Mean value analysis of product form solution queueing networks with repetitive service blocking

Transcription

1 Performance Evaluaton (1999) Mean value analyss of product form soluton queueng networks wth repettve servce blockng Matteo Sereno Dpartmento d Informatca, Unverstà d Torno, Corso Svzzera 185, Turn, Italy Abstract Queueng network models wth fnte capacty queues and blockng are used to represent systems wth resource constrants, such as producton, communcaton and computer systems. Varous blockng mechansms have been defned n the lterature to represent the dfferent behavours of real systems wth lmted resources. Queueng networks wth blockng have a product form soluton under specal constrants, for dfferent blockng mechansms. In ths paper we present a Mean Value Analyss for the computaton of performance measures n product form soluton queueng networks wth repettve servce blockng. Basc to the dervaton of ths algorthm are recursve expressons for the performance ndces that are a non trval generalsaton of those derved for the Mean Value Analyss of product form queueng networks wthout blockng. In ths paper we gve a formal dervaton of several recursve relatons as well as detals on ther mplementaton. A few basc examples are evaluated wth the technques dscussed n ths paper to show the advantages of ths approach Elsever Scence B.V. All rghts reserved. Keywords: Queueng networks wth blockng; Repettve servce blockng; Product form soluton; Computatonal algorthms; Mean value analyss algorthm 1. Introducton Queueng networks are used to model complex servce systems such as producton, communcaton and computer systems. Networks of queues wth blockng are used to represent resource constrants such as memory constrants or software constrants n computer systems, as well as wndow flow control n communcaton networks. Varous types of blockng have been reported n the lterature (e.g., [4,5, 13,15]) n order to model dfferent system behavours. Each blockng mechansm descrbes how a node becomes blocked, what happens n blockng stuaton and how the blocked node becomes unblocked. The most commonly used blockng mechansms are the blockng after servce, theblockng before servce,andtherepettve servce also called rejecton blockng [4,5,13,15]. Queueng networks wth blockng are n general dffcult to treat. Product form solutons have been derved only under partcular constrants and dependng on the consdered blockng type. A dscusson of product form queueng networks wth blockng can found n [5]. In ths paper we concern wth product form queueng networks wth repettve servce blockng /99/$ see front matter 1999 Elsever Scence B.V. All rghts reserved. PII: S (99)

2 20 M. Sereno / Performance Evaluaton (1999) Even f the product form equlbrum dstrbuton s apparently smlar to that of queueng networks wthout blockng, nevertheless the algorthms derved for these type of networks, such as convoluton algorthm [7] and Mean Value Analyss (MVA) algorthm [16], cannot be mmedately appled to the queueng networks wth blockng. Normalsaton constant algorthms for product form queueng networks wth repettve servce blockng have been proposed n [1,3,15]. In [9] a MVA algorthm for the sub-class of cyclc product form queueng networks wth repettve servce blockng has been proposed. Ths algorthm can be consdered a hybrd algorthm because t uses recurrence equatons together wth the normalsaton constant for the computaton of the performance measures. In ths paper we derve a MVA algorthm that allows to compute performance measures for any product form queueng network wth repettve servce blockng. Moreover n the recursve equatons that are the bass of the MVA the dependency on the normalsaton constant dsappears. Ths feature could be used for dervng approxmaton technques nspred by smlar heurstc technques that have been developed for queueng networks wthout blockng (e.g. see [8,14,17]). The balance of the paper s the followng: Secton 2 presents the class of product form soluton queueng network wth repettve servce blockng. Secton 3 contans the actual contrbuton of ths paper, presentng a set of recursve equatons that yeld a Mean Value Analyss algorthm for the class of queung networks under nvestgaton. Secton 4 presents three examples of applcaton of ths algorthm for the evaluaton of performance measures. Secton 5 concludes the paper outlnng possble future works on ths topc. 2. Product-form queueng networks wth RS-RD blockng We consder a queueng network wth arbtrary topology. The blockng dscplne s the repettve servce blockng wth random destnaton (RS-RD): a customer upon completon of ts servce at queue attempts to enter destnaton queue j. If the node j s full, the customer s looped back nto the queue where t receves a new ndependent servce accordng to the server queue dscplne. The customer, after t receves a new servce, chooses a new destnaton staton ndependently of the one that t had selected prevously. In a dfferent defnton of the repettve servce blockng dscplne the customer s destnaton s determned after the frst servce and can not be modfed (for detals are references on the these types of blockng and related results see [4,5,13]). Let M denote the number of statons and N the number of customers. Each node has a sngle exponental server wth mean servce rate ¼ (wth 1 M). Wth n D [n 1 ; n 2 ;:::;n M ] we denote the network state when there are n customers at node. Wth b D [b 1 ; b 2 ;:::;b M ] we denote the vector of the capactes of the nodes, when customers stay n node queue, the node s full and the blockng occurs, D1means that node has nfnte capacty. In the followng we denote by n N the number of avalable postons n the buffer of staton. Obvously we have that n C n N D. We consder closed sngle class queueng networks wth RS-RD blockng mechansm. P Let a.n/ D max 0; N M jd1; j6d b j denote the mnmum feasble queue length at staton, for any 1 M. The state space of the queueng network s the set of all the feasble states, that s, ( ) M E.N/ D n: a.n/ n ; n D N :

3 M. Sereno / Performance Evaluaton (1999) Let P Dkp j k be the routng matrx of the network, where p j denotes the probablty that a customer leavng node tres to enter node j. We consder deadlock-free QN and assume that the routng matrx P s rreducble. In ths case the Markov process s rreducble on the state space E.N/ and hence exsts the equlbrum dstrbuton ³ D f³.n/ : n 2 E.N/g. The queueng networks we are consderng exhbt product form of the jont steady-state dstrbuton of the queue length under some constrants on the network parameters: ³.n/ D 1 G MY f.n /; where the normalsaton constant G s gven by G D P n2e.n/ Q M f.n /, and the defnton of the functon f.ð/, wth 1 M, depends on the network parameters, the th queue length, and the product form type. A complete defnton of the functon f.ð/ for dfferent blockng mechansms (ncluded the RS-RD) can be found n [4,5]. In the followng we recall these results for the RS-RD blockng mechansm. Let denote by x the vector that s a postve soluton of the followng system: x D x Ð P; where P s the routng matrx. In the lterature x s often called vst ratos vector. For any staton, wth 1 M, the functon f.ð/ can be defned as follows: f.n / D ² n : To compute the functon ² we have to dstngush two cases: pf (): ² D x ¼ n the case of RS-RD queueng networks wth reversble routng, staton wth arbtrary servce tme dstrbuton and a symmetrc schedulng dscplne [11] or exponental servce tme dstrbuton and arbtrary schedulng; pf (): ² D 1 " ; where ε D [" 1 ;" 2 ;:::;" M ] s obtaned by solvng the system ε D ε P,whereP Dkpj 0 k, wth 1, j M, pj 0 D ¼ j p j for j 6D and p 0 P D 1 j6d p0 j. Ths form of the functon ² holds for RS-RD queueng networks wth arbtrary topology, where the number of customers N satsfes the followng condton: M N > mn : (2) 1M For networks wth cyclc topology the prevous constrant s relaxed (½ nstead of >). (1)

4 22 M. Sereno / Performance Evaluaton (1999) The mean value analyss algorthm The computaton of the normalsaton constant developed for ths knd of PF-QNs by [1,3], allows, n prncple, the evaluaton of the steady state probablty dstrbuton of the states of the PF-QNs wth RS-RD blockng from whch several nterestng margnal probablty dstrbutons and performance ndces can be derved. In ths secton we frst defne the quanttes that are useful for obtanng our Mean Value Analyss algorthm for PF-QNs wth RS-RD blockng. The steady state probablty that n staton (wth 1 M) thereareh customers s defned as follows: Prfn D hi Ng D ³.n/: (3) n2e.n/ n Dh The steady state probablty that n staton there are at least h customers s defned as follows: Prfn ½ hi Ng D ³.n/: n2e.n/ n ½h (4) Usng the relaton n C n N D we can derve the steady state probablty that n the buffer of staton there are l avalable postons: Prfn N D li Ng DPrfn D li Ng: (5) We can also defne the steady state probablty that there are h customers at staton and k customers at staton j: Prfn D h; n j D ki Ng D ³.n/; (6) n2e.n/ n Dh; n j Dk and n the same manner of Eq. (4) the steady state probablty that there are at least h customers at staton and at least k customers at staton j: Prfn ½ h; n j ½ ki Ng D ³.n/: (7) n2e.n/ n ½h; n j ½k The probablty dstrbuton provded by Eq. (3) allows to compute the average number of customers at staton when there are N customers n the network. In the followng we denote ths quantty by E.N/. The throughput Ł.N/ of a staton s the sum of the average number of customers that depart per unt of tme from staton towards staton j, for any staton j such that p j > 0 Ł.N/ D M Prfn > 0; n j N > 0I Ngp j¼ : (8) jd1 j6d

5 M. Sereno / Performance Evaluaton (1999) For the MVA algorthm we need a relaton among the throughput of the statons. In case of pf () ths relaton s gven by the job flow balance equatons Ł.N/ D M Ł j.n/p j ; jd1 and hence the vector of the throughput of the statons s proportonal to the vst rato vector x. An mportant consequence s that for any par of statons and j Ł.N/ Ł j.n/ D x : (10) x j In case of pf () ths relaton must be obtaned usng the features of ths product form type. In partcular we use the non-empty condton that s mpled by Inequalty (2). Ths condton ensures that n the network there s no empty staton. If nstead of the moton of the customers we consder the moton of the empty postons of the buffers (also called holes) we can derve the routng matrx P that descrbes the moton of the holes (see [2]). The element Np j denotes the probablty that a hole leavng node tres to enter node j,where Np j D ¼ j Ð p j : M Np r rd1 In ths case usng the results of [10] (see also [18]), we can fnd a relaton between the throughput of the statons usng the routng matrx P. Let x be vst rato vector defned as a postve soluton of the followng system x D x Ð P: (11) The relaton of the throughput of the statons n case of pf () s Ł.N/ Ł j.n/ D Nx : (12) Nx j In the followng we wll wrte Ł.N/ Ł j.n/ D v ; (13) v j where v (wth 1 M) s defned as follows: 8 < x f pf./; v D : Nx f pf./: Lettng!.N/ represent the mean response tme of a customer at staton, ths quantty can be computed usng Lttle s formula [12] that n ths case assumes the followng expresson:!.n/ D E.N/ Ł.N/ ; 1 M: (14) The mean response tme for blockng queueng networks has been defned n [4]. (9)

6 24 M. Sereno / Performance Evaluaton (1999) Recursve expresson for probablty dstrbutons Let us prove now few lemmas that provde some recursve relatons for PF-QNs wth RS-RD blockng. The method used for dervng these relatons s based on the normalsaton constant calculus presented n [1,3]. In the followng we denote as G.N 0 / the normalsaton constant of the network wth N 0 customers and as G fg.n 0 / the normalsaton constant for the network wth all the statons except staton and N 0 customers. Lemma 1 (from [3]). In a RS-RD PF-QN wth M statons and N customers the steady state probablty that at staton there are h customers, s expressed wth the followng formula: Prfn D hi Ng D G fg.n h/ ² h G.N/ : (15) Usng the prevous lemma and Eq. (5) we can derve a relaton for the steady state probablty that n the buffer of staton there are l avalable postons: Prfn N D li Ng D G fg.n C l/ ² l : (16) G.N/ Eqs. (15) and (16) can be used to derve recursve expressons for dfferent dstrbutons of customers or of avalable buffer postons n a partcular staton of the QN. These expressons have the nce property that the dependences on the normalsaton constant dsappear, whle they are defned n terms of expresson of other customer dstrbutons computed for smaller number of customers crculatng n the network. An alternatve expresson for the result of Eq. (15) can be easly obtaned. Lemma 2. In a RS-RD PF-QN wth M statons and N customers the steady state probablty that at staton there are h customers s provded by the followng relaton: Prfn D hi Ng DH.N/ Prfn D h 1I N 1g; (17) wth H.N/ D Prfn D a.n 1/ C 1: Ng Prfn D a.n 1/I N 1g ; (18) and where a.n 1/ s the mnmum number of customers at staton when there are N 1 customers n the network. Proof. From Lemma 1 we have that Prfn D hi Ng D G fg.n h/ ² h G.N/ D G fg.n 1.h 1// G.N/ D G fg.n 1.h 1// G.N/ ² h 1 ² G.N 1/ G.N 1/ ²h 1 ²

7 D Prfn D h 1I N 1g M. Sereno / Performance Evaluaton (1999) G.N 1/ ² G.N/ G.N 1/ G fg.n.a.n 1/ C 1// D Prfn D h 1I N 1g G.N/ G fg.n.a.n 1/ C 1// ² : From Eq. (15) we have that Prfn D a.n 1/ C 1I Ng D G fg.n.a.n 1/ C 1// ² a.n 1/C1 : G.N/ From the prevous relaton we can derve that G fg.n.a.n 1/ C 1// G.N/ We can also derve that G fg.n.a.n 1/ C 1// ² a.n 1/ G.N 1/ By combnng the prevous relatons we have that ² D Prfn D a.n 1/ C 1I Ng ² a :.N 1/ D G fg.n 1..a.N 1/ C 1/ 1/ G.N 1/ D G fg.n 1 a.n 1// ² a.n 1/ G.N 1/ D Prfn D a.n 1/I N 1g: Prfn D hi Ng D Prfn D a.n 1/ C 1I Ng Prfn D a.n 1/I N 1g Prfn D h 1I N 1g D H.N/ Prfn D h 1I N 1g: We can easly generalse Eq. (17) n the followng way: ² a.n 1/ Prfn D h; n j D li NgDH.N/ Prfn D h 1; n j D li N 1g D H j.n/ Prfn D h; n j D l 1I N 1g: (19) Usng Eq. (17) we can derve a relaton for the steady state probablty that n staton there are at least h customers. Lemma 3. In a RS-RD PF-QN wth M statons and N customers the steady state probablty that at staton there are at least h customers s provded by the followng relaton: Prfn ½ hi Ng DH.N/.Prfn ½ h 1I N 1g Prfn D I N 1g/: (20) Proof. Usng Eq. (17) Eq. (4) becomes

8 26 M. Sereno / Performance Evaluaton (1999) Prfn ½ hi Ng D Prfn D li Ng ldh D H.N/ Prfn D l 1I N 1g D H.N/ ldh 1 tdh 1 Prfn D ti N 1g D H.N/ Ð.Prfn ½ h 1I N 1g Prfn D I N 1g/ : All the prevous relatons allow to prove the followng result. Lemma 4. In a RS-RD PF-QN wth M statons and N customers the steady state probablty that at staton there s at least one customer and n the buffer of staton j there s at least one avalable poston s provded by the followng relaton:! b Prfn > 0; n j N > 0I Ng DH 1.N/ 1 Prfn D I N 1g Prfn D t; n j D b j I N 1g : Proof. We can derve that Prfn > 0; n j N > 0I Ng DPrfn ½ 1; n j N ½ 1I Ng D D D hd1 hd1 hd1 b j Prfn D h; n j N D li Ng ld1 b j Prfn D h; n j D b j ld1 b j 1 sd0 Prfn D h; n j D si Ng D Prfn ½ 1; n j ½ 0I Ng D Prfn ½ 1I Ng hd1 li Ng td0 Prfn D h; n j D b j I Ng hd1 Prfn D h; n j D b j I Ng D H.N/.1 Prfn D I N 1g/ H.N/ H.N/ 1 Prfn D t; n j D b j I N td0 D H.N/ 1 Prfn D I N 1g 1 1g! Prfn D t; n j D b j I N td0 1g (21)!

9 M. Sereno / Performance Evaluaton (1999) Recursve expressons for average performance ndces The expressons obtaned n the prevous secton are the bass for the dervaton of recursve formulas for the average performance ndces that wll allow the development of the Mean Value Analyss algorthm for PF-QNs wth RS-RD blockng consdered n ths paper. These results are summarsed n the followng theorem. Theorem 1. In a RS-RD PF-QN wth M statons and N customers the average number of customers at a staton at the steady state s gven by: E.N/ D H.N/.1 C E.N 1/ Prfn D I N 1g. C 1// ; (22) the throughput of a staton s gven by 0! 1 M b 1 Ł.N/ D 1 Prfn D I N 1g Prfn D t; n j D b j I N 1g p j ¼ A (23) jd1 and the mean response tme of a customer at staton s gven by!.n/ D M jd1 td0 1 C E.N 1/ Prfn D I N 1g. C 1/ 1 Prfn D I N 1g 1 td0 Proof of Eq. (22). Wth some algebra we can derve that: E.N/ D Prfn ½ li Ng ld1 D H.N/ D H.N/ Prfn ½ l 1I N 1g ld1 Prfn D t; n j D b j I N Prfn D I N ld1! b 1 Prfn ½ mi N 1g Prfn D I N 1g md0 D H.N/.E.N 1/ C Prfn ½ 0I N 1g Prfn D I N 1g. C 1// D H.N/.1 C E.N 1/ Prfn D I N 1g. C 1// : Proof of Eq. (23). From Eq. (8) we have that Ł.N/ D Lemma 4 D M Prfn > 0; n j N > 0I Ngp j¼ jd1 0 M jd1 1 Prfn D I N 1g 1g! : (24) 1g p j ¼! b 1 Prfn D t; n j D b j I N Proof of Eq. (24). The proof of ths equaton follows from Lttle s law, Eq. (22), and Eq. (23). td0! 1 1g p j ¼ A :

10 28 M. Sereno / Performance Evaluaton (1999) From local to global relatons The am of next steps s the composton of the local recursve relatons derved n Theorem 1 to obtan the MVA algorthm. Let and r two statons, from Eq. (13) we have that v D Ł.N/ v r Ł r.n/ : Lemma 5. In a RS-RD PF-QN wth M statons and N customers the throughput of a staton r can be expressed wth the followng formula: Ł r.n/ D M N v v r Ð!.N/ Proof. Startng from ths obvous relaton N D M E.N/ we can apply Lttle s law, and hence N D M!.N/ Ð Ł.N/ D Ł r.n/ D Ł r.n/ M M!.N/ Ð Ł.N/ Ł r.n/ :!.N/ Ð v v r ; where r s a staton of the QN. From the prevous dervaton t follows that Ł r.n/ D M N v v r Ð!.N/ : 3.4. Recursve equatons: dfferences and smlartes between the blockng and the non-blockng queueng networks The recursve equatons presented before represent a generalsaton of the correspondng equatons developed for PF-QNs wthout blockng (see for nstance [6]). For a node wth nfnte capacty we have that a.n/ D 0, for any N D 0; 1;:::. In ths case we have the followng relaton: Prfn D hi Ng DPrfn > 0I NgÐPrfn D h 1I N 1g; (26) (25)

11 when h D 1wehavethat M. Sereno / Performance Evaluaton (1999) Prfn D hi Ng Prfn > 0I Ng D Prfn D h 1I N 1g : From the prevous equaton we can see that for a node havng a buffer wth nfnte capacty the term H.N/ corresponds to the probablty that there s at least one customers n the node. For a node wth nfnte capacty buffer ( D1)networkswehavethatPrfn D I Ng D0, hence Eq. (20) and (21) and consequently Eq. (22), (23), and (24), assume the same form of the correspondng equatons derved for non-blockng queueng networks Mean Value Analyss for RD-RD networks: the algorthm The relatonshps derved n ths secton can now be combned n a complete recursve scheme for the computaton of all the performance ndces of PF RD-RD networks whose general organsaton follows that of the MVA for product form Queueng Networks wthout blockng. Eq. (24) provdes the way of computng the sojourn tmes for all the staton of the network when there are L customers n the network n terms of quanttes computed when n the network there are L 1 customers. Wth these quanttes t s possble to compute the throughput of the reference staton (arbtrarly chosen) when there are L customers n the network usng Lemma 5. From the throughput of the reference staton, usng Eq. (13), we can compute the throughputs of all the staton of the network. Usng Eq. (23) of Theorem 1 from the throughput of staton we can compute the term H.L/. The knowledge of these terms allows to compute the average number of customers for all the staton (Eq. (22) of Theorem 1). In order to complete our recursve scheme we need to compute the probabltes that appear n most of these formulas. In partcular, for any staton, we must evaluate the probabltes Prfn D hi Lg, for 0 ½ h ½. Usng Eq. (17) of Lemma 2 we have that Prfn D hi Lg DH.L/ Prfn D h 1I L 1g for 1 ½ h ½ : We can compute Prfn D 0I Lg as Prfn D 0I Lg D1 Prfn D hi Lg: (27) hd1 For any par of statons and j such that p j > 0 we need to compute the probabltes Prfn D h; n j D b j I Lg,for0 h.wehavethat Prfn D h; n j D b j I Lg DH.L/ Prfn D h 1; n j D b j I L 1g for 1 h : We can compute Prfn D 0; n j D b j I Lg as Prfn D 0; n j D b j I Lg DPrfn j D b j I Lg Prfn D h; n j D b j I Lg: (28) hd1 The recursve equatons that have been derved n ths paper hold for all the types of RS-RD product form queueng networks. However, as t has been ponted out n Secton 2 there are two dfferent cases

12 30 M. Sereno / Performance Evaluaton (1999) of product form soluton (pf () and pf () of Secton 2). In the second case the exstence of the product form soluton depends on the number of customers n the network (see Inequalty (2)). To account n the MVA algorthm ths fact we ntroduce the concept of least product form number of customer (mn PF ). Ths value s the mnmum number of customers that allows to have the product form soluton and t s defned as follows: 8 0 fpf./; M >< mn PF D M >: mn 1M mn C 1 1M f pf./ and cyclc topology; fpf./ and arbtrary topology: The ntalsaton step takes nto account the type of product form soluton. The MVA algorthm computes the performance ndces of the RS-RD PF-QN n the followng way: begn end Intalse(mn PF ) for L :D mn PF C1 to N do One-Step-MVA(L) The detals of the procedures Intalse(mn PF )andone-step-mva(l) are provded n Tables 1, 2 and 3. The procedure Intalse(mn PF ) depends on the type of product form soluton. In the case pf () the ntalsaton can be done usng ether a drect method (soluton of the Markov chan) or usng the normalsaton constant. In both cases we have to pont out that the computatonal effort s constant wth respect to the number of customers because t depends only on the value of mn PF. Table 1 Procedure Intalse for pf() Procedure Intalse(0) begn for D 1 to M do n.0/ D 0 end for D 1 to M do for h D 1 to do Prfn D hi 0g D0 Prfn D 0I 0g D0 for D 1 to M do for j D 1 (1 j M) wthp j > 0 do for h D 0 to do Prfn D h; n j D b j I 0g D0 Intalse Table 2 Procedure Intalse for pf() Procedure Intalse(mn PF ) begn end for D 1 to M do n.0/ D 0 for D 1 to M do for h D 1 to do Computaton of Prfn D hi mn PF g P Prfn D 0I mn PF gd1 b hd1 Prfn D hi mn PF g for D 1 to M do for j D 1 (1 j M) wthp j > 0 do for h D 0 to do Computaton of Prfn D h; n j D b j I mn PF g Intalse

13 M. Sereno / Performance Evaluaton (1999) Table 3 Procedure One-Step-MVA Procedure One-Step-MVA(L) (1) begn (2) for D 1 to M do Computaton of!.l/ (Theorem 1, Eq. (24)) Let r be the reference staton (arbtrarly chosen) (3) Computaton of Ł r.l/ (Lemma 5, Eq. (25)) (4) for D 1 to M do Computaton of Ł.l/ (Eq. (13)) H.l/ (Theorem 1, Eq. (23)) E.l/ (Theorem 1, Eq. (22)) (5) for D 1 to M do for h D 1 to do Computaton of Prfn D hi Lg (Eq. (17)) Computaton of Prfn D 0I Lg (Eq. (27)) (6) for D 1 to M do for each j D 1 (1 j M) wthp j > 0 do for h D 1 to do Computaton of Prfn D h; n j D b j I Lg (Eq. (19)) Computaton of Prfn D 0; n j D b j I Lg (Eq. (28)) (7) end One-Step-MVA The dervaton of the tme complexty for the MVA algorthm s qute smple. The procedure One-Step-MVA s executed at most N tmes. It s possble to see that the tme complexty of the One-Step-MVA s O.maxfb 1 ; b 2 ;:::;b M gðm/. From ths t follows that the tme complexty of the algorthm s O.maxfb 1 ; b 2 ;:::;b M gðm Ð N/. 4. Numercal results In ths secton we present three examples to llustrate the MVA algorthm for PF-QNs wth RS-RD blockng. The frst example s a queueng network wth server topology. Ths type of networks have product form soluton of type pf () when only the central server has fnte capacty buffer (see [4,5]). The parameters of the example network are M D 5, N D 8, p 1 j D 0:25, 2 j 5, ¼ D 1, 1 5, b 1 D 3, and D1,2 5. Table 4 summarses some performance measures obtaned for ths network. The second example s a queueng network wth cyclc topology. Ths type of networks have product form soluton of type pf () when Inequalty (2) holds (wth ½ nstead of >) (see [4,5]). The parameters of the example network are M D 5, N D 22, ¼ 1 D 1:5, ¼ 2 D 3:0, ¼ 3 D 2:5, ¼ 4 D 0:5, ¼ 5 D 6:0, D 5, 1 5. Table 5 summarses some performance measures obtaned for ths network.

14 32 M. Sereno / Performance Evaluaton (1999) Table 4 Performance measures for the central server network Node Ł.N/!.N/ n.n/ Table 5 Performance measures for the cyclc network Node Ł.N/!.N/ n.n/ Table 6 Performance measures for the general topology network Node Ł.N/!.N/ n.n/ Fg. 1. RS-RD queueng network wth general topology. Last example s a queueng network wth general topology llustrated n Fg. 1. Ths type of networks have product form soluton of type pf () when Inequalty (2) holds (see [4,5]). The parameters of the example network are M D 5, N D 22, ¼ 1 D 5:0, ¼ 2 D 2:0, ¼ 3 D 11:0, ¼ 4 D 3:0, ¼ 5 D 1:0, D 5, 1 5. The routng probabltes of are p 12 D 0:5, p 13 D 0:5, p 24 D 0:5, p 25 D 0:5, p 35 D 1, p 41 D 1, and p 51 D 1 (customer moton). The vst rato vector computed wth respect to the matrx P s x D [1:0; 0: ; 0: ; 0:75; 0:25]. Table 6 summarses some performance measures obtaned for ths network. 5. Conclusons In ths paper a Mean Value Algorthm for queueng networks wth repettve servce blockng mechansm and product form soluton has been proposed. Basc to the dervaton of ths algorthm are recursve expressons of the performance ndces that are the generalsaton of those derved for the Mean Value Analyss of product form queueng networks wthout blockng. An appealng feature of the recursve formulaton of MVA could be the possblty of developng approxmaton technques nspred by smlar heurstc technques that have been developed for queueng networks wthout blockng (e.g. see [8,14,17]). Ths possblty s currently under nvestgaton. The algorthm presented n the paper has been mplemented and tested usng a few smple examples. More work needs be done n order to solve the numercal problems that can arse because the presence of subtracton operatons n some of the recursve equatons. Another drecton of research s the dervaton of recursve equatons for product form queueng networks wth other blockng mechansms.

15 M. Sereno / Performance Evaluaton (1999) Acknowledgements Ths work has been supported n part by the 60% projects, and n part by the Esprt Human Captal and Moblty project MATCH. References [1] I.F. Akyldz, H. von Brand, Computatonal algorthm for networks of queues wth rejecton blockng, Acta Inf. 26 (1989) [2] I.F. Akyldz, H. von Brand, Dual and selfdual networks of queues wth rejecton blockng, Computng 43 (1989) [3] S. Balsamo, M.C. Clò, Convoluton algorthm for product-form queueng networks wth blockng, n: Proc. Thrd Internatonal Workshop on Queueng Networks wth Fnte Capacty, Bradford, England, July [4] S. Balsamo, V. De Ntto-Personé, Closed queueng networks wth fnte capactes: Blockng types, product-form soluton and performance ndces, Performance Evaluaton 12 (1991) [5] S. Balsamo, V. De Ntto-Personé, A survey of product-form queueng networks wth blockng and ther equvalences, Ann. Oper. Res. 48 (1994) [6] S.C. Bruell, G. Balbo, Computatonal Algorthms for Closed Queueng Networks, Elsever North-Holland, New York, [7] J.P. Buzen, Computatonal algorthms for closed queueng networks wth exponental servers, Commun. ACM 16 (9) (1973) [8] K.M. Chandy, D. Neuse, Lnearzer: A heurstc algorthm for queueng network models of computng systems, Commun. ACM 25 (2) (1982) [9] M.C. Clò, Mva for product-form cyclc queueng networks wth RS blockng, n: Proc. Thrd Internatonal Workshop on Queueng Networks wth Fnte Capacty, Bradford, England, July [10] A. Hordjk, N. van Djk, Networks of queues wth blockng, n: Proc. Performance 81, 1981, pp [11] F.P. Kelly, Reversblty and Stochastc Networks, Wley, London, [12] J.D.C. Lttle, A proof of the queueng formula l D ½w, Oper. Res. 9 (1961) [13] R.O. Onvural, Survey of closed queueng networks wth blockng, ACM Comput. Surv. 22 (1990) [14] K.R. Pattpat, M.M. Kostreva, J.L. Teele, Approxmate mean value analyss algorthms of queueng networks: Exstence, unqueness and convergence results, J. ACM 3 (1990) [15] H.G. Perros, Queueng Networks wth Blockng, Oxford Unversty Press, Oxford, [16] M. Reser, S.S. Lavenberg, Mean value analyss of closed multchan queueng networks, J. ACM 27 (2) (1980) [17] P.J. Schwetzer, A survey of mean value analyss, ts generalzatons, and applcatons, for networks of queues, Techncal report, Unversty of Rochester, Rochester, USA, [18] N.M. van Djk, Queueng Networks and Product Forms A Systems Approach, Wley, New York, Matteo Sereno was born n Nocera Inferore, Italy. He receved hs Doctor degree n Computer Scence from the Unversty of Salerno, Italy, n 1987, and hs Ph.D. degree n Computer Scence from the Unversty of Torno, Italy, n Snce November 1992, he s an assstant professor at the Computer Scence Department of the Unversty of Torno. Hs current research nterests are n the area of performance evaluaton of computer systems, modellng of communcaton networks and parallel archtectures, queueng network and stochastc Petr net models.