Joural of Applied Mathematics ad Computatioal Mechaics 5, 4(), 3-43 www.amcm.pcz.pl p-issn 99-9965 DOI:.75/amcm.5..4 e-issn 353-588 THE POTENTIALS METHOD FOR A CLOSED QUEUEING SYSTEM WITH HYSTERETIC STRATEGY OF THE SERVICE TIME CHANGE Yuriy Zherovyi, Bohda Kopytko Iva Frako Natioal Uiversity of Lviv, Lviv, Ukraie Istitute of Mathematics, Czestochowa Uiversity of Techology Częstochowa, Polad yu_zherovyi@yahoo.com, bohda.kopytko@im.pcz.pl Abstract. We propose a method for determiig the characteristics of a sigle-chael closed queueig system with a expoetial distributio of the time geeratio of service requests ad arbitrary distributios of the service times. I order to icrease the system capacity, two service modes (the mai mode ad overload mode), with the service time distributio fuctios F( x ) ad F ( x ) respectively, are used. The overload mode starts fuctioig if at the begiig of service of the ext customer the umber of customers i the system ξ( t ) satisfies the coditio ξ ( t) > h. The retur to the mai mode carried out at the begiig of service of the customer, for which ξ ( t) = h, where h < h. The Laplace trasforms for the distributio of the umber of customers i the system durig the busy period ad for the distributio fuctio of the legth of the busy period are foud. The developed algorithm for calculatig the statioary characteristics of the system is tested with the help of a simulatio model costructed with the assistace of GPSS World tools. Keywords: queueig system with a fiite umber of sources, hysteretic strategy for service time, potetials method, statioary characteristics. Itroductio Closed queueig systems are widely used as models to evaluate characteristics of the iformatio systems, data etworks ad queueig processes i productio, trasport, trade, logistics ad service systems []. The closed system is also called the system with a fiite umber of sources or Egset system. Suppose that a sigle-chael queueig system receives service requests from m idetical sources. Each source is alterately o ad off. A source is off whe it has a service request beig served, otherwise the source is o. A source i the o-state geerates a ew service request after a expoetially distributed time (the geeratio time) with mea / λ. The sources act idepedetly of each other.
3 Y. Zherovyi, B. Kopytko The service time of a service request has a arbitrary distributio. A service request, that is geerated whe chael is occupied, waits i the queue. The arrival rate of customers for a closed system at time t depeds o the umber of customers i the system ξ ( t ) ad is equal to λ( mξ ( t )). Such a flow of customers is called the Poisso flow arrivals of the secod kid [, p. 35]. I the case of a expoetial service time distributio, the formulas are kow for the statioary distributio of the umber of customers for a sigle-chael closed system [, p. 37], ad for a multichael closed system [3, p. 34]. I [4] usig the method of embedded Markov chai, a algorithm is costructed to determie the statioary distributio of the umber of customers for a sigle-chael closed system with a arbitrary distributio of the service time. I order to icrease the system capacity, threshold strategies of the service itesity (service time) chage are used i queueig systems. I the geeral case, the essece of this strategy is that the service time distributio depeds o the umber of customers i the system at the begiig of each customer service [5, 6]. With the help of the potetials method, we have developed a efficiet algorithm for computig the statioary distributio of the umber of customers i the systems with threshold fuctioig strategies [5-], icludig a sigle-chael closed system with service time, depedet o the queue legth [5]. The potetials method is a geeralizatio of a approach proposed by V. Korolyuk [] for the study of a lower semicotiuous radom walk to the case of several radom walks. The potetial method with a sigle basic radom walk was used to α study the M /G//N system with a sigle fixed service time distributio []. For the study of systems with several modes of operatio, it is ecessary to use as may basic radom walks ad their potetials as there are differet modes of fuctioig. Sice the arrival rate of customers for a closed system depeds o the umber of customers i the system, usig the potetials method eve i the case of a covetioal closed system with a sigle fuctio of the service time distributio F( x ), for each value of the umber of customers ξ ( t) = {,, K, m }, it is ecessary to cosider a selected basic radom walk. I this paper we apply the potetials method to study a sigle-chael closed system with two-threshold hysteretic switchig of the service times. Two service modes (the mai mode ad overload mode), with the service time distributio fuctios F( x ) ad F ( x ) respectively, are used. The overload mode starts fuctioig if at time t of the begiig of service of the ext customer, the umber of customers i the system satisfies the coditio ξ ( t) > h. The retur to the mai mode carried out at the begiig of service of the customer, for which ξ ( t) = h, where h < h < m.. Basic radom walks Deote by P the coditioal probability, provided that at the iitial time the umber of customers i the queueig system is equal to {,,, K, m }, ad
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 33 by E (P) the coditioal expectatio (the coditioal probability) if the system starts to work at the time of arrival of the first customer. Let η( x ) be the umber of customers arrivig i the system durig the time iterval [; x ). Let sx ( ) = ( ), = ( ) < ; sx f s e df x M xdf x f = e df ( x), M = xdf ( x) <, F( x) = F( x), F ( x) = F ( x). For Re s ad {,, K, h } cosider the sequeces π i ( s ), qi( s ) ad R ( s ), defied by the relatios: sx π i = { ( ) } ( ), {,,,, }; ( ) e P η x = i + df x i K m f s () q = e { ( x) = i} F( x) dx, i {,,,, m }; R =. f π sx i P η K Similarly, for Re s ad { h +, h +, K, m } we set the sequeces π i ( s ), q i ( s ) ad R ( s ) : sx π ( ) = { ( ) } ( ), {,,,, }; ( ) P η = + i s e K x i df x i m f s () q = e { ( x) = i} F( x) dx, i {,,, K, m }; R =. f π sx P η i The sequece π i ( s ) ( π i ( s )) with s > ad a fixed ca be treated as the distributio of umps of some lower semicotiuous radom walk defied by the distributio fuctio F( x ) ( F ( x )) of the correspodig mode of service ad probabilities P { η ( x) = i + }. Let T iλ deote a expoetially distributed radom variable with parameter i λ. The we have,, ( m) λ ( m) λx P { η( x) = } = P{ T > x} = e, {,, K, m}; P { η( x) = } = P T( mi) λ < x < T( mi) λ = i= i= ( mi) λ P ( mi) λ i= i= k = = P T > x T > x = ( m k) (3)
34 Y. Zherovyi, B. Kopytko ( mi) λx i+ e ( ), {,, K, m }, {,, K, m }; i!( i)! i= m m i i iλx P{ η( x) = m } = P T( mi) λ < x = + ( ) Cme, i= i= {,, K, m }. With regard to (3) the expressios () for π i ( s ) ad qi( s ) are reduced to the form f (( m ) λ + s) π, = ; f π π, i+ f (( m i) λ + s) = ( m k) ( ), f i!( i)! k= i= m i i, m m λ i= k = i= {,, K, m }; f (( m ) λ + s) = + ( ) C f ( i + s); q = ; f ( m ) λ + s q = ( m k) i= i+ f (( m i) λ + s) ( ), i!( i)!(( m i) λ + s) {,, K, m }; m f i i f ( iλ + s) q, m = + ( ) Cm, s iλ + s where {,, K, h }. Similarly, from () usig (3) we obtai: f (( m ) λ + s) π, =, { h +, h +, K, m}; f i+ f (( m i) λ + s) π, = ( m k) ( ), f i!( i)! k = i= { h +, h +, K, m }, {,, K, m }; m i i π, ( ) = + ( ) m s C ( λ + ), {, m f i s, K, m }; f i= f (( m ) λ + s) q =, { h +, h +, K, m}; ( m ) λ + s + i f (( m i) λ + s) q = ( m k) ( ), i!( i)!(( m i) λ + s) k = i= { h +, h +, K, m }, {,, K, m }; m f i i f ( iλ + s) q, m = + ( ) Cm, { h +, h +, K, m }. s iλ + s i= (4) (5)
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 35 Note that lim f ( s ) = lim f ( s ) =, lim =, lim =. f M f M s + s + s + s + We itroduce the otatio: π = lim π, q = lim q, R = lim R, i i i i s + s + s + π = lim π, q = lim q, R = lim R. i i i i s + s + s + With the help of equalities ()-(6) we ca obtai expressios for the members of the sequeces π i, q i, R, π i, q i ad R. Note that π m, =, q = m M, R m =. 3. Distributio of the umber of customers i the system durig the busy period Deote by P F, ( P F, ) the coditioal probability, provided that at the iitial time the umber of customers of the system is equal to ad the service begis with the service time distributed accordig to the law F( x ) ( F ( x )). Let τ ( m) = if{ t : ξ ( t ) = } deote the legth of the first busy period for the cosidered queueig system, ad for k {,, K, m } ψ ( t, k) = P { ξ ( t) = k, τ ( m) > t}, h ; F, ψ ( t, k) = P { ξ ( t) = k, τ ( m) > t}, h + m; F, ψ ( t, k), h ; ϕ ( t, k) = ψ ( t, k), h + m; st st Φ ( s, k) = e ϕ ( t, k) dt, Φ ( s, k) = e ψ ( t, k) dt, Re s >. h h It is evidet that ϕ ( t, k ) =, ψ ( t, k) = ϕ ( t, k ). With the help of the formula of total probability we obtai the equalities: (6) m t ϕ ( t, k) = P { η( x) = } ϕ ( t x, k) df( x) + + = + I{ k m} P { η( t) = k } F( t), {,, K, h }; m t ψ ( t, k) = P { η( x) = } ψ ( t x, k) df ( x) + + = + I{ k m} P { η( t) = k } F ( t), { h +, h +, K, m}. (7)
36 Y. Zherovyi, B. Kopytko Here I{ A } is the idicator of a radom evet A; it equals or depedig o whether or ot the evet A occurs. Itroduce the otatio: f ( s, k, m) = I{ k m} q, f ( s, k, m) = I{ k m} q ( s ). ( ), k ( ), k Takig ito accout the relatios () ad (), from (7) we obtai the system of equatios for the fuctios Φ ( s, k ) ad Φ ( s, k ) : m Φ ( s, k) = f π Φ + ( s, k) + f ( s, k, m), {,, K, h }, (8), ( ) = m Φ ( s, k) = f π Φ ( s, k) + f ( s, k, m), { h +, h +, K, m}, (9), + ( ) = with the boudary coditios Φ ( s, k) =, Φ ( s, k) = Φ ( s, k). () h h For solvig the systems of equatios (8)-() we will use the fuctios Ri( s ) ad R, defied by the recurrece relatios: i R = R+ ; R, + = R+ R+, f π +, i R++ i, i, i= {,,, K, h }, {,, K, m }; () R = R ; R +, + = R R + +, f +, i R π ++ i, i, i= { h, h +, h +, K, m }, {,, K, m }. Itroduce the otatio: h C = R f R π, C = R ;, h i + i, hi h h, i= h m D =f R π R, D ; i + i, i, m h i= = h+ h D ( s, k) = R f ( s, k, m) i ( + i) i= h m f R i i= = h+ m π R f ( s, k, m), D ( s, k). + i, i u ( + u) h u=
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 37 Theorem. For all k {,, K, m } ad Res > the equalities ( ( ) Φ ) Φ ( s, k) = C D ( s, k) + C D C D ( s, k) () m C m, m m i ( + i) i= m D ( s, k), h ; Φ ( s, k) = R Φ ( s, k) R f ( s, k, m), h + m ; Φ ( s, k) = R Φ ( s, k) R f ( s, k, m), h + h, m m i ( + i) i= are fulfilled, where ( ) h h, m h h (3) mh Ch D ( s, k) + C R h f ( h+ i) ( s, k, m) Dh ( s, k) i= Φ m ( s, k) =. (4) C D + C R D Proof. Sice Φ ( s, k) = Φ ( s, k) for { h +, h +,, K, m }, the the equalities (3) ca be writte as m Φ ( s, k) = R Φ ( s, k) R f ( s, k, m), h + m. (5), m m i ( + i) i= Usig the method of mathematical iductio, equatios (9), relatios (), ad arguig as i the proof of Theorem of [5], we obtai the proof of equalities (5). The system of equatios (8) ca be writte as h m Φ ( s, k) f π Φ ( s, k) = f π Φ ( s, k) +, + + = = h + f ( s, k, m), h. ( ) (6) We have received the equalities (5) from the equatios (9). Similar argumets allow us to obtai from (6) the followig relatios: Φ ( s, k) = R Φ ( s, k), h h h m (7) Ri f + i, i ( s, k) + f( + i) ( s, k, m), h. π Φ i= = h After defiig with the help of (5) all Φ ( s, k ) for { h +, h +,, K, m }, the system (7) is rewritte as Φ ( s, k) = C Φ ( s, k) + D Φ ( s, k) D ( s, k), h. h m
38 Y. Zherovyi, B. Kopytko Usig the boudary coditios (), we ca fid As a result, we obtai the equality Φ h ( s, k ) ad Φ (, ). m s k Φ ( s, k) = D ( s, k) D ( s, k) ( Φ m ) h C ad relatios (), (4). The theorem is proved. 4. Busy period ad statioary distributio If the system starts fuctioig at the momet whe the first customer arrives, the st e P{ ξ ( t) = k, τ ( m) > t} dt = Φ ( s, k) = ( ( ) Φm ) = C D ( s, k) + C D C D ( s, k) D ( s, k). C (8) To obtai a represetatio for st e P{ τ ( m) > t} dt we sum up equalities (8) for k ruig from to m +. Give the defiitios of f( ) ( s, k, m ), f( ) ( s, k, m ), q( s ) ad q ( s ), it is ot difficult to ascertai that f f ( s, k, m) = f ( s, k, m) = q =, {,, K, h }; m m m ( ) ( ) k= k = = s ( ) (,, ) = f s f s k m f ( s, k, m) = q =, { h +, h +, K, m}. m m m ( ) ( ) k= k = = s Itroduce the otatio: h h m m f f D = R f R π R ; Φ i i + i, i u s i= s i= = h+ u= m mh f Ch D + C R h ( ) ( ) i s Dh s s i= =. C D + C R D ( ) h h, m h h Thus, (8) cofirms the followig statemet.
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 39 Theorem. The Laplace trasform of the distributio fuctio of the legth of the busy period is defied as st e P{ τ ( m) > t} dt = ( ( ) Φm ) = C D + C D C D D. C (9) To fid P { τ ( m) > t} dt= E τ ( m ) we eed to pass to the limit i (9) as s +. We use the sequeces π i, R, π i ad R, as well as sequeces R i ad R i, obtaied by limit passages: Ri= lim Ri, R = lim R. For R i ad R () implies the recurrece relatios: s + i s + i i R = R+ ; R, + = R+ R+, π +, ir+ + i, i, i= {,,, K, h }, {,, K, m }; R = R ; R +, + = R + R +, π, ir + + + i, i, i= { h, h +, h +, K, m }, {,, K, m }. () Usig the relatios () ad takig ito accout the equalities m π = by mathematical iductio we ca prove that =, { h +, h +, K, m }, R, m =, { h +, h +, K, m }. () Give (6) ad (), usig (9) we obtai the followig statemet. Theorem 3. The mea legth of the busy period is determied i the form where E τ ( m) = MT ( m, h, h ) + MT ( m, h, h ), ()
4 Y. Zherovyi, B. Kopytko h h hh T ( m, h, h ) = R R R( h, h ) R ; i i h i i= i= i= h m h m m T ( m, h, h ) = R i i+, i R i i, i Ru + π π i= = h+ i= = h+ u= hh m m mh R, h+ R h h iπ h+ i, h i u Rh i R h h = i= = h+ u= i= Rh, h h+ + R( h, h ) R R + ; (, ). Itroduce the otatio: lim P{ ξ ( t) = k} = p ( m ), k {,,, K, m }. Reasoig t as i the paper [6], ad takig ito accout that the legth of the idle period durig oe cycle is distributed expoetially with parameter m λ, from (8) we obtai formulas for the statioary distributio of the umber of customers i the system. Theorem 4. The statioary distributio of the umber of customers i the system is give by k p ( m) = ; + mλeτ ( m) k pk ( m) = mλp ( m) Rkqk + ( R iqi, k i R iqi+, k i), k {,, K, h}; i= k k h pk ( m) = mλp ( m) R iqi, k i + R( h, h ) ( R h iq h + i, kh i Rh iqh + i, kh i) i= i= k i= R iqi+, ki, k { h +, h +, K, h }; k h pk ( m) = mλp ( m) D ( k) + R( h, h ) R h iqh + i, kh ( ) i Dh k D ( k), i= k { h +, h +, K, m}, (3) where h k k D ( k) = Ri q+ i, k i I{ h + k m} + i, i Ruq + u, k u. π i= = h+ u= We fid the statioary queue characteristics - the average queue legth EQm ( ) ad average waitig time Ew( m ) - by the formulas
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 4 m EQ( m) EQ( m) = kp ( m); Ew( m ) =. (4) k+ k= λav Here λ av is a steady-state value of the arrival rate of customers, defied by the equality λ av m k = = ( m k) p ( m ). (5) The parameter λ av is characteristic of the system capacity, because for the steady- -state regime we have the equality of the itesities of flows of customers arrivig ad served. If F ( x) = F( x ), the we obtai a closed system with the distributio of the service time that is idepedet of the umber of customers i the system. Puttig h= h= m i () ad (3), we obtai formulas for Eτ ( m ) ad pk( m ) for this system. k 5. Examples for calculatig of statioary characteristics Let us cosider two examples for the calculatio of statioary characteristics of closed queueig systems. Example. Assume that m = 6, λ=, h =, h = 4, the uiform distributio o the itervals (;.5] ad (;.5] correspods to the distributio fuctios of the service time F( x ) ad F ( x ) respectively. Thus,.5 ( ).5 ( ) 4 =.5, y y M M =.5, f ( y) = e, f ( y) = e. y y Example. Cosider a closed system for which F ( x) = F( x ). Assume that the distributio fuctio F( x ) ad the values of parameters m, λ, h ad h are the same as i Example. Let us call system i a system with parameters correspodig to the example i, i =,. The row " p k (6)" of Tables ad cotais steady-state probabilities p k (6), calculated by the formulas (3) for the systems ad respectively. For the sake of compariso, the same tables cotai the correspodig probabilities evaluated 6 by the GPSS World simulatio system [3] for the time value t =. The values of the statioary characteristics foud by the formulas (), (4) ad (5) ad calculated with the help of GPSS World, are show i Tables 3 ad 4 respectively.
4 Y. Zherovyi, B. Kopytko Statioary distributios of the umber of customers i the system Table Number of customers (k) 3 4 5 6 p k (6).79.343.376.656.336.9865.4455 p k (6) (GPSS World, t = 6 ).76.3444.346.6.393.993.44 Statioary distributios of the umber of customers i the system Table Number of customers (k) 3 4 5 6 p k (6).45.76.7848.7.385.859.8435 p k (6) (GPSS World, t = 6 ).457.83.7839.9.3779.888.845 Statioary characteristics of the system Table 3 Characteristics E τ (6) EQ (6) Ew (6) λ av Aalytical value.673.654.564 4.75 Value accordig to GPSS World, t = 6.63.653.564 4.77 Statioary characteristics of the system Table 4 Characteristics E τ (6) EQ (6) Ew (6) λ av Aalytical value 8.456 3.3.757 3.98 Value accordig to GPSS World, t = 6 8.64 3.3.756 3.98 Aalyzig the obtaied results, we see that the applicatio of the hysteretic strategy of the service time chage allows oe to reduce the queue legth ad the legth of the busy period ad, therefore, to icrease the capacity of the system λ av. 6. Coclusios With the help of the potetials method, we have obtaied simple ad suitable for umerical realizatio formulas for fidig the statioary characteristics of sigle-chael closed systems with a arbitrary distributio of the service time.
The potetials method for a closed queueig system with hysteretic strategy of the service time chage 43 I order to icrease the system capacity, the hysteretic strategy of the service time chage are applied. Refereces [] Nesterov Yu.G., Aalysis of characteristics of a closed queuig system with relative priorities, Nauka i Obrazovaie, MGTU im. N. Baumaa 4, 3, 4-54 (i Russia). [] Bocharov P.P., Pechiki A.V., Queueig Theory, RUDN, Moscow 995 (i Russia). [3] Zherovyi Yu.V., Markov Queueig Models, Vydavychyi Tsetr LNU im. Ivaa Fraka, Lviv 4 (i Ukraiia). [4] Zherovyi Yu., Statioary probability distributio of states for a sigle-chael closed queuig system, Visyk Lviv. Uiver. Series Mech.-Math. 7, 67, 3-36 (i Ukraiia). [5] Zherovyi Yu.V., Zherovyi K. Yu., Potetials method for a closed system with service times depedet o the queue legth, Iformatsioye Protsessy 5, 5,, 4-5. θ θ [6] Zherovyi K.Yu., Zherovyi Yu.V., M /G//m ad M /G/ systems with the service time depedet o the queue legth, J. of Commuicat. Techology ad Electroics 3, 58,, 67-75. [7] Zherovyi Yu., Zherovyi K., Potetials Method for Threshold Strategies of Queueig, LAP Lambert Academic Publishig, Saarbrűcke 5 (i Russia). θ [8] Zherovyi K.Yu., Statioary characteristics of the M /G//m system with the threshold fuctioig strategy, J. of Commuicat. Techology ad Electroics, 56,, 585-596. [9] Zherovyi Yu., Isesitivity of the Queueig Systems Characteristics, LAP Lambert Academic Publishig, Saarbrűcke 5. [] Zherovyi Yu.V., Zherovyi K.Yu., Probabilistic characteristics of a M /G//m queue with two-loop hysteretic cotrol of the service time ad arrival rate, J. of Commuicat. Techology ad Electroics 4, 59,, 465-474. [] Korolyuk V.S., The Boudary Problem for the Compoud Poisso Processes, Naukova Dumka, Kyiv 975. α [] Bratiychuk M., Borowska B., Explicit formulae ad covergece rate for the system M /G//N as N, Stochastic Models, 8,, 7-84. [3] Zherovyi Yu., Creatig Models of Queueig Systems i the Eviromet GPSS World, Palmarium Academic Publishig, Saarbrűcke 4 (i Russia). θ