Optimal Threshold Control by the Robots of Web Search Engines with Obsolescence of Documents

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1 Optmal Threshold Control by the Robots of Web Search Engnes wth Obsolescence of Documents Konstantn Avrachenkov, Alexander Dudn, Valentna Klmenok, Phlppe Nan, Olga Semenova Abstract A typcal web search engne conssts of three prncpal parts: crawlng engne, ndexng engne, and searchng engne. The present work ams to optmze the performance of the crawlng engne. The crawlng engne fnds new web pages and updates web pages exstng n the database of the web search engne. The crawlng engne has several robots collectng nformaton from the Internet. We frst calculate varous performance measures of the system (e.g., probablty of arbtrary page loss due to the buffer overflow, probablty of starvaton of the system, the average tme watng n the buffer). Intutvely, we would lke to avod system starvaton and at the same tme to mnmze the nformaton loss. We formulate the problem as a mult-crtera optmzaton problem and attrbutng a weght to each crteron we solve t n the class of threshold polces. We consder a very general web page arrval process modeled by Batch Marked Markov Arrval Process and a very general servce tme modeled by Phase-type dstrbuton. The model has been appled to the performance evaluaton and optmzaton of the crawler desgned by INRIA Maestro team n the framework of the RIAM INRIA-Canon research project. 1 Introducton The problem of control by the robots (crawlers) that traverse the web and brng web pages to the ndexng engne that updates the data base of a web INRIA Sopha Antpols, France, K.Avrachenkov@sopha.nra.fr Belarusan State Unversty, Belarus, dudn@bsu.by Belarusan State Unversty, Belarus, klmenok@bsu.by INRIA Sopha Antpols, France, nan@sopha.nra.fr Belarusan State Unversty, Belarus, olgasmnv@tut.by 1

2 search engne s formulated and analyzed n [13]. Ths problem s formulated n [13] as the controlled queueng system. The system has a sngle server wth the exponental servce tme dstrbuton, fnte buffer of capacty K 1, K 2. There are N avalable robots and each of these robots, when actvated, brngs pages to the server n a Posson stream at fxed rate. These N statonary Posson processes are mutually ndependent and ndependent of servce tmes. The number of actve robots may be modfed at any arrval or departure event. When an arrval occurs, one or several actve robots can be de-actvated at once. When a departure occurs the controller may decde ether to actvate one or several avalable non-actve robots. Of course, the controller can choose to do nothng(.e. the number of actve robots remans the same). In [13], the problem of fndng a polcy that mnmzes a weghted sum of the loss rate and starvaton probablty (probablty of the empty system) s consdered. It s solved by means of the tools of the Markov Decson Processes theory. The followng possble generalzatons of the model, whch are certanly worthwhle analyzng, are mentoned n [13]: More general nput processes, e.g., an M M P P (Markov Modulated Posson Process) should be consdered so as to reflect more accurately travelng tmes of robots n the network; Because of the obsolescence of stored documents ssue, the watng tme should be bounded, even f the buffer sze s effectvely nfnte; Other cost functons could be nvestgated, for nstance, cost functons ncludng response tmes. In ths paper, we made all the mentoned and some further generalzatons. We assume that, under the fxed number of currently actve robots, the arrval process s of the BMAP type. The BMAP s a more general process comparng to the MMPP and allows delverng a batch of web pages to be ndexed whle the MMPP assumes that the pages are delvered one-byone. It s very typcal for a computer system to operate n batch mode. Also BMAP allows delverng pages at the phase change tme moments. We assume that the servce tme dstrbuton s of the PH (Phase) type whch s much more general comparng to the exponental dstrbuton assumed n [13]. The class of phase type dstrbutons s dense n the feld of 2

3 all postve-valued dstrbutons and practcally we can deal wth any real dstrbuton [2]. Snce web pages can become obsolete, we bound the watng tme stochastcally. Watng tme of each web page n a buffer s restrcted by a random varable havng P H dstrbuton dentcal and mutually ndependent for all web pages. The phase type dstrbuton has been used to model obsolescence tmes for nstance n [15]. We suppose that the cost functon can have a more general form than n [13] and nclude the cost of the obsolescence and the response tme. In the next secton we formulate the model and the optmzaton problem. Secton 3 contans the steady-state analyss of the mult-dmensonal Markov chan whch defnes dynamcs of the system under the fxed values of the parameters defnng the control strategy. In Secton 4 the man performance measures of the system are computed. In Secton 5 the condtonal sojourn tme dstrbutons are calculated. In Secton 6 a partcular case of ordnary arrvals s dscussed. In Secton 7 the theoretcal results are llustrated by numercal examples. In partcular, the mathematcal model s appled to the performance evaluaton and optmzaton of the web crawler desgned by INRIA Maestro team n the framework of the RIAM INRIA- Canon research project. Secton 8 concludes the paper. 2 Mathematcal Model We consder a sngle server system wth the fnte buffer of capacty K 1, K 2. So, the total number of web pages whch can stay n the system s restrcted by the number K. Web pages are served by a server n the order of ther arrvals. Servce tmes of web pages are ndependent dentcally dstrbuted random varables havng P H dstrbuton wth rreducble representaton(β, S). It means the followng. Servce of a web page s defned as a tme untl the contnuous-tme Markov chan m t, t 0, havngthestates (1,...,M) as the transent and state 0 as absorbng one reaches the absorbng state. An ntal state of the chan s selected n a random way, accordng to the probablty dstrbuton defned by the row-vector (β, 0), where β s the stochastc row vector of dmenson M. Transtons ( ) of the Markov chan m t, t 0, S S0 are descrbed by the generator where the matrx S s a subgenerator and the column vector S 0 s defned by S 0 = Se M and has all 0 0 non-negatve and at least one postve components, e M s the column vector 3

4 of dmenson M consstng of all 1 s. The average servce tme b 1 s gven by b 1 = β( S) 1 e M. For more detals about the PH type dstrbuton, ts propertes, specal cases and applcatons see [11, 12]. Web pages can be delvered nto the system by N avalable robots. The number of actve robots vares n the set {1,...,N}. We assume that the process of web pages delverng under l, l = 1,N, actve robots s descrbed as follows. Let ν t, t 0, be an rreducble contnuous tme Markov chan havng fnte state space {0,1,...,W}. Sojourn tme of the chan ν t, t 0, n the state ν has exponental dstrbuton wth a parameter λ (l) ν. After ths tme expres, wth probablty p (l) 0 (ν,ν ) the chan jumps nto the state ν wthout generaton of web pages and wth probablty p (l) k (ν,ν ) the chan jumps nto the state ν and a batch consstng of k web pages s generated, k 1. The ntroduced probabltes satsfy condtons: p (l) 0 (ν,ν) = 0, W k=1ν =0 p (l) k (ν,ν )+ W ν =0 p (l) 0 (ν,ν ) = 1, ν = 0,W, l = 1,N. TheparametersdefnngthsflowarekeptnthesquarematrcesD (l) k, k 0, l = 1,N, of sze W = W +1 defned by ther entres: (D (l) 0 ) ν,ν = λ (l) ν, (D(l) 0 ) ν,ν = λ(l) ν p 0(ν,ν ), (1) (D (l) k ) ν,ν = λ(l) ν p (l) k (ν,ν ), ν,ν = 0,W, k 1, l = 1,N. Denote the generatng functon D (l) (z) = k=0 D (l) k zk, z 1. The matrx D (l) (1) s the nfntesmal generator of the process ν t,t 0, under the fxed number l of actve robots. The statonary dstrbuton vector θ (l) of ths process satsfes the equatons θ (l) D (l) (1) = 0,θ (l) e = 1. Here and n the sequel, 0 s the zero row vector. The average ntensty λ (l) (fundamental rate) of the BMAP under the fxed number l of actve robots s defned by λ (l) = θ (l)dd(l) (z) z=1 e, dz and the ntensty λ (l) g of group arrvals s defned by λ (l) g = θ (l) ( D (l) 0 )e. 4

5 The varance v (l) of ntervals between group arrvals s calculated as follows: v (l) = 2(λ (l) g ) 1 θ (l) ( D (l) 0 ) 1 e (λ (l) g ) 2, and the correlaton coeffcent c cor (l) of ntervals between successve group arrvals s gven by c (l) cor = 1 v(λ (l) g ) 2 ( ) λ g (l) θ (l) ( D (l) 0 ) 1 (D(1) (l) D (l) 0 )( D(l) 0 ) 1 e 1. The ntroduced representaton of the arrval process va the matrces D (l) k, k 0, l = 1,N, unfes several possble nterpretatons of the process of web pages delvered by a fxed number of actve robots: 1. The processes of web pages delverng by all robots are ndependent BMAP processes. Let the process of web pages delverng by the lth robotbethebmap whchs governed by thecontnuous tmemarkov chan ν (l) t, t 0, havng fnte state space {0,1,...,W l } and defned by the matrces D (l) k, k 0, of sze W l = W l + 1. See [10] for more detals about the BMAP, ts propertes and specal cases. We denote D (l) (z) = k=0 D (l) k zk, z 1. Let us assume that the robots are arranged n such a way that the frst robot s always actve, then, when a queue decreases, the second robot can be actvated, etc, the Nth robot s the most rare actvated. The matrces D (l) k, k 0, l = 1,N, of sze W = N k=1 W k defned by formulae (1) are expressed va the matrces D (l) k, k 0, of sze W l, l = 1,N, descrbng the BMAPs n the followng way: D (l) 0 = D (1) 0 D (2) 0 D (l) 0 D(l+1) (1) D (N) (1), D (l) k = D(1) k D (2) k D (l) k I W l+1 I WN, k 1. Here and denote Kronecker product and sum of matrces correspondngly (see, e.g., [7]), I L denotes dentty matrx of sze L. If the sze of the matrx s clear from context the suffx can be omtted. def I 0 = The common process of web pages delvered by all robots together s the BM AP process drected by the contnuous tme Markov chan 5

6 ν t, t 0, havng fnte state space {0,1,...,W} and defned by the matrces D k, k 0, of sze W. Some set of thnnng probabltes q 1,...,q N, 0 < q 1 < < q N = 1, s fxed. When l robots are actve, procedure of thnnng the BM AP process wth the thnnng probablty q l s appled. It means that an arbtrary arrvng batch s accepted wth probablty q l and s rejected wth the complmentary probablty 1 q l. We denote D(z) = D k z k, z 1. The matrces D (l) k, k 0, l = 1,N, defned by formulae (1) are expressed va the matrces D k, k 0, descrbng the common BMAP and va the thnnng probablty q l n the followng way: k=0 D (l) 0 = D 0 q l +D(1)(1 q l ), D (l) k = D kq l, k Let the process of web pages delvered by robots s descrbed by a BMMAP.TheBMMAP sdrectedbycontnuoustmemarkov chan ν t, t 0, havng fnte state space {0,1,...,W}. Sojourn tme of the chan ν t, t 0, n the state ν has exponental dstrbuton wth a parameter λ ν. After ths tme expres, wth probablty p 0 (ν,ν ) the chan jumps nto the state ν wthout generaton of web pages and wth probablty p (l) k (ν,ν ) the chan jumps nto the state ν and a batch consstng of k, k 1 web pages are delvered by the lth robot. The ntroduced probabltes satsfy condtons: p 0 (ν,ν) = 0, N W l=1 k=1ν =0 p (l) k (ν,ν )+ W p 0 (ν,ν ) = 1, ν = 0,W. ν =0 The matrces D (l) k, k 0, l = 1,N, defned by formulae (1) are expressed va the matrces D 0, D (l) k, k 1, l = 1,N, formed by the probabltes p 0 (ν,ν) and p (l) k (ν,ν ) n the followng way: D (l) 0 = D 0 + N j=1 m=l+1 D (m) j, D (l) k = l m=1 D (m) k, k The process of web pages delvered by all robots s the BMAP process drected by the contnuous tme Markov chan ν t, t 0, havng fnte state space {0,1,...,W} and defned by the transton ntensty matrces D (l) k, k 0, l = 1,N, dependng on the number of actve robots. 6

7 Interpretaton 3 seems be the most attractve because t assumes that the work of the robots can be dependent, whch s qute realstc. Because the total amount of web servers from whch new pages should be brought s more or less constant, reducton of the number of actve robots causes the ncrease of the feld n the Internet, whch s crawled by each robot, and correspondngly causes a change of tme for nformaton retreval by robots. So, the BMMAP looks to be the most realstc model for the web page delvery process. If a batch of delvered web pages meets free server one web page starts the servce mmedately whle the rest moves to the buffer. If the server s busy at an arrval epoch, all web pages of the batch are placed nto the buffer f there s enough free space n the buffer. If the number of free places n the buffer s less than the number of web pages n the batch, the correspondng number of web pages s lost. It means that we consder so called partal admsson strategy. The alternatve strateges of complete rejecton or complete admsson can be nvestgated n analogous way. For each webpage placednto thebuffer, thewatng tmesrestrcted by the random varable (so called obsolescence tme) havng P H dstrbuton wth rreducble representaton (γ,γ). It means the followng. Avalable watng tme of the th web page n the buffer s defned as a tme untl the contnuous-tme Markov chan r () t, t 0, havng the states (1,...,R) as the transent and state 0 as absorbng one, reaches the absorbng state. Transton of ths process nto the absorbng state means that ths web page gets out of date(obsolescence occurs). An ntal state of the chan s selected n a random way, accordng to the probablty dstrbuton defned by the row vector (γ,0), where γ s the stochastc row vector of dmenson R. Transtons ( ) of the Markov chan r () t, t 0, are descrbed by the generator Γ Γ0 where the matrx Γ s sub-generator and the column vector Γ s defned by Γ 0 = Γe. The average tme untl obsolescence g 1 s gven by g 1 = γ( Γ) 1 e. If the obsolescence tme expres before a web page s pcked-up from the buffer to the server, t s assumed that ths web page mmedately leaves the buffer and s lost. The obsolescence tmes of dfferent web pages are ndependent of each other and dentcally dstrbuted. It s worth to note that the analyss presented below could be drastcally smplfed f we suggest that the obsolescence tme s exponentally dstrbuted. However, ths suggeston rarely holds true n the real world systems because ths suggeston means that, wth hgh probablty, nformaton obsoletes very quckly. 7

8 A reasonable class of robots control strateges s the class of the threshold strateges defned as follows. Integers j 1,...,j N 1 are fxed such that 1 = j 0 < j 1 < < j N 1 < j N = K. If the number of web pages n the system satsfes nequalty j r 1 +1 j r, then N r +1 robots are actve and r 1 robots are de-actvated, r = 1,N. Note that the descrbed threshold strateges are popular n lterature n controlled queues, see, e.g., [1, 3, 5, 6, 14]. For some systems, t s proven that the optmal strategy n the class of all Markovan strateges belongs to the class of threshold strateges. For some other systems such a result s not proven, but the optmal strategy s sought n the class of threshold strateges. Advantage of such strateges s ther ntutve justfcaton and relatve smplcty of mplementaton n real-lfe systems. Numercal examples presented n the paper [13] for a partal case of our model confrm that the threshold strateges are optmal n the class of all Markovan strateges, although the authors cannot prove ths fact. Our system s much more complcated and we also cannot prove optmalty of the optmal threshold strategy n wder classes of strateges. We just try to fnd an optmal threshold strategy and beleve that t s optmal or sub-optmal n wder classes as well. We also menton that the descrpton of the gven above threshold strategy suts only for the case when N K. Whle the numercal examples presented n [13] address, e.g., the case N = 16 and K = 5. However, f we look at the optmal strategy gven by Fgure 2 n [13], we see that the optmal number of the actve robots vares between 4 and 14 wth 4 swtchng ponts where two or three robots are actvated or de-actvated. Because, n contrast to [13], we do not assume that each actve robot generates a statonary Posson process of arrvals at a fxed rate but we assume that each robot generates a batch Markovan arrval process, we see that our strategy suts for the case N > K as well. We could acheve ths formally by allowng non-strct nequaltes: 1 = j 0 j 1 j N 1 j N = K n fxng the thresholds. Thus, speakng below about a robot we may thnk about a vrtual robot as a group of several avalable real robots whch are actvated and de-actvated smultaneously. We wll solve the problem of choosng the optmal threshold strategy. The cost functon s assumed to be of the followng form: J = λ(c loss P loss +c obs P obs )+a V (1) +c rob N act +c star P star (2) where λ s the mean number of pages delvered nto the system by robots durng a unt of tme (fundamental rate of the arrval process), P loss s the probablty of arbtrary page loss due to the buffer overflow, P obs s the probabltyofarbtrarypageobsolescencedurngwatngnthequeue, P star sthe 8

9 probablty of the system starvaton, V (1) s the response tme (the average sojourn tme of web pages whch are not lost or deleted dueto obsolescence), N act s the average number of actve robots, and c loss,c obs,a,c rob,c star are the correspondng non-negatve cost coeffcents. The values of the cost coeffcents can be set up by experts n the doman. Alternatvely, the cost coeffcents can be vewed as Lagrange multplers n the constraned problem and can be found from the dual problem formulaton. It s clear that f the average number of actve robots s ncreasng then the three frst summands n (2) are ncreasng whle the last summand, the cost of system starvaton, s decreasng. The last summand s mportant because starvaton means that the ndexng machne s dle and so freshness of the data base suffers. The problem of mnmzaton of the cost crteron (2) s not trval. To solve ths problem, we wll use socalled drect approach. To ths end, we wll calculate the statonary dstrbuton of the system state under an arbtrary fxed set of thresholds (j 1,...,j N 1 ). It wll allow us to calculate the man performance measures of the system and the value J of the cost crteron as a functon J(j 1,...,j N 1 ). Problem of fndngthe optmal set (j 1,...,j N 1 ) s then easly solved wth the help of a computer, e.g., by enumeraton. 3 Statonary dstrbuton of the number of web pages n the system Let some set of thresholds (j 1,...,j N 1 ) be fxed. We are nterested n the statonary dstrbuton of theprocess t, t 0, where t s thenumberof web pages n thesystem at the epocht, t = 0,K. Ths processs non-markovan. To nvestgate ths process, we wll consder the followng mult-dmensonal contnuous tme Markov chan ξ t = { t,ν t,m t,r (1) t,...,r (t 1) t }, t 0, where ν t s the state of the drectng process of arrval process at epoch t, ν t = 0,W; m t s the state of the process whch drects the servce at epoch t, m t = 1,M; r () t s the state of the process, whch drects the obsolescence of the th web page n a queue at epoch t, r () t = 1,R, = 0,K 1. We assume that the web pages n the buffer are enumerated n the order of ther arrval nto thesystem. If abatch of web pages arrves to thesystem, the accepted web pages are enumerated n a unform random manner. When a web page s pcked up for the servce or s deleted from the queue because 9

10 ts admssble watng tme expres, the rest of web pages s mmedately enumerated respectvely. Denote p(0,ν) = lm t P{ t = 0,ν t = ν}, p(1,ν,m) = lm t P{ t = 1,ν t = ν,m t = m}, (3) p(,ν,m,r 1,...,r 1 ) = lm P{ t =,ν t = ν,m t = m,r (1) t t = r 1,...,r ( 1) t = r 1 }, 2. Because the state space of the Markov chan ξ t,t 0, s fnte and due to the assumpton about rreducblty of the processes defnng arrval, servce and obsolescence processes, lmts (3) exst. Enumerate the states of the Markov chan ξ t,t 0, n the lexcographc order and form the probablty row vectors p, = 0,K, of probabltes correspondng to the state of the frst component of the process ξ t,t 0. Denote also p = (p 0,...,p K ). Let Q be the nfntesmal generator of the Markov chan ξ t,t 0, and Q, be the block of the generator Q consstng of enumerated n lexcographc order rates of transton of the Markov chan ξ t,t 0, from the states wth the value of the component t to the states wth the value of ths component,, 0. Dmenson of the block Q, s defned by ( WM a R b ) ( WM a R b ) where a l = mn{l,1}, b l = max{(l 1),0}, l = 0,K. Lemma. Non-zero blocks Q, of the nfntesmal generator Q of the Markov chan ξ t,t 0, are defned by Q 0,j = D (N) 0, j = 0, D (N) j β γ (j 1), j = 1,K 1, (D (N) (1) K 1 r=0 D (N) r ) β γ (K 1), j = K, { I Q, 1 = W S 0, = 1, I W B 1, > 1, { D (χ()) Q, = 0 A 1, = 1,K 1, D (χ()) (1) A 1, = K, D (χ()) l I MR 1 γ l, l = 1,K 1, < K 1, Q,+l = (D (χ()) (1) K 1 D r (χ()) ) I MR 1 γ l, l = K, < K, r=0 10

11 Here > 0. B = S 0 β e R I R 1 +I M Γ 0, A = S Γ, = 0,K 1, l def γ = γ... γ }{{} l l def, Γ = } Γ... Γ {{}, Γ l 0 l def = Γ 0... Γ }{{} 0, l 1, l and the value χ(), whch corresponds to the number of actve robots when web pages stay n the system, s calculated by χ() = N ψ() where the value ψ(), ψ() = 0,N 1, s defned by relatons j ψ() +1 j ψ()+1, = 0,K. Proof of the lemma s mplemented by analyzng the probabltes of transton of the mult-dmensonal Markov chan ξ t,t 0, under the fxed set of thresholds (j 1,...,j N 1 ) durng an nterval of nfntesmal length. It s rather transparent and s omtted. It s well-known that the vector p of statonary probabltes satsfes the followng equlbrum equatons pq = 0, pe = 1. (4) The dmenson of the vector p s equal to W(1+M RK 1 1 R 1 ). It can be rather hgh. For nstance, f W = M = R = 2 ths dmenson s equal to 2 K+1 2. For K = 10 ths equals to So, drect soluton of equatons (4) by brute force can be very tme and computer memory demandng. An effectve and numercally stable algorthm for computng the blocks p, = 0,K, of the vector p, whch explots a structure of generator Q s presented n [9], and conssts of the followng steps: 1. Compute the matrces G, = 0,K 1, from the backward recurson K 1 G = ( Q +1,+1 Q +1,+1+l G +l G +l 1...G +1 ) 1 Q +1,, l=1 = K 1,K 2,...,0. 2. Compute the matrces Q,l from the backward recursons Q,K = Q,K, = 0,K, Q,l = Q,l +Q,l+1 G l, = 0,l,l = K 1,K 2,...,0. 11

12 3. Compute the matrces F l, l = 0,K, by recurson l 1 F 0 = I, F l = F Q,l ( Q l,l ) 1,l = 1,K. =0 4. Compute the vector p 0 as the unque soluton to the followng system of lnear algebrac equatons: K p 0 Q 0,0 = 0, p 0 F l e = 1. l=0 5. Compute the vectors p, = 1,K, by formula p = p 0 F, = 1,K. Thus, the problem of computng the statonary dstrbuton p, = 0,K, of the consdered queueng system under an arbtrary fxed set of thresholds (j 1,...,j N 1 ) can be effcently solved. 4 Man Performance Measures of the System As the man performance measures of the system we consder the values λ, P loss, P obs, P star, N act whch appear n the cost crteron (2). The calculaton of the values P star, N act s the easest. Theorem 1. The probablty P star of the system starvaton (the probablty of the dle state of the ndexng machne) s calculated by P star = p 0 e W. (5) The average number N act of actve robots s calculated by N act = N nϕ n, (6) n=1 where probablty ϕ n that n robots are actve at an arbtrary epoch s computed by ϕ n = j N+1 n =j N n +1 p e WM a R b,n = 1,N. (7) 12

13 Theorem 2. The probablty P loss of an arbtrary web page loss due to the buffer overflow s calculated by P loss = 1 1 λ N 1 j n+1 n=0 =j n+1 (k K +)(D (N n) k I M a R b )e, (8) p K k=0 where the average ntensty λ of the nput flow s computed by λ = N 1 j n+1 n=0 =j n+1 p k=1 k(d (N n) k I M a R b )e. (9) Proof. It follows from the formula of total probablty that P loss = 1 K =0 k=1 P (k) P k Φ (,k), (10) where P (k) s the probablty that there are web pages n the system at an arrval moment of a batch consstng of k web pages, P k s the probablty that an arbtrary web page arrves n a batch consstng of k web pages, Φ (,k) s the probablty that an arbtrary web page wll be accepted nto the system condtoned on the fact that t arrves n a batch consstng of k web pages and web pages present n the system at the arrval moment. It can be shown that the above lsted probabltes are calculated by P (k) = N 1 m=0r=j m+1 p (D (N n) k I M a R b )e j m+1 p r (D (N m) k I M arr br )e j n +1 j n+1, n = 1,N 1,, (11) P k = N 1 j n+1 n=0 =j n+1 N 1 j n+1 n=0 =j n+1 p k(d (N n) k I M a R b )e p l=1 { Φ (,k) 1, k K, =, (12) l(d (N n) l I M a R b )e K k, k > K, (13) = 0,K,k 1. 13

14 By substtutng expressons (11) - (13) nto formula (10) we get formulae (8) and (9). The theorem s proven. Theorem 3. The probablty P obs of an arbtrary web page obsolescence s calculated by P obs = 1 λ K =2 p (I W I M Γ ( 1) 0 )e. (14) The probablty P success of an arbtrary web page successful servce n the system s calculated by P success = 1 λ K p (I W S 0 I R 1)e. (15) =1 The statement of the theorem s clear because the rght hand sde of (14) represents the rato of the obsolescence rate and the arrval rate nto the system. The rght hand sde of (15) represents the rato of the rate of web pages successfully served n the system and the arrval rate nto the system. 5 Sojourn Tme Dstrbuton Let V(x), V 1 (x), and V 2 (x) be dstrbuton functons of sojourn tme of an arbtrary web page n the system under study, an arbtrary web page, whch wll get successful servce, and arbtrary web page, whch wll be deleted from the system due to ts obsolescence, and v(u), v (1) (u) and v (2) (u) be the correspondng Laplace-Steltjes transforms: v(u) = 0 e ux dv(x), v (k) (u) = 0 e ux dv k (x), k = 1,2, Re u > 0. Theorem 4. The Laplace-Steltjes transforms v(u), v (1) (u) and v (2) (u) are calculated by v(u) = v (1) (u)p success +v (2) (u)p obs +P loss, (16) v (1) (u) = 1 P success K 1 v (2) (u) = 1 K 1 P obs =0 k=1 =0 k=1 14 p,k v (1),k (u), (17) p,k v (2),k (u), (18)

15 where p,k = 1 λ p k(d (χ()) k e W I M a R b ), the column vectors v (m),k (u), = 0,K 1, k 1, m = 1,2, are computed by mn{k,k },k (u) = 1 C,l k v(m) +l 1 (u), m = 1,2, (19) v (m) l=1 C,l = β δ,0 I M a R b γ (l δ,0), the vectors v (m) (u), = 0,K 1, m = 1,2, are computed recursvely by v (1) 0 (u) = (ui S) 1 S 0, (20) where v (1) (u) = (ui A ) 1 ˆB 1 v (1) 1 (u), = 1,K 1, v (2) 1 0 (u) = 0, v(2) (u) = l 1 (ui A k ) 1 ˆB k 1 l=0 k=0 (ui A l ) 1 (I M I R l 1 Γ 0 )e, = 1,K 1, (21) ˆB = B I R, = 0,K 1. { 1, = l, δ,l s Kronecker delta: δ,l = 0, l. Proof. To prove the theorem, we use the method of collectve marks (method of catastrophes). We nterpret the parameter u as an ntensty of some magnary statonary Posson flow of catastrophes. So, v(u) has the meanng of probablty that no catastrophe arrves durng the sojourn tme of an arbtrary web page, v (1) (u) s probablty that no catastrophe arrves durng the sojourn tme of an arbtrary web page condtonal that ths web page wll get servce successfully, and v (2) (u) s probablty that no catastrophe arrves durng the sojourn tme of an arbtrary web page condtonal that ths web page wll be deleted from the system due to ts obsolescence. So, the formula (16) evdently stems from the formula of total probablty. It s obvous that the sojourn tme of an arbtrary (tagged) web page does not depend on the arrval process after the tagged web page arrval moment. Thus, n analyss we can gnore transtons of the drectng process of the arrval process after the moment of the tagged web page arrval. 15

16 The formulae (17) and (18) also follow from the formula of total probablty. Here row vector p,k defnes the probablty of an arbtrary web page arrval at the moment when there are web pages n the system n a batch of sze k and the probablty dstrbuton of the drectng processes of servce and obsolescence at ths moment. Column vector v (m),k (u) defnes the probablty of no catastrophe arrval durng the condtonal sojourn tme of a tagged web page who arrves at the moment when there are web pages n the system n a batch of sze k under the fxed value of the drectng processes of servce and obsolescence at the arrval moment. For m = 1 the condton s that the tagged web page wll successfully obtan servce. For m = 2 the condton s that the tagged web page wll be deleted from the system due to ts obsolescence. The relaton (19) s based agan on the formula of total probablty. The matrx C,l defnes the probablty dstrbuton of nstallaton, upon the arrval of the tagged web page, of the ntal states of the drectng process of servce (f = 0,.e., the system was empty at the arrval moment) and of the drectng processes of obsolescence of the web pages, whch arrve at the same batch as the tagged web page and are placed n the buffer before the tagged web page, and of ths web page. Recall that we assume that f the batch conssts of k web pages then the tagged web page wll be the lth n the batch, l = 1,k, wth probablty 1 k, k 1. The vector v (m) (u) defnes the probablty of no catastrophe arrval durng the condtonal sojourn tme of the tagged web page who sees web pages n the system before her n the queue and the correspondng states of the drectng processes of servce and obsolescence after the moment of arrval. The recurrent formulae (20) for the vector Laplace-Steltjes transforms v (1) (u) s clear f we take nto account that: () v (1) 0 (u) s the vector Laplace- Steltjes transform of the servce tme dstrbuton, () (ui A ) 1 defnes the probablty that no catastrophe arrves durng the tme nterval between the moment of the tagged web page arrval, at whch the number of web pages n the queue before the tagged web page s equal to, and the moment when the number of web pages n the queue before the tagged web page s decreased to 1, () the matrx ˆB defnes the transton of the drectng processes of servce and obsolescence of the web pages at the moment of decreasng. The formulae (21) for the vector Laplace-Steltjes transforms v (2) (u) takes nto account reasonngs (), () presented above as well as consderaton that no catastrophe should arrve untl the obsolescence moment of a 16

17 tagged web page and that l, l = 0, 1, web pages can depart from the system untl the obsolescence moment. The theorem s proven. Corollary 1. The average sojourn tme (response tme) V of an arbtrary web page, the average sojourn tme V (1) of an arbtrary web page who wll get successful servce, and the average sojourn tme V (2) of an arbtrary web page who wll be deleted from the system due to ts obsolescence are computed by V = V (1) P success + V (2) P obs, (22) V (1) = 1 P success K 1 V (2) = 1 K 1 P obs where the column vectors w (m) 1,2, are computed by mn{k,k } w (m),k = =0 k=1 =0 k=1 dv(m),k =,k (u) l=1 the vectors w (m), = 0,K, m = 1,2, are computed by p,k w (1),k, (23) p,k w (2),k, (24) du u=0, = 0,K 1, k 1, m = 1 C,l k w(m) +l 1, m = 1,2, (25) w (1) 0 = S 1 e M, w (1) = (A ) 1 [ v (1) (0)+ ˆB 1 w (1) 1 ], = 1,K 1, v (1) 0 (0) = e M, v (1) (0) = (A ) 1 ˆB 1 v (1) 1 (0), = 1,K 1, 1 [ l 1 m 1 = ( 1) l (A k ) 1 ˆB k 1 w (2) l=0 m=0 k=0 l 1 (A m ) 2 ˆB m 1 k=m+1 (A k ) 1 ˆB k 1 + l 1 + (A k ) 1 ˆB k 1 (A l ) ](A 1 l ) 1 (I MR l 1 Γ 0 )e M, k=0 = 1,K 1. 17

18 Proof of corollary evdently follows from the well-known expresson for the mean value of a random varable va the dervatve of the Laplace-Steltjes transform of ts dstrbuton functon. Note that expressons for the varance and the hgher order moments of the sojourn tme dstrbuton can be also easly derved based on equatons (16)-(18). 6 Case of the Ordnary Arrval Process Consder the specal case when web pages arrve not n batches, but one-byone. It means that D (l) k = 0, k 2, l = 1,N. In ths case the generator Q of the Markov chan ξ t,t 0, s the three block dagonal matrx whch n turn means that ths chan s a fnte space Quas-Brth-and-Death-Process. Thus, n ths case the algorthm for solvng equlbrum equatons (4) for the vector p of statonary probabltes and formulae for some performance measures smplfy. The algorthm has the followng form 1. Compute the matrces G, = 0,K 1, from the backward recurson G = [ (Q +1,+1 +Q +1,+2 G +1 )] 1 Q +1,, = K 2,K 3,...,0, wth the termnal condton G K 1 = (Q K,K ) 1 Q K,K Compute the matrces F, = 0,K, by recurson F 0 = I, F = F 1 Q 1, [ (Q, +Q,+1 G )] 1, = 1,K. 3. Compute the vector p 0 as the unque soluton to the followng system of lnear algebrac equatons: K p 0 (Q 0,0 +Q 0,1 G 0 ) = 0, p 0 F l e = 1. l=0 4. Compute the vectors p, = 1,K, by formula p = p 0 F, = 1,K. 18

19 The formula for the loss probablty P loss s gven by P loss = 1 λ p K(D (1) 1 I MR K 1)e, where λ = N 1 j n+1 n=0 =j n+1 p (D (N n) 1 I M a R b )e. The Laplace-Steltjes transforms v (1) (u) and v (2) (u) are computed by v (1) (u) = 1 λp success K 1 =0 p (D (χ()) 1 e W I M a R b )C,1 v (1) (u), v (2) (u) = 1 K 1 p (D (χ()) 1 e W λp I M a R b )C,1 v (2) (u). obs =0 The average sojourn tmes V (1) and V (2) are computed by V (1) = 1 λp success K 1 =0 p (D (χ()) 1 e W I M a R b )C,1 w (1), V (2) = 1 K 1 p (D (χ()) 1 e W I λp M a R b )C,1 w (2). obs =0 7 Numercal example To demonstrate feasblty of the developed algorthms for calculatng the statonary state dstrbuton of the system under the fxed parameters of the control strategy and calculatng the optmal set of these parameters, let us consder numercal examples. Frst we suppose that the system can have any number of actve robots between one and four at any tme moment (N = 4) and the buffer capacty s equal to 5 (K = 5). We assume that the arrval process s formed accordng to Model 4 from Secton 2. Namely, when r robots are actvated the BMAP-nput s descrbed by the matrces D (r) k, k 0, r = 1,N, gven by 19

20 D (1) 0 = D (2) 0 = ( ) ( ) ( ) 10 2, D (1) =,D (1) =, ( ) ( ) ( ) ,D (2) =, D (2) =, ( ) ( ) D (2) =, D (3) =, ( ) ( ) D (3) =, D (3) =, ( ) ( ) D (4) =, D (4) =, ( ) ( ) D (4) 2 = D (4) =, D (4) = The ntenstes λ (r) of the BMAP when r robots are actve, r = 1,N, are computed by λ (1) = 1.28, λ (2) = 2.41, λ (3) = 3.125, λ (4) = The coeffcents of correlaton c (r) cor and the ntenstes of batches arrval λ (r) g are the followng: c (1) cor = 0.218, c (2) cor = 0.111, c (3) cor = 0.02, c (4) cor = 0.035, λ (1) g = 0.853, λ (2) g = 1.208, λ (3) g = 2.5, λ (4) g = Let the PH dstrbuton of servce tme( be defned ) by the row vector 3 1 β = ( ) and the sub-generator S =. The mean servce 2 3 tme s equal to Let the PH dstrbuton of a web page obsolescence tme ( be descrbed ) by the row vector γ = ( ) and the sub-generator Γ = The mean tme untl obsolescence s equal to 5. Let the cost coeffcents be fxed to c loss = 5, c obs = 10, a = 2, c rob = 20, c star = 300. We have chosen the cost coeffcents n ths way to obtan commensurable optmal values n the optmal soluton and non-trval optmal polcy. Note that n ths example we fxed the cost coeffcents based on some heurstc reasonngs or common sense. In general, the rght choce of the cost coeffcents s the mportant and dffcult task. It requres good knowledge of the real world system, whch s descrbed by the mathematcal model under 20

21 study, and clear understandng what s most undesrable for the concrete system (loss of the delvered web pages, obsolescence of a page, starvaton of the system, long watng n the queue, keepng many robots actve, etc), and what s less mportant. So, the help of experts s requred for the rght choce of the cost coeffcents. If such a choce does not seem to be possble, some alternatve formulaton of the optmzaton problem, e.g., mult-crtera problem or problem wth constrants may be consdered. In the latter approach the cost coeffcents are Lagrange multplers n the constraned problem and can be found from the dual problem formulaton. Let us fnd the optmal strategy of control by the system under the fxed above values of the system parameters and the cost coeffcents. The thresholds j 1,...,j N 1 n the problem formulaton are fxed such as 1 = j 0 < j 1 < < j N 1 < j N = K,.e., the thresholds cannot concde and the use of all N modes of operaton (the number of the mode s characterzed by the number of the actve robots) s mandatory. It s ntutvely clear that actually t can happen that the optmal strategy does not need to use some modes of operaton at all. So, to fnd the optmal strategy, we have to compare the values of the optmal values of the cost crteron when all N modes are used, when N 1 modes are used whle 1 mode s gnored,..., two modes are used whle N 2 modes are gnored, when only one mode s used (.e., the number of the actve robots s not vared). Let us denote by C r the value of the cost crteron when exactly r robots are always actve, r = 1,4. Thevalues C r, r = 1,4, aregven by C 1 = , C 2 = 110.0, C 3 = , C 4 = So, f there s no possblty to control the number of robots, and one has to decde how many robots should permanently work, the best choce s to have permanently three actve robots. Next we consder the threshold type strateges for controllng the number of actve robots. Table 1 contans the optmal value of the cost crteron for varous combnatons of the used modes and the optmal threshold strategy for each such combnaton. Table 1: The Value of the Optmal Cost Crteron for varous threshold strateges 21

22 possble numbers of actve robots Optmal thresholds Optmal value of the cost crteron or or or or or or or 2 or 1 2, or 2 or 1 1, or 3 or 1 0, or 3 or 2 0, or 3 or 2 or 1 0,2, Let us explan the entres of the table. For nstance, the hghlghted lne corresponds to the control wth one threshold at two. When the queue length does not exceed two, there are three actve robots and when the queue length exceeds two, the number of actve robots decreases to one. As another example, the control correspondng to the last lne of the table has the followng structure: when the system s empty, four robots are actve; when the queue length s greater than zero and smaller than three, three robots are actve; when the queue length exceeds two, only one robot s actve. As t s seen from Table 1, the optmal strategy assumes that only two among four avalable operaton modes (modes wth one and three actve robots) should be used. The optmal value C of the cost crteron s equal to It s evdent that C gves the relatve proft R more than 28% comparng to the case wthout control. The value of R s computed as ( C ) R = 1 100%. mn{c 1,C 2,C 3,C 4 } The dependence of the cost crteron on the threshold when the strategy of control uses only two modes, for all possble combnatons of the modes, s shown on Fgure 1. 22

23 modes 1 and 2 modes 1 and 3 modes 1 and 4 modes 2 and 3 modes 2 and 4 modes 3 and 4 cost crteron threshold Fgure 1: Dependence of the cost crteron on the threshold The obtaned results llustrate the necessty of the nput control and possblty to reduce the cost of the system operaton by means of the threshold type control. Now let us vary the buffer sze K from 1 to 100. Table 2 contans the optmal crteron value for the cases wth and wthout control (C and C r, r = 1,4), the optmal threshold j and the relatve proft R for dfferent buffer capacty K. Note that n all cases the optmal control only uses the modes wth one and three actve robots. The results for K > 30 are approxmately the same as for K =

24 Table 2: Varable Buffer Sze K K j C C 1 C 2 C 3 C 4 R, % Let the mean servce tme b 1 be vared by means of the matrx S multplcaton by the value s whch vares from 0.1 to 15. Table 3 contans the value s, the mean servce tme b 1, the optmal set of possble numbers of actve robots, the optmal value of the threshold, the optmal cost crteron value and the value of the cost crteron when only one of the modes s n use (when the number of actve robots s fxed), and the relatve proft R. Note that when s s equal to or grater than 15,.e. b 1 s less than 0.04, no dynamc control s requred. Now we vary the mean tme g 1 untl obsolescence by the same way as t was done for the mean servce tme. Table 4 shows obtaned results. Note that n the optmal set of modes only one and three actve robots are used n all consdered cases. Consder the effect of the servce tme varaton on the cost crteron value gven that the mean servce tme b 1 s constant and s equal to Let the matrx S of servce tme dstrbuton have the form ( ) α1 0 S = 0 α 2 and β = ( ). The varance of the servce tme s calculated as follows: var s = 2βS 2 e (β( S) 1 e) 2. To mantan the mean servce tme b 1 at the same value, the entres α 1 and α 2 of matrx S must be related through the formula b 1 = β( S) 1 e M as 24

25 follows: α 2 = 0.1α 1 b 1 α Note that α 1 should be grater than to keep α 2 postve. Let us vary the value of α 1 from to 9. The servce tme varaton var s takes the values from to In the case α 1 = the servce tme dstrbuton s exponental. In the optmal set of operaton modes only one and three actve robots are used. Fgure 2 shows dependence of the cost crteron value on the threshold under dfferent values of the servce tme varaton (α 1 {1.521,3,5,7,9}) var = var = var = var = var = cost crteron threshold Fgure 2: Dependence of the cost crteron on the servce tme varaton Note that when α 1 becomes larger than 4 (var s > 3.415), not the modes wth one and three actve robots but the modes wth one and four actve robots are the optmal set of exploted modes. Fgure 3 shows the dependence of the cost crteron when only optmal set of modes s n use. In the fgure, two lower curves correspond to the modes wth one and three actve robots and other curves correspond to the modes wth one and four actve robots. Now let us vary the varance of the tme untl obsolescence n a smlar same way. Let the matrx G have the form ( ) α1 0 G = 0 α 2 25

26 var = var = var = var= var = cost crteron threshold Fgure 3: Dependence of the cost crteron on servce tme varaton and γ = ( ). To mantan the average tme untl obsolescence g 1 = 5, the value α 2 needs to be related to α 1 as follows: α 2 = 0.1α 1 g 1 α We vary α 1 from 0.2 to 7. Thus the varance takes the values from 25 to 449. Note that n the case α 1 = 0.2, the tme untl obsolescence s exponentally dstrbuted. The optmal strategy conssts of modes wth one and three actve robots n all consdered cases. Fgure 4 shows dependence of the cost crteron value on the varance of the tme untl obsolescence (α 1 {0.2,0.5,1,3,7}). Note that as the varance grows the optmal threshold decreases from 2 to 0. Now let us consder the example based on real data obtaned from the web crawler desgned by INRIA Maestro team n the framework of RIAM INRIA-Canon Research Project. Data about the nformaton delvery process by the robot to the data base was presented n the form of the text fle whch contaned more than tmestamps defnng epochs of nformaton delvery. Ths data was processed n the followng way. Inter-arrval tmes were computed. The obtaned sample was censored: very long ntervals, whch actually 26

27 cost crteron var = 25 var= 187 var = 313 var = 417 var = threshold Fgure 4: Dependence of the cost crteron on varaton of tme untl obsolescence correspond to the perods when the crawlng process was stopped due to some reason, are deleted from the sample. The obtaned sample was transformed nto two samples n such a way as the very short nter-arrval tmes were deleted from the ntal sample and the correspondng nformaton about the number of successvely deleted ntervals was placed nto another sample. As the result of these manpulatons, we stated that the arrval process s the batch arrval process. One sample defnes the ntervals between the epochs of batches arrval, the second sample defnes the number of the nformaton unts n the correspondng batch. By nformaton unts we mean ether the prncpal part of a web page (e.g., web page man html fle) or ts embedded resources (e.g., mage and audo fles). Based on the second sample, dstrbuton of the number of the nformaton unts n a batch was computed as follows. The sze of a batch vares n the nterval [1,8] and probabltes d k that the batch sze s equal to k = 1,2,...,8 are the followng: d 1 = , d 2 = , d 3 = , d 4 = , d 5 = , d 6 = , d 7 = , d 8 = The mean batch sze s equal to Based on the frst sample, we computed estmaton of the mean value 27

28 of an nterval between arrvals of batches as , estmaton of the varance of such an nterval as and the estmatons of lagk correlaton of nter-arrval tmes for k equal to 1,2,...,6 are gven by 0.622; 0.574; 0.555; 0.537; 0.523; Thus, the flow defned by a sample under study has slowly decreasng correlaton. In ths stuaton, t s reasonable to apply method by Damond and Alfa [4] orented to such flows. As the result, the process of the arrval of the batches was defned by the M AP whch s characterzed by the matrces ( ) D 0 =, ( ) D 1 = A, A = BasedonthsMAP forbatch arrvalsandnformaton aboutthebatch sze dstrbuton, the BMAP of web pages s constructed. It s defned by the matrces D 0 and D k = d k A, k = 1,8. To estmate a web page servce tme dstrbuton, the followng nformaton about the sze of an arbtrary nformaton unt delvered by a crawler (content sze) n the used data base was exploted: the mean content sze s equal to bytes, the mean squared content sze s equal to E+11, and the mean cubed content sze s equal to E+18. Based on ths nformaton, the servce tme dstrbuton of a web page was descrbed, up to some normalzng constant defned by the content processng rate (n our applcaton the processng rate s constant), by the hyperexponental dstrbuton whch s the partal case of the P H dstrbuton defned by the vector β = ( ) and by sub-generator ( ) S = The mean servce tme s 8.2. The squared coeffcent of varaton of the servce tme dstrbuton s equal to Because the exponental dstrbuton has the squared coeffcent of varaton equal to 1, t s clear that ths dstrbuton cannot be consdered as a good approxmaton of the servce tme dstrbuton. We assume that the system has four avalable operaton modes. Our web crawler has been used wth the Internet lnk whch became saturated wth fve robots. The buffer capacty s K =

29 When the r robots are actvated, the BMAP-nput s descrbed by the matrces D (r) k, D (r) k = rd k, k = 0,8,r = 1,4, where the matrces D k, k = 0,8, are defned above. The ntenstes λ (r) of the BMAP when r robots are used, r = 1,4, are as follows: λ (1) = , λ (2) = , λ (3) = 0.046, λ (4) = The dstrbuton of a web page obsolescence tme s exponentally dstrbuted wth parameter ,.e., the mean tme untl obsolescence s equal to The cost crteron coeffcents are taken as c loss = 200, c obs = 250, a = 3, c rob = 20, c star = 600. The cost crteron values C r when r, r = 1,4, robots are always actvated aregven byc 1 = , C 2 = , C 3 = andc 4 = When all four modes of operatons are exploted the optmal cost crteron value s and the optmal set of thresholds s [2,2,2],.e., the optmal strategy assumes that four robots should be actve untl the number of web pages n the system does not exceed 2. When the number of web pages n the system exceeds 2, three robots should be deactvated and only one robot should be actve. Table 5 contans the optmal values of the cost crteron for the dfferent combnatons of operaton modes. Table 5: The Values of the Optmal Cost Crteron for the Fxed Combnaton of Operaton Modes 29

30 possble numbers of actve robots Optmal thresholds Optmal value of the cost crteron , , , , , , ,2,3 0, ,2,4 2, ,3,4 2, ,3,4 2, ,2,3,4 2,2, The relatve proft of operaton mode control exceeds 9% comparng to the case when no control s appled and all four robots are always actve. 8 Concluson In ths paper we provde performance evaluaton and optmzaton of the crawlng part of a web search engne. We model the crawler wth a fnte buffer, monotoncally controlled arrval rate (controlled number of crawlng robots), and wth stochastcally bounded watng tme. The system s consdered under rather general assumptons about the arrval process, servce and obsolescence tme dstrbutons. Statonary dstrbuton of the system state, sojourn tme dstrbuton and man performance measures of the system are calculated under any fxed set of thresholds defnng the control strategy. Ths allows us to reduce the problem of optmal control to mnmzaton of a known functon of several nteger varables. Numercal results are presented. They show that the dynamc nput control can gve essental proft. Effects of buffer sze changes and changes of average servce and obsolescence tmes and ther varances are nvestgated. In partcular, we llustrate that the assumpton about the exponental dstrbuton of servce and obsolescence tmes can gve poor estmaton of the 30

31 system performance measures and optmal values of the thresholds when actually these tmes have hgh varaton. The model has been appled to the performance evaluaton and optmzaton of the crawler desgned by INRIA Maestro team n the framework of the RIAM INRIA-Canon research project. Acknowledgment The authors would lke to thank Mnstry of Scence of France for fnancal support of ths research va ECO-NET programme and RIAM INRIA-Canon Research Project. References [1] J. Artalejo, D.S. Orlovsky, and A.N. Dudn, Multserver retral model wth varable number of actve servers, Computers and Industral Engneerng, vol. 28, pp , [2] S. Asmussen, O. Nerman and M. Olsson, Fttng phase-type dstrbutons va the EM algorthm, Scandnavan Journal of Statstcs, vol. 23, pp , [3] T. Crabll, Optmal control of a servce faclty wth varable expenental servce tmes and constant arrval rate, Management Scence, vol. 9, pp , [4] J.E. Damond and A.S. Alfa, On approxmatng hgher order M AP s wth M AP s of order two, Queueng Systems, vol. 34, pp , [5] A. Dudn, Optmal mult-threshold control for a BM AP/G/1 queue wth N servce modes, Queueng Systems, vol. 30, pp , [6] A.N. Dudn and V.I. Klmenok, Optmal admsson control n a queueng system wth heterogeneous traffc, Operatons Research Letters, vol. 28, pp , [7] A. Graham, Kronecker products and matrx calculus wth applcatons, Chchester: Ells Horwood,