {,,,, m };, m = P η q ( ) e { ( x) m } F ( x) dx; R ( ) = f ( ) ( ) π, () Let T ad T deote the expoetially ditributed radom variable with parameter λ ad λ= λβ repectively, ad Z i a radom variable ditributed accordig to the law of Pacal, that i, P{ Z = } = β ( ), =,, It i ow [], β that ZT = T, that i, a a reult of a radom decimatio of the implet flow we obtai a implet flow
Y Zherovyi, B Kopyto Give the above, for h m we fid ( j λx) λx P{ η( x) = j} = e, j m ; j! j ( λx) P{ η( x) m } = { ( x) j} e j! For h we obtai j ( λx) λx P{ η( x) = j} = e, j h ; j! m m λx P η = = j= j= { } P { η( x) = j} = P ( h ) T ( h j ) T < x< ( h ) T ( h j) T = = G ( x) G ( x), h j m ; h, h j h, h j m P { η( x) m } = P { η( x) = j} =, r, j= P{ } { T rt x} G ( x) = P < = { T x} = ( h ) T ( m h) T < x = G ( x); r = h, m h r r ( ) λ λ = Cr ( r )!( λ λ) ( )!( r )!( λ λ) r j j i i j ( j)!( λ λ) λx ( λx)! x ( λ λx) ( ) e e, j j= j! λ i= i! λ i= i! r m h; G ( x) = P < λx ( λx) = e! = Taig ito accout the expreio for P { η ( x) = j} ad equalitie i ( λx) ( λ) x λ λ ( i) e F ( x) dx= g ( λ) = f ( ), λ! ( λ) i= λ i ( λx) ( λ) x λ λ ( i) e F ( x) dx= g ( λ) = f ( ), λ! ( λ) i= λ by (), () calculate the term of the equece π ( ), q ( ), π ( ) ad q ( ) i i i i
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy For h h we fid π ( ) =, q ( ) = g ( λ), j m ; ( j) f ( λ), j j j f( ) f ( λ) π ( ) = f ( λ), q ( ) = g ( λ) m m ( j), m, m j f( ) j= λ j= (3) For h h ad h h we obtai π π ( j) f ( λ), j j j f( ) ( ) =, q ( ) = g ( λ), j h ;, j = ( h, h j h, h j ) f( ) ( ) e G ( x) G ( x) df ( x), j ( h, h j h, h j ) q ( ) = e G ( x) G ( x) F ( x) dx, h j m ; π, m = e Gh, m h x df x q, m = e Gh, m h f( ) ( ) ( ) ( ), ( ) ( x) F ( x) dx; e G ( x) df ( x) = f ( ) C ( r )! r r ( ) λ λ, r r r ( )!( r )!( λ λ) = r j j j ( j)!( λ λ) ( i)! ( i) ( λ λ) ( ) f ( ) ( ), j λ f λ j= j! λ i= λ i= r m h; ( ), = e G ( x) df ( x) = f ( ) f ( λ); (4) f ( ) ( ) e G ( x) F ( x) dx= C ( r )! r r λ λ, r r r ( )!( r )!( λ λ) = r j j j ( j)!( λ λ)! ( λ λ) ( ) g ( ) ( ), j i λ g i λ j= j! λ i= λ i= r m h; f( ) e G,( x) F ( x) dx= g ( λ) =
Y Zherovyi, B Kopyto Expreio for π ( ) ad q ( ) are imilar with preeted i (3) ad (4) for i i h h or h h, ad for h h repectively, if we ( j) ( j) replace f, f, g j ad F by f, f, g j ad F Note that lim f ( ) = lim f ( ) =, lim = M, lim = M f ( ) f( ) (5) Itroduce the otatio: π = lim π ( ), q = lim q ( ), R = lim R ( ), i i i i π = lim π ( ), q = lim q ( ), R = lim R ( ) i i i i With the help of equalitie ()-(5) we ca obtai expreio for the member of the equece π i, q i, R, π i, q i ad R 3 Ditributio of the umber of cutomer i the ytem durig the buy period ( F, Let ξ ( t) be the umber of cutomer i the ytem at time t Deote by P, P ) the coditioal probability, provided that at the iitial time the umber of cutomer of the ytem i equal to ad the ervice begi with the ervice time ditributed accordig to the law F ( x ) ( F ( x)) Let τ = if{ t : ξ( t) = } deote the legth of the firt buy period for the coidered queueig ytem, ad for {,,, m } F ψ ( t, ) = P { ξ( t) =, τ > t}, h ; F, ψ ( t, ) = P { ξ( t) =, τ > t}, h m ; F, ψ ( t, ), h; ϕ( t, ) = ψ ( t, ), h m ; t t ϕ ψ Φ (, ) = e ( t, ) dt, Φ (, ) = e ( t, ) dt, Re> It i evidet that ϕ ( t, ) =, ψ h ( t, ) = ϕ (, ) h t With the help of the formula of total probability we obtai the equalitie:
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy 3 m ϕ ( t, ) = P { η( x) = j} ϕ ( t x, ) df ( x) j j= t P { η( x) m } ϕ ( t x, ) df ( x) I{ m } P { η( t) = } F ( t), h ; m ψ ( t, ) = P { η( x) = j} ψ ( t x, ) df ( x) ψ j j= t P { η( x) m } ψ ( t x, ) df ( x) t m t m m m m m I{ m } P { η( t) = } F ( t), h m; t ( t, ) = ψ ( t x, ) df ( x) I{ = m } F ( t) (6) Here I{ A } i the idicator of a radom evet A; it equal or depedig o whether or ot the evet A occur Itroduce the otatio: f (, ) = I{ m } q ( ), f (, ) = I{ m } q ( ) ( ), ( ), Taig ito accout the relatio () ad (), from (6) we obtai the ytem of equatio for the fuctio Φ (, ) ad Φ (, ) : m Φ (, ) = f ( ) π ( ) Φ (, ) f (, ), h ; (7), j j ( ) j= m Φ (, ) = f ( ) π ( ) Φ (, ) f (, ), h m; (8), j j ( ) j= f m ( ) Φ m (, ) = f m ( ) Φ m(, ) I{ = m } with the boudary coditio Φ (, ) =, Φ (, ) =Φ (, ) (9) h h
4 Y Zherovyi, B Kopyto For olvig the ytem of equatio (7)-(9) we will ue the fuctio R i( ) ad R i( ), defied by the recurrece relatio: j R ( ) = R ( ); R, j ( ) = R ( ) R, j( ) f( ) π, i( ) R i, j i( ), i= h, j m ; j R ( ) = R ( ); R, j ( ) = R ( ) R, j( ) f ( ) π, i( ) R i, j i( ), i= h m, j m () Itroduce the otatio: h C ( ) = R ( ) R ( ) f ( ) π ( ), C ( ) = R ( ) ;, h i i i, h i h h, i= h m D ( ) = Ri ( ) f i( ) π i, m i( ) π i, j i( ) A j( ) ; i= j= h m A ( ) = R ( ) R ( ) f ( ) π ( ); D ( ) = D (, ) ;, m i i i, m i h h i= h m m j D(, ) = Ri ( ) f( i) (, ) f i( ) π i, j i( ) ju( ) f( j u) (, ) R i= j= h u= Reaoig a i the proof of Theorem of [7], we obtai the followig tatemet Theorem For all {,,, m } ad Re> the equalitie ( ( ) ) Φ (, ) = C ( ) D (, ) C ( ) D ( ) C ( ) D ( ) Φ (, ) m C( ) m m i ( i) i= D (, ), h ; Φ (, ) = A ( ) Φ (, ) R ( ) f (, ), h m ; m Φ (, ) = A ( ) Φ (, ) R ( ) f (, ), h h ; m i ( i) i= f m ( ) Φ m (, ) =Φ m (, ) = f m ( ) Φ m(, ) I{ = m }
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy 5 are fulfilled, where m h C ( ) Dh (, ) ( ) hi f( h i) (, ) Ch ( ) D (, ) R i= Φ m(, ) = C ( ) D ( ) C ( ) D ( ) A ( ) ( ) h h h 4 Buy period ad tatioary ditributio If the ytem tart fuctioig at the momet whe the firt cutomer arrive, the t e P{ ξ( t) =, τ > t} dt=φ(, ) = ( ( ) m ) = C( ) D (, ) C( ) D( ) C( ) D ( ) Φ (, ) D (, ) C ( ) () To obtai a repreetatio for t e P { τ > t} dt we um up equalitie () for ruig from to m Give the defiitio of f( ) (, ), f ( ) (, ), q ( ) ad q ( ), it i ot difficult to acertai that j f ( ) (, ) (, ) ( ), ; m m m f( ) = f( ) = q j = h = = j= f ( ) f (, ) = f (, ) = q ( ) =, h m m m m ( ) ( ) j = = j= Itroduce the otatio: h m m j f ( ) ( ) i f j u D( ) = Ri ( ) f i( ) π i, j i( ) ju( ) ; R i= j= h u= m h f h ( ) i C( ) Dh ( ) ( ) ( ) hi Ch D ( ) R i= Φ m( ) = C ( ) D ( ) C ( ) D ( ) A ( ) ( ) h h h Thu, () cofirm the followig tatemet j
6 Y Zherovyi, B Kopyto Theorem The Laplace traform of the ditributio fuctio of the legth of the buy period i defied a t e P{ τ > t} dt= ( ( ) m ) = C( ) D( ) C( ) D( ) C( ) D( ) Φ ( ) D( ) C ( ) () To fid P{ τ > t} dt= E ( τ) we eed to pa to the limit i () a We ue the equece π i,, obtaied by limit paage: π ad R, a well a equece R i ad R i, = lim R i( ), R i = lim R i( ) For R i ad R i R i R () implie the recurrece relatio: i j R = R ; R, j = R R, j π, ir i, j i, i= h, j m ; j R = R ; R, j = R R, j π, ir i, j i, i= h m, j m (3) Note that π h, = Rh = R h,= Uig the relatio (3) ad taig ito accout the equalitie m m π =, h ; π =, h m; j j j= j= we ca prove that R R m R π =, h ;, m i i, m i i= m R π =, h m, m i i, m i i= (4) Give (5) ad (4), uig () we obtai the followig tatemet
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy 7 Theorem 3 The mea legth of the buy period i determied i the form where m h E( τ) = D D R( h, h) Dh, himh i R (5) i= h m m j, h, h D = i M i π i, j i jum j u ; R( h, h ) = R R R R i= j= h u= Rh, h h Itroduce the otatio: lim P { ξ( t) = } = p, {,,,, m } Reaoig t a i the paper [4], from () we obtai formula for the tatioary ditributio of the umber of cutomer i the ytem Theorem 4 The tatioary ditributio of the umber of cutomer i the ytem i give by p= ; λe( τ) p= λp Rq ( R iqi, i Riq i, i ), h; i= i= ( ) h p= λp iqi, i R( h, h ) hiqh i, h i hiqh i, h i R R R i= i= p p Riq i, i, h h; h = λp D( ) D ( ) R( h, h ) hiq h i, h ( ), i Dh R i= h m, m = m h = λp D( m ) D ( m ) R( h, h ) hiq h i, m h ( ), i Dh m R i= where h j D( ) = Ri q i, i I{ h m} π i, j i juqj u, j u R i= j= h u= (6)
8 Y Zherovyi, B Kopyto Uig (5) we fid the ratio of the mea umber of cutomer erved per uit of time to the mea umber of all arrivig cutomer per uit time ad obtai the formula for the tatioary ervice probability m h Pv = p T T R( h, h) hi T, h R (7) i= where T h m m j = R π R i i, j i ju i= j= h u= We fid the tatioary queue characteritic - the average queue legth E ( Q) ad average waitig time E ( W) - by the formula m E( Q) E( Q) = p, E( W) = λ = Pv (8) 5 Example for calculatig of tatioary characteritic Aume that m= 6, λ=, h =, h = h= 4, β = 4, the uiform ditributio o the iterval (; 5] ad (; 5] correpod to the ditributio fuctio of the ervice time F x F x ( h ) ( ) = ( ) ad F ( x) = F( x) repectively Thu, ( h ) m 5y 5 ( ) y ( ) 4 M= M = 5, M = M = 5, f ( y) = e, f ( y) = e y y The row " p " of Table cotai teady-tate probabilitie p, calculated by the formula (6) For the ae of compario, the ame table cotai the correpodig probabilitie evaluated by the GPSS World imulatio ytem [, ] 6 for the time value t= The value of the tatioary characteritic foud by the formula (5), (7) ad (8), ad calculated with the help of GPSS World, are how i Table
The potetial method for the M/G//m queue with cutomer droppig ad hyteretic trategy 9 Statioary ditributio of the umber of cutomer i the ytem Table Number of cutomer 3 4 5 6 7 p 373 54 5757 35 4539 33756 5697 5369 p (GPSS World) 37 56 5768 35 457 3373 5693 548 Statioary characteritic of the ytem Table Characteritic E(τ) E(Q) E(W) P v Aalytical value 674 35 54 65 Value accordig to GPSS World 6894 35 54 649 Calculatio how that if the radom droppig of cutomer i ot ued, the for the coidered data the value of the capacity of the ytem P v i icreaed by 3%, but E ( Q) ad E ( W) are alo icreaed by 56% ad % repectively 6 Cocluio With the help of the potetial method, we have obtaied imple ad uitable formula for umerical realizatio for fidig the tatioary characteritic of the M/G//m queueig ytem with the radom droppig of cutomer ad hyteretic chage of the ervice time We have examied a fairly geeral tatemet of the problem, becaue we aume that the ervice time deped o the umber of cutomer i the ytem ad the droppig probability i a fuctio of the queue legth at the time of arrival of a cutomer Our approach, ulie mot of the method ued to tudy the emi-marov model of queueig, allow to ivetigate ot oly tatioary, but alo the traiet regime of the ytem, i particular, to fid the Laplace traform for the ditributio of the umber of cutomer i the ytem durig the buy period ad for the ditributio fuctio of the legth of the buy period Referece [] Chydzińi A, Nowe modele olejowe dla węzłów ieci paietowych, Pracowia Komputerowa Jaca Salmieriego, Gliwice 3 [] Tihoeo O, Kempa WM, Queue-ize ditributio i M/G/-type ytem with bouded capacity ad pacet droppig, Commuicatio i Computer ad Iformatio Sciece 3, 356, 77-86
Y Zherovyi, B Kopyto [3] Kempa WM, A direct approach to traiet queue-ize ditributio i a fiite-buffer queue with AQM, Applied Mathematic ad Iformatio Sciece 3, 7, 3, 99-95 [4] Zherovyi KYu, Zherovyi YuV, M θ /G//m ad M θ /G/ ytem with the ervice time depedet o the queue legth, Joural of Commuicat Techology ad Electroic 3, 58,, 67-75 [5] Zherovyi Yu, Zherovyi K, Potetial Method for Threhold Strategie of Queueig, LAP Lambert Academic Publihig, Saarbrüce 5 (i Ruia) [6] Zherovyi KYu, Statioary characteritic of the M θ /G//m ytem with the threhold fuctioig trategy, Joural of Commuicat Techology ad Electroic, 56,, 585-596 [7] Zherovyi Yu, Kopyto B, The potetial method for a cloed queueig ytem with hyteretic trategy of the ervice time chage, Joural of Applied Mathematic ad Computatioal Mechaic 5, 4(), 3-43 [8] Zherovyi YuV, Zherovyi KYu, Method of potetial for a cloed ytem with queue legth depedet ervice time, J of Commuicat Techology ad Electroic 5, 6,, 34-347 [9] Zherovyi Yu, Kopyto B, Zherovyi K O characteritic of the M θ /G//m ad M θ /G/ queue with queue-ize baed pacet droppig, Joural of Applied Mathematic ad Computatioal Mechaic 4, 3(4), 63-75 [] Wetzel ES, Ovcharov LA, Theory of Stochatic Procee ad it Egieerig Applicatio, Vyhaya Shola, Mocow (i Ruia) [] Zherovyi Yu, Creatig Model of Queueig Sytem Uig GPSS World, LAP Lambert Academic Publihig, Saarbrüce 5 [] Boyev VD, Sytem Modelig, Tool of GPSS World, BHV-Peterburg, St Peterburg 4 (i Ruia)