M/G /n/0 Erlang queueing system with heterogeneous servers and non-homogeneous customers

Transcription

1 BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 66, No. 1, 18 DOI: 1.445/11959 M/G // Erlg queueig system with heterogeeous servers d o-homogeeous customers M. IÓŁKOWSKI* Fculty of Applied Iformtics d Mthemtics, Wrsw Uiversity of Life Scieces, 159 Nowoursyows St., 787 Wrsw, Pold Abstrct. I the preset pper, we ivestigte multi-server queueig system with heterogeeous servers, ulimited memory spce, d o-homogeeous customers. The rrivig customers pper ccordig to sttiory Poisso process. Service time distributio fuctios my be differet for every server. Customers re dditiolly chrcterized by some rdom volume. O every server, the service time of the customer depeds o their volume. The umber of customers distributio fuctio is obtied i the clssicl model of the system. I the model with o-homogeeous customers, the sttiory totl volume distributio fuctio is determied i the term of Lplce Stieltjes trsform. The sttiory first d secod momets of totl customers volume re clculted. A lysis of some specil cses of the model d some umericl exmples re lso icluded. Key words: multi-server queueig systems, queueig systems with o-homogeeous customers, queueig systems with heterogeeous servers, totl volume distributio, Lplce Stieltjes trsform. 1. Itroductio The first lysis of the clssicl M/M// queueig system ws mde by Erlg i [3] d [4]. I this model, the uthor ssumes tht the ivestigted system is composed of ideticl servers. More precisely, service time is expoetilly distributed with the sme prmeter μ for every server. I dditio, customers rrive t the system with Poisso etrce flow, which mes tht the time itervls betwee successive customers rrivls re expoetilly distributed with the sme prmeter. There is o queue i this system, so rrivig customer is lost if they fid tht the system is full (i.e. every server is busy). The mi chrcteristic obtied durig the lysis of this system is the umber of customers distributios i the sttiory mode p. It c be esily prove [11] tht the obtied formule lso remi vlid for more uiversl M/G// queueig system with ideticl servers if we replce the costt μ by 1 /β 1, where β 1 is the me vlue of service time. O the other hd, queueig systems with heterogeeous servers re rrely lyzed. The first ivestigtios of M/M // queueig systems with heterogeeous servers ppered i [6, 7], d [8]. Iterestig lyses of M/M // queueig system with heterogeeous servers my lso be foud i [1] d [1]. Mewhile, some ivestigtios of M/G // systems with heterogeeous servers my be foud i [5] d [9]. If we dditiolly ssume tht rrivig customers hve some rdom volume, the we obti very iterestig ew re of reserch tht is coected with the cocept of totl volume, which is the sum of the volumes of ll of the customers preset i the system. These queueig systems re clled queueig sys- * e-mil: mrci_ziolowsi@sggw.pl Muscript submitted , revised d , iitilly ccepted for publictio , published i Februry 18. tems with o-homogeeous customers d the mi purpose, i this cse, is to obti the totl volume chrcteristics or loss chrcteristics, t lest i sttiory mode. The results coected with M/G// queueig system with ideticl servers d o-homogeeous customers re preseted i [11]. The preset pper ims to lyze M/G // queueig system with heterogeeous servers d o-homogeeous customers. The rest of this pper is orgized s follows. Sectio cotis the lysis of the clssicl M/G // queueig system with heterogeeous servers. The mi purpose of this prt is to obti the umber of customers distributio fuctios i the sttiory mode. Sectio 3 cotis lysis of o-clssicl M/G // queueig system with o-homogeeous customers. I this sectio, we obti the totl volume distributio chrcteristics i sttiory mode. Sectio 4 will ivestigte some specil cses of the model lyzed i Sectio 3 d it will show tht the chrcter of the service time d customer volume depedecy hs ifluece o the totl volume chrcteristics. Filly, Sectio 5 cotis some cocludig remrs.. The clssicl model lysis This sectio will ivestigte the modifictio of the clssicl M/G// queueig system i which the service time chrcteristics my differ for ech server. The customers choose free servers i rdom wy, which mes tht rrivig customer will be serviced by oe of the l free servers with probbility 1/l. Our purpose is to obti the umber of customers distributio fuctios i the sttiory mode. To do this, we will use the geerlized method of the uxiliry vrible []. I the lyzed model, the umber of customers is limited by vlue d ll preset customers i the system re beig serviced (i.e. there is o queue). We deote the prmeter of etrce Poisso flow s ; the service time distributio fuctio for j-th server s B j (t); Bull. Pol. Ac.: Tech. 66(1) 18 59

2 M. iółowsi d its first momet s β j, j = 1,. Let η(t) be the umber of customers preset i the system t time istt t. I dditio, we ssume tht there exist service time desities b j (t). This ssumptio is techicl becuse it llows us to use, durig system behvior lysis, the service itesity fuctio tht is defied b j (x) 1 B j (x) for j-th server by the formul μ j (x) =, lthough we my obti the sme results without this ssumptio. Let A(t) be the set of busy servers t time istt t. We deote the legth of the time itervl from the begiig of the service of the customer (tht is still serviced by j-th server t time t) to time t s ξ j (t), j A(t). It is esy to prove tht process (η(t), A(t), ξ j (t), j A(t)) (1) is Mrovi process tht describes the behvior of the system. I the cse of empty system (η(t) = ), the process reduces to η(t). Now we itroduce the followig fuctios: P (t) = P{η(t) = }, =, ; () If we lyze, for simplicity, the system behvior i the specil cse (M/G // system), the we c write dow the followig equtios: µ t {1} P (t + t) = P (t)(1 t) + t G1 (x, t)μ1 (x)dx + (9) t {} P (t + t) + G1 (x, t)μ (x)dx + o( t); {1} {1} G 1 (x + t, t + t) = G1 (x, t)[1 ( + μ1 (x)) t] + t + t G (x, u, t)μ (u)du + o( t); (1) {} {} G 1 (x + t, t + t) = G1 (x, t)[1 ( + μ (x)) t] + t (11) + t G (u, x, t)μ 1 (u)du + o( t); G (x 1 + t, x + t, t + t) = = G (x 1, x, t)[1 (μ 1 (x 1 ) + μ (x )) t] + o( t); (1) P 1, i,, i } (t) = P{η(t) =, A(t) = 1, i,, i }}, = 1, ; (3) t {1} G1 (u, t + t)du = P (t) (t) + o( t); (13) G 1, i,, i } (x 1,, x, t)dx 1 dx = = P{η(t) =, A(t) = 1, i,, i }, ξ j (t) [x j, x j + dx j ), j A(t)}, = 1,. If η(t) =, the the fuctio described i (4) my be deoted simply s G (x 1,, x, t). I the sttiory mode (which exists if β j < 1, j = 1, ) we c itroduce logies tht re idepedet of the time vrible t: p = limp t!1 (t), =, ; (5) (4) p 1, i,, i } = lim t!1 P 1, i,, i } (t), = 1, ; (6) g 1, i,, i } (x 1,, x ) = lim t!1 G 1, i,, i } (x 1,, x, t), = 1,. If =, the we my simply deote g (x 1,, x ) isted of, i,, i } (x 1,, x ). The fuctios itroduced i (4) d (7) re ot symmetric cosiderig ll of the permuttios of the vribles x j, j A(t), s it is i clssicl M/G// Erlg system with ideticl servers. It is cler tht p 1, i,, i } 1 1 =, i,, i } (x 1,, x )dx 1 dx, (8) = 1,. (7) t {} G1 (u, t + t)du = P (t) (t) + o( t); (14) t {1} G (x + t, u, t + t)du = G 1 (x, t) t + o( t); (15) t {} G (u, x + t, t + t)du = G 1 (x, t) t + o( t). (16) If t! the from equtios (9 16) we obti the followig equtios: dp (t) dt G 1 {1} (x, t) t G 1 {} (x, t) t t {1} = P (t) + G1 (x, t)μ1 (x)dx + t {} + G1 (x, t)μ (x)dx; + G {1} 1 (x, t) {1} = ( + μ 1 (x))g x 1 (x, t) + t + G (x, u, t)μ (u)du; + G {} 1 (x, t) {} = ( + μ (x))g x 1 (x, t) + t + G (u, x, t)μ 1 (u)du; (17) (18) (19) 6 Bull. Pol. Ac.: Tech. 66(1) 18

3 M/G // Erlg queueig system with heterogeeous servers d o-homogeeous customers G (x 1, x, t) t + G (x 1, x, t) x 1 + G (x 1, x, t) x = = (μ 1 (x 1 ) + μ (x ))G (x 1, x, t); () G 1 {1} (, t) = P (t); (1) G 1 {} (, t) = P (t); () G (x,, t) = G 1 {1} (x, t); (3) G (, x, t) = G 1 {} (x, t). (4) I the sttiory mode (t! 1) from (17 4), we esily obti: 1 1 {1} {} = p + g1 (x)μ1 (x)dx + g1 (x)μ (x)dx; (5) {1} (x) x {} (x) x 1 {1} = ( + μ 1 (x)) (x) + g (x, u)μ (u)du; (6) 1 {} = ( + μ (x)) (x) + g (u, x)μ 1 (u)du; (7) g (x 1, x ) x 1 + g (x 1, x ) x = (μ 1 (x 1 ) + μ (x ))g (x 1, x ); (8) {1} () = p ; (9) {} () = p ; (3) g (x, ) = {1} (x); (31) g (, x) = {} (x). (3) Now we dd the ormliztio coditio: 1 1 {1} {} p + g1 (x)dx + g1 (x)dx g (x 1, x )dx 1 dx = 1. (33) By direct substitutio, we c chec tht the solutios of equtios (5 3) hve the form: {1} (x) = p (1 B 1(x)); (34) {} (x) = p (1 B (x)); (35) g (x 1, x ) = p (1 B 1(x 1 ))(1 B (x )). (36) Usig (8) d the well-ow formul β j = 1 [1 B j (u)]du, we filly obti: {1} 1 p p 1 = (1 B 1(x))dx = β 1p ; (37) {} 1 p p 1 = (1 B (x))dx = β p ; (38) p 1 = p 1 {1} + p1 {} = p (β 1 + β ); (39) 1 1 p = p (1 B 1(x 1 ))(1 B (x ))dx 1 dx = = 1 1 p (1 B1 (x 1 ))dx 1 (1 B (x ))dx = = p β 1 β. (4) Ad by the help of (33): p = (β 1 + β ) + β 1β. (41) I the sme wy, we c lyze the M/G // system with heterogeeous servers for the rbitrry. The problem is tht the umber of equtios describig the system behvior icreses expoetilly together with the icresig vlue of, so i this cse we use some set ottios. Let {C } deote the set of ll of the -elemet combitios of the -elemet set. The equtios i the sttiory mode c be preseted i the followig form: + = ( + j/ 1,, i } = p + { j} g1 (x)μj (x)dx; (4) 1 1 j 1,, i } j =1 g 1,, i } (x1,, x ) = x j μ j (x j )),, i } (x 1,, x ) + j 1,, i } g 1,, i, j} + 1 (x 1,, x j 1, u, x j,, x )μ j (u)du, 1,, i } {C }, = 1, 1; (43) g 1,, i } (x1,, x ) = μ j (x j )g 1,, i } (x 1,, x ); (44) j =1 x j j =1 Bull. Pol. Ac.: Tech. 66(1) 18 61

4 M. iółowsi = { j} () = p, j = 1, ; (45) g 1,, i } (x 1,, x j 1,, x j+1,, x ) = + 1,, i j 1, i j+1,, i } 1 (x 1,, x j 1, x j+1,, x ), j = 1,, =,. (46) By direct substitutio, we my obti the solutios i the followig form: Thus: g 1, i,, i } (x 1, x,, x ) = = ( )!p (1 B! j (x j )); = 1,. j 1,, i } Usig (8) i logous wy to (4), we obti: (47) p 1,, i } = ( )!p β! j, = 1, ; (48) j 1,, i } p = ( )!p β! j, = 1,. (49) i {C } From the ormliztio coditio, we obti: p = ( )!! =1 i {C } β j 1. (5) If we itroduce the ottio: y j = β j the formule (48 5) te the form: p 1,, i } = ( )!p y! j, = 1, ; (51) j 1,, i } p = ( )!p y! j, = 1, ; (5) i {C } p = ( )!! =1 i {C } y j 1. (53) As ws ivestigted, the umber of customers distributio fuctio i the sttiory mode depeds oly o the first momets of fuctios B j (x) d does ot deped o the formule tht defie them. Formule (51 53) re iterestig ot oly from the theoreticl poit of view. O the bsis of (5) d (53), we c clculte some very prcticl chrcteristics, icludig the me vlue of the umber of customers preset i the system i the sttiory mode (Eη = = p ) d the loss probbility ( p ). O the other hd, we my lso ivestigte the usge of ech server. For exmple, if the lyzed system is composed of three servers, the we my compute the usge of ech server q i, i = 1, 3 s follows: q 1 = p 1 {1} + p {1, } + p {1, 3} + p3 {1,, 3}. q = p 1 {} + p {1, } + p {, 3} + p3 {1,, 3}. q 3 = p 1 {3} + p {1, 3} + p {, 3} + p3 {1,, 3}. Let us ow cosider the followig umericl exmple. Assume tht we del with M/G /3/ queueig system with heterogeeous servers. The mi vlues of service time o ech server re equl to: β 1 =, β = 3, β 3 = 4 cosequetly d =. The, by usig (51 53), we my obti the umericl results coected with the umber of customers distributio fuctio. We preset them i the first colum of Tble 1, together with the results obtied by simultio for three distributios of service time. I the ext colums, we preset the results for expoetil distributio (service time is expoetilly distributed for ech server with prmeters μ 1 = 1 /, μ = 1 / 3, μ 3 = 1 / 4 ), uiform distributio (o the itervl [1, 3] for first server, [, 4] for secod server d [3, 5] for third server), d costt distributio (service time is costt for ech server with prmeters t 1 =, t = 3, t 3 = 4). The results show tht the umber of customers distributio fuctio depeds oly o the me vlues of service time o ech server, d does ot deped o the formule of the service time distributio fuctios. Tble 1 The umber of customers distributio fuctio i the M/G /3/ system p theoret. sim.-expoetil sim.-uiform sim.-costt p p p p The model with o-homogeeous customers This sectio will lyze the modifictio of the clssicl M/G // queueig system with heterogeeous servers i which the rrivig customers dditiolly hve, idepedetly of the other customers, some rdom volume ζ tht is o-egtive rdom vrible. I geerl, service time o j-th server ξ j depeds o customer volume ζ. I other words, for ech server we hve the followig distributio fuctio: F j (x, t) = P{ζ < x, ξ j < t}, j = 1,. The im of our ivestigtio is to obti the chrcteristics of the totl volume σ(t), which is the sum of the volumes of ll customers preset i the system i time istt t. I the stedy stte, the process σ(t) coverges to rdom vrible σ. We ssume tht the totl volume is ulimited. Let us itroduce the followig ottios: D(x) = P{σ < x} is the totl volume distributio fuctio i the stedy stte; δ(s) = 6 Bull. Pol. Ac.: Tech. 66(1) 18

5 M/G // Erlg queueig system with heterogeeous servers d o-homogeeous customers = 1 e sx dd(x) is the Lplce Stieltjes trsform of the rdom vrible σ; δ i is its i-th momet; α j (s, q) = 1 1 e sx qt df j (x, t), j = 1, deotes the double Lplce Stieltjes trsform of the rdom vector (ζ, ξ j ); d α j l is the mixed momet of the (l + )-th order of this vector. To obti the chrcteristics of the totl volume distributio fuctio i the stedy stte, we itroduce some coditiol distributio fuctios of the totl volume, d we the use the totl probbility theorem. We itroduce the coditiol totl volume distributio fuctio s follows: H 1,, i } (x, y 1,, y ) = P{σ < xjη =, A = 1,, i }, ξ i1 = y 1,, ξ i = y }, 1,, i } {C }, = 1,, (54) where η is the umber of customers preset i the system i sttiory mode, A is the set of busy servers, d ξ j, j 1,, i } re the sttiory logies of the fuctios ξ j (t) (which were itroduced i Sectio ). By usig the totl probbility theorem, we obti the followig formul: D(x) = p + =1 1 1 g 1,, i } (y 1,, y ) 1,, i } {C } (55) H 1,, i } (x, y 1,, y )dy 1 dy. Now we use Lplce Stieltjes to trsform to both sides of (55), obtiig: δ(s) = p g 1,, i } (y 1,, y ) =1 1,, i } {C } (56) where h 1,, i } (s, y 1,, y )dy 1 dy. h 1,, i } (s, y 1,, y ) = 1 e sx dh 1,, i } (x, y 1,, y ) is the Lplce Stieltjes trsform of the distributio fuctio tht ws defied i (54). Now we fid the formul for h 1,, i } (s, y 1,, y ). We deote s χ j, j A the volume of the customers serviced by j-th server. Let E j (x) = P{χ j < xjξ j = y j }, j A be the coditiol distributio fuctio of the rdom vrible χ j uder ssumptio tht its service lsts y j time uits. Now we use well-ow formul [11]: de j (x) = [1 B j (y j )] 1 1 u = y j df j (x, u). (57) If we itroduce the Lplce Stieltjes trsform e j (s) = 1 e sx de j (x), the trsformtio of (57) leds to the followig result: e j (s) = [1 B j (y j )] 1 1 x = e sx 1 u = y j df j (x, u). (58) It is rther cler tht if A = 1,, i } the totl volume σ is the sum of the idepedet rdom vribles χ j, j A. The coditiol distributio fuctio H 1,, i } (x, y 1,, y ) is the covolutio of the distributios E j (x), mely: H 1,, i } (x, y 1,, y ) = E i1 E i E i (x). (59) O the bsis of the properties of the Lplce Stieltjes trsform, we quicly obti: h 1,, i } (s, y 1,, y ) = e j (s). (6) j 1,, i } From (56, 58, 6), d (47) we filly obti: δ(s) = p + =1 i {C } x = e sx ( )!p! (61) df j (x, u) dy 1 dy. u = y j Give tht -dimesiol itegrl preset i (61) my be preseted i the form of product of oe-dimesiol itegrls, the formul (61) my be rewritte i simple form: δ(s) = p + 1 x = e sx ( )!p! dz 1 df j(x, u). z = u = z =1 i {C } 1 Now we compute the itegrl i (6): 1 1 x = e sx 1 1 z = dz 1 u = z df j(x, u) = u = x = u = e sx df j (x, u) dz = z = 1 1 = ue sx df j (x, u) = α j(s, q) j q q =. Filly, usig (63), we obti the followig result: δ(s) = p ( ( )!! =1 i {C } α j(s, q) q (6) (63) j q = ). (64) The obtied formul let us compute the first two momets δ 1 = Eδ, δ = Eδ of the totl volume i sttiory mode. We use the well-ow formule tht re coected with the Lplce Stieltjes trsform properties: δ 1 = δ (), δ = δ (). O the other hd, the mix momets of the (l + )-th order of the rdom vector (ζ, ξ j ) c be computed usig the properties of the double Lplce Stieltjes trsform α j (s, q): α j l = E(ζ l ξ j ) = ( 1) l + l + α j (s, q) s l q j s =, q =. (65) Bull. Pol. Ac.: Tech. 66(1) 18 63

6 M. iółowsi By usig the bove properties of sigle d double Lplce Stieltjes trsforms, d (64) we filly obti: δ 1 = δ () = p ( 1 α ( )!! =1 i {C } 11 α j β l ). (66) l i{j} δ = δ () = p ( 1 α ( )!! =1 (67) 1 α j β l + α j α m β l i {C ). } l i{j} {j, m}½i l i{j, m} If the set of idexes of the sum or product symbol is empty, the we omit these sums or products i computtio. I the process of computig formule (66) d (67), we use the followig properties of the derivtives: ( i =1 ( i =1 f i (x)) = f i (x) i =1 f i (x)) = f i (x) i =1 f l (x). (68) l{1,, }} f l (x) + l{1,, }} + f i (x)f j (x) f l (x)., j}½{1,, } l{1,, }, j} (69) The obtied results c be used to desig computer or commuictio systems of the logous type but with limited memory spce. For exmple, we c cosider more prcticl system M/G //(, V) i which the volume of ll customers is limited by the V vlue. This mes tht the rrivig customer my lso be lost whe there re free servers i the system but their volume is too high (i.e. the sum of the volume of rrivig customer d the volumes of other customers tht re served is bigger th V vlue). This limittio leds to dditiol losses of customers. I the process of desigig computer system, we my choose the V vlue i such wy tht miimizes these dditiol losses. The, we use the (66, 67) formule d, for exmple, choose the vlue of V from the itervl [δ 1 mδ, δ 1 + mδ], where δ = δ δ 1 is the stdrd devitio of totl customers volume i the sttiory mode, d m costt, m = (1,, 3, ) my be chose by the computer system desiger. This choice hs ifluece o the loss probbility i the M/G //(, V) system. From the prcticl poit of view, it is best to choose the followig vlue: V = δ 1 + 3δ. 4. Specil cse lysis This sectio will ivestigte two prcticl specil cses of the lyzed model. I the first cse, customer volume d service times for j-th server, j = 1, re idepedet. The secod cse presets situtio i which service time for j-th server is proportiol to customer volume with coefficiet c j i.e. ξ j = c j ζ. For these specil cses, we obti the formule for the totl volume distributio fuctio i sttiory mode d for its first two momets Service time d customer volume re idepedet. Assume tht customer volume d service time re idepedet for every server. More precisely, every pir (ζ, ξ j ), j = 1, preset two idepedet rdom vribles. Let L(x) = P{ζ < x} d B j (t) = P{ξ j < t} be the distributio fuctios of the rdom vribles ζ d ξ j. We deote the first two momets of the rdom vrible ζ s φ 1 d φ. I this cse, we hve the obvious formul: α j (s, q) = φ(s)β j (q), (7) where φ(s) = 1 e sx dl(x) d β j (q) = 1 e qt db j (t) re the Lplce Stieltjes trsforms of the rdom vribles ζ d ξ j, respectively. The, we hve the equlity: α j (s, q) j q q = = φ(s)β j. (71) Now we substitute the obtied formul ito (64) d we the obti the followig: Thus: δ(s) = p ( ( )!(φ(s))! =1 i {C } δ 1 = p φ 1 ( 1 ( )!! =1 i {C } δ = p ( 1 ( )![φ + ( 1)φ! 1 ] =1 β j ). (7) β j ) ; (73) i {C } β j ). (74) Also ssume tht customer volume is expoetilly distributed with prmeter f d the service times o ech of servers re lso expoetilly distributed with prmeters μ 1,, μ. The, we obti the followig formul: δ(s) = p ( 1 + 1! ( f =1 f + s) ( )!i {C } 1 μ j ). (75) The formule for first two momets c be obtied from (73) d (74) by mig substitutios: φ 1 = 1 / f, φ = / f, β j = 1 /μ j. Usig the Lplce trsform iversio of the fuctio δ(s)/ s, with the help of computer lgebr systems [1], we my obti formul for totl volume distributio fuctio i form: D(x) = p ( ( )!! =1 1 e fx 1 (fx)l l! l = i {C } 1 (76) μ j ). The totl volume distributio fuctio is lier combitio of Erlg distributios with prmeter f. 64 Bull. Pol. Ac.: Tech. 66(1) 18

7 M/G // Erlg queueig system with heterogeeous servers d o-homogeeous customers 4.. Service time is proportiol to the customer volume. Assume ow tht the service time o j-th server is proportiol to the customer volume with coefficiet c j, j = 1, i.e ξ j = c j ζ. I this cse, we obti the formul: Thus: α j (s, q) = φ(s + c j q). (77) α j (s, q) j q q = = c j φ (s). (78) If we substitute this formul ito (64), we obti: δ(s) = p ( ( φ (s)) ( )!! =1 i {C } c j ). (79) The first two momets my be computed bsig o the followig formule: δ 1 = p φ ( 1 ( )!(φ 1 ) 1 c j! =1 i {C ) ; (8) } δ = p ( 1 φ! 1 ( )! =1 [φ 3 φ 1 + ( 1)φ ] (81) c j ), i {C } where φ 3 is the third momet of the customer volume. Also ssume tht customer volume is expoetilly distributed with prmeter f d the service times o ech of servers re proportiol to their volumes with coefficiets c j. The, the service times re lso expoetilly distributed with prmeters f / c1,, f / c. The, we obti the followig formul: δ(s) = p ( (f)! =1 ( f + s) ( )! i {C } c j ). (8) The formule for the first two momets c be obtied from (8) d (81) by mig substitutios: φ 1 = 1 / f, φ = / f, φ 3 = 6 / f 3. Usig the Lplce trsform iversio, we filly obti the totl volume distributio fuctio i form: D(x) = p ( ( )!! =1 1 e fx 1 (fx)l l! l = i {C } (83) c j ). This formul is similr to (76) but here we hve more Erlg distributios i the sum. This mes tht eve if we choose coefficiets i such wy tht from the clssicl poit of view time service distributio fuctios re the sme i these two situtios the chrcteristics of the totl volume distributio vry eve o the level of first two momets. So, the chrcter of the lyzed depedecy hs ifluece o the totl volume chrcteristics. I fct, if i the first model we substitute μ j = f / cj, j = 1,, the from the clssicl poit of view two models re equivlet; tht is, we hve, for exmple, the sme service times distributio fuctios but the chrcteristics of totl volume distributio re ot the sme. I the first model, fter rther esy computtios, we obti: δ 1 = p f ( 1! =1 δ = p f ( ( )! ( f ) i {C } c j ) ; (84) 1 ( + 1)( )!! =1 ( f ) c j i {C ). (85) } I the secod model, we obti: δ 1 = p f ( 1! =1 δ = p f ( ( )! ( f ) i {C } c j ) ; (86) 1 ( + 1)( )!! =1 ( f ) c j i {C ). (87) } Comprig the bove formule, we otice tht the vlue of first momet is exctly two times greter i the secod model d the vlue of the secod momet is lso greter i the secod model but i this cse the coefficiet is ot costt d depeds o the umber of servers. We will ow illustrte the theoreticl results with some umericl exmples. Let us cosider M/G /3/ queueig system i two versios M1 d M. I the first versio of the system, service times re expoetilly distributed with prmeters μ 1 = 1 /, μ = 1 / 3, μ 3 = 1 / 4. Service times d customer volumes re idepedet d the customer volumes re expoetilly distributed with prmeter f = 1. I the secod versio, the customer volumes re lso expoetilly distributed with the sme prmeter f = 1 but service times re proportiol to customer volumes with coefficiets c 1 =, c = 3, c 3 = 4. I Tble, we preset the results coected with the first two momets of the summry volume i sttiory mode obtied usig (84 87), together with the results obtied by simultio tht cofirm the theoreticl results. Tble Momets of the summry volume i M/G /3/ system M1-theoret. M1-sim. M-theoret. M-sim. δ δ Coclusios I the preset pper, we hve ivestigted the modifictio of M/G// queueig system with heterogeeous servers d o-homogeeous customers. I the begiig, we obti umber of customers distributio fuctios i the sttiory Bull. Pol. Ac.: Tech. 66(1) 18 65

8 M. iółowsi mode for the clssicl queueig system M/G // i which service time chrcteristics my be differet for every server. Both the theoreticl d the simultio results show tht the umber of customers distributio fuctio does ot deped o the form of the service time distributios, but o their me vlues. Lter o, we lyzed o-clssicl M/G // queueig system with o-homogeeous customers i which service times d customer volumes re depedet d the totl volume is ulimited. For this system, we obti the formule for the Lplce Stieltjes trsform of the totl volume distributio fuctio d its first two momets. The, we ivestigte two specil cses i which we obti the formule for the totl volume distributio fuctio i exct form. We the show tht the chrcter of the depedecy of service time d customer volume hs ifluece o the totl volume chrcteristics. Our simultio results lso cofirm this fidig. Refereces [1] M.L. Abell, J.P. Brselto, The Mthemtic Hdboo, Elsevier, 199. [] P.P. Bochrov, C.D Apice, A.V. Pechii, S. Slero, Queueig Theory, VSP, Utrecht-Bosto, 4. [3] A. Erlg, The theory of probbilities d telephoe coverstios, Nyt Tidssrift for Mtemti B, (199). [4] A. Erlg, Solutio of some problems i the theory of probbilities of sigificce i utomtic telephoe exchges, The Post Office Electricl Egieers Jourl 1, (1918). [5] D. Fios, The geerlized M/G/ blocig system with heterogeeous servers, The Jourl of the Opertiol Reserch Society 33 (9), (198). [6] H. Gumbel, Witig lies with heterogeeous servers, Opertios Reserch 8 (4), (196). [7] V.P. Sigh, Two-server Mrovi queues with blig: Heterogeeous vs. homogeeous servers, Opertio Reserch 18 (1), (197). [8] V.P. Sigh, Mrovi queues with three heterogeeous servers, AIIE Trsctios 3 (1), [9] J. Sztri, O the /G/M/1 queue d Erlg s loss formuls, Serdic 1 (1986). [1] J. Sztri, Bsic Queueig Theory, Uiversity of Debrece, Fculty of Iformtics, 1. [11] O. Tihoeo, Probbility Methods of Iformtio Systems Alysis, Ademic Oficy Wydwicz EXIT, Wrszw, 6 (i Polish). [1] M. iółowsi, M/M//m Queueig Systems with No-Homogeeous Servers, J Długosz Uiversity i Częstochow, Scietific Issues, Mthemtics XVI, Bull. Pol. Ac.: Tech. 66(1) 18