Int'l Conf. cientific Coputing CC'7 3 Queuing Netwok Appoxiation Technique fo Evaluating Pefoance of Copute ystes with Hybid Input ouce Nozoi iyaoto, Daisuke iyake, Kaoi Katsuata, ayuko Hiose, Itau Koike, Toshiyuki Kinoshita chool of Copute cience, Tokyo Univesity of Technology Hachioji Tokyo, 9-098, Japan Abstact - Queuing netwok techniques ae effective fo evaluating the pefoance of copute systes. We discuss a queuing netwok technique fo copute systes with hybid input souce. We consideed a queuing netwok with two input souces, a finite input souce and an open input souce. We call this as hybid input souce. Fo a finite input souce, the finite nube of teinals exists in the netwok and a job is dedicated to its own teinal. Afte a think-tie at the teinal, the job oves to the seve and acquies a pat of the eoy, and executes CPU and Input / Output I/O) pocessing. On the othe hand, in the open input souce, the job aives at the seve andoly fo the outside, acquies a pat of the eoy, and executes the CPU and I/O pocessing. When the job is copleted at the CPU and I/O pocessing, it eleases the eoy and goes back to outside. Howeve, because the eoy esouce can be consideed as a seconday esouce fo the CPU and I/O equipent, the queuing netwok odel has no poduct fo solution and cannot calculate the exact solutions. We poposed an appoxiation queuing netwok technique to calculate the pefoance easues of copute systes with hybid input souce in which ultiple types of jobs exist. This technique involves dividing the queuing netwok into two levels; one is "inne level" in which a job executes CPU and I/O pocessing, and the othe is "oute level" that includes teinals and counication lines. By dividing the netwok into two levels, we can pevent the nube of states of the netwok fo inceasing and appoxiately calculate the pefoance easues of the netwok. We evaluated the poposed appoxiation technique by using nueical expeients and claified the chaacteistics of the syste esponse tie and the ean nube of jobs in the inne level. Keywods: pefoance evaluation, queuing netwok, cental seve odel, finite input souce, open input souce. Intoduction Queuing netwok techniques ae effective fo evaluating the pefoance of copute systes. In copute systes, two o oe jobs ae geneally executed at the sae tie, which causes delays due to conflicts in accessing hadwae o softwae esouces such as the CPU, I/O equipent, o data files. We can evaluate how this delay affects the copute syste pefoance by using a queuing netwok technique. oe queuing netwoks have an explicit exact solution, which is called a poduct fo solution []. With this solution, we can easily calculate the pefoance easues of copute systes, fo exaple the busy atio of hadwae and the job esponse tie, and so on. Howeve, when the exclusion contols ae active o when a eoy esouce exists, the queuing netwok does not have the poduct fo solution. To calculate an exact solution of a queuing netwok that does not have the poduct fo solution, we have to constuct a akov chain that descibes the stochastic chaacteistics of the queuing netwok and nueically solve its equilibiu equations. When the nube of jobs o the aount of hadwae in the netwok inceases, the nube of Figue : Cental seve odel in hybid input souce IBN: -603-45-9, CEA Pess
4 Int'l Conf. cientific Coputing CC'7 states of the queuing netwok dastically inceases, ince the nube of unknown quantities in the equilibiu equations is equal to the nube of states of the queuing netwok, the nube of unknown quantities in the equilibiu equations also dastically inceases. Theefoe, we cannot pefo calculation of the exact solution of the queuing netwok nueically. Hee we conside a queuing netwok with two input souces, a finite input souce and an open input souce Figue ). We call this as hybid input souce. The finite input souce assues online eal-tie pocessing in the copute syste, and the open input souce assues batch pocessing. Fo a finite input souce, the finite nube of teinals exist in the netwok and a job is dedicated to its own teinal. Afte a think-tie at the teinal, the job oves to the seve and acquies a pat of the eoy, and executes CPU and I/O pocessing. When the job copletes CPU and I/O pocessing at the seve, it eleases the eoy and goes back to its oiginal teinal. In an open input souce, the job aives at the seve andoly fo the outside, acquies a pat of the eoy, and executes the CPU and I/O pocessing. When the job is copleted at the CPU and I/O pocessing, it eleases the eoy and goes back to outside. Thus, this odel is constucted with two levels, one is outside the seve teinals, outside of syste and counication lines) and the othe is inside the seve CPU, I/O equipent, and eoy). We call the "oute level" and "inne level" espectively. ince a job executes CPU and I/O pocessing occupying the eoy, the eoy can be consideed as a seconday esouce fo the CPU and I/O equipent in inne level. Geneally, when a queuing netwok includes a seconday esouce, it does not have poduct fo solutions and an appoxiation technique is equied to analyze the netwok. We have poposed hee an appoxiation technique fo calculating the pefoance easues of copute systes with hybid input souce. We peviously epoted the esults fo copute systes with eoy esouce in open input souce, in which jobs aive fo and depat to outside of the syste. In this pape, we conside the hybid input souce odel. We aleady epoted the open input souce odel and the finite input souce odel in [6] and [7] espectively. In ode to pevent the nube of states of the akov chain fo inceasing, we divide the odel into two levels, one is oute level that includes the outside of the syste, the teinals, and counication lines, and the othe one is inne level that includes CPU, I/O equipent and eoy esouces Figue ). iilaly, in [6] and [7], two diffeent types of jobs exist in the netwok. Both jobs behavio in the inne and the oute level diffes fo each job class. When thee is a single job class, both the inne and oute level has a poduct fo Figue : Concept of appoxiation solution. Howeve, if thee ae ultiple job classes with a finite input souce and an open input souce, the inne level does not have a poduct fo solution. Theefoe, an appoxiation is needed to analyze the inne level. Dividing the odel into two levels is one of two-laye queuing netwok techniques [3][4]. Ou poposed technique is also a two-laye technique fo copute systes with hybid input souce. eanwhile, the akov chain involving two diensional state tansition siila to ou poposed odel was discussed in [5].. odel desciption The CPU and I/O odel in the inne level is equivalent to the odinay cental seve odel with ultiple job types each of which is called a job class). In this odel, f job classes of finite input souce and o job classes of open input souce exist, and job classes of finite input souce ae nubeed =,,, f by affixing and job classes of open input souce ae nubeed = f +, f +,, f + o by also affixing. The inne level consists of a single CPU node and ultiple I/O nodes. We denote as the nube of I/O nodes. The I/O nodes ae nubeed =,,, by affixing, and the CPU node is nubeed = 0 by also affixing. The sevice ate of job class at the CPU node is 0 and the sevice ate of job class at an I/O node is. The sevice tie at each node is a utually independent ando vaiable subject to coon exponential distibutions. Jobs ae scheduled on a fist coe fist seved FCF) pinciple at all nodes. At the end of CPU pocessing, a job pobabilistically selects an I/O node and oves to it, o copletes CPU and I/O pocessing and goes back to its own teinal o to outside of the syste. The selection IBN: -603-45-9, CEA Pess
Int'l Conf. cientific Coputing CC'7 5 pobability of I/O node of job class is p and the copletion pobability of job class is p 0. Theefoe, 0 p,,, f, f,, f o ). eoy esouces ae added to this cental seve odel. We denote as the nube of eoy esouces. In oute level, a job of finite input souce stays at the teinal fo shot while. The staying tie is called think-tie of a job. The think-tie is utually independent ando vaiable subject to coon exponential distibution with paaete of job class =,,, f ). This equals to job depatue ate fo the teinal. Afte the think-tie, the job oves to the inne level. eanwhile a job of open input souce aives in the inne level fo the outside at ando at aival ate of job class = f +, f +,, f + o ). When a job aives in the inne level, it equests and acquies a pat of the eoy esouces befoe enteing the cental seve odel. If all the pats of the eoy ae occupied, the job joins the syste waiting queue and waits fo a pat of the eoy to be eleased by anothe job. When the job copletes CPU and I/O pocessing, it eleases the eoy and leaves the inne odel and goes back to its own teinal o to the outside. ince the job has to acquie a pat of eoy befoe enteing the cental seve odel, the nube of jobs occupying a eoy is always equal to the nube of jobs in the cental seve odel. Theefoe, at ost jobs can execute CPU and I/O pocessing at the sae tie. That is, the axiu job ultiplicity in the cental seve odel is. When the nube of jobs of job class in the cental seve odel f o is denoted by n,. n By eplacing CPU oute level tansition with CPU CPU tansition, the cental seve odel is odified to a closed odel in which the nube of jobs is constant Figue ). In this odel, when CPU CPU tansition occus, the job teinates and a new job is bon. Theefoe, the ean job esponse tie is the ean tie between two successive CPU CPU tansitions. This ean job esponse tie can be consideed as a job lifetie. 3. Appoxiation odel To obtain the exact solution of the cental seve odel with hybid input souce, we have to descibe the entie odel with a single akov chain fo each job class. Howeve, this causes the nube of states of the akov chain to dastically incease when the nube of jobs and the nube of nodes in the netwok incease. By dividing the netwok into two levels, and descibing each level with two akov chains, we can pevent the nube of states of the odel fo inceasing Figue ). We set the following notations. t : ean think-tie of jobs in job class =,,, f ) : depatue ate fo the teinal of job class =,,, f ) K : nube of teinals of job class =,,, f ) : aival ate of job class = f +, f +,, f + o ) : total ean sevice tie at node- in a job lifetie of job class =,,..., f + o ; =0,,..., ) n : nube of jobs in job class at node- n = n, n,..., n f, n f +,..., n f +o ) : vecto of nube of jobs n =0,,,..., K fo =,,, f ; n =0,,, 3,... fo = f +, f +,, f + o ) n * = n 0, n,..., n, n 0, n,..., n,..., n f +o0, n f +o,..., n f +o ) : state vecto of the cental seve odel F n) = {n * n n, n 0 =0,,..., )} 0 n +n +...+n f +o ) : set of all feasible states of the cental seve odel when the nube of jobs of job class is n P s n * ) : steady-state pobability of state n * T n : ean job esponse tie of the cental seve odel when the vecto of nube of jobs is n n : sevice ate fo the cental seve odel of job class T : syste esponse tie of job class ince the cental seve odel in inne level is equivalent to the odinay cental seve odel with ultiple job classes, it has the poduct fo solution. Then the steady-state pobability P s n * ) is epesented by the following foula [][]. P s n * ) = 0 n, n,, n whee n, n,, n, n, ) ) nf n) 0 n is the noal- IBN: -603-45-9, CEA Pess
6 Int'l Conf. cientific Coputing CC'7 izing constant of steady-state pobabilities when the nube of jobs of job class in the cental seve odel is n =,,..., f + o ). Fo these steadystate pobabilities, we can calculate the ean job esponse tie T n of job class as follows when the nube of jobs is n. n n,, n,, n, ) Tn n,, n,, n, ) The eoy esouce in ou odel can be consideed as an // queuing odel with seves. In an odinay // queuing odel, the sevice ate at a seve is constant, egadless of the nube of guests in the sevice. In the eoy esouce of ou odel, howeve, the sevice ate changes depending on the nube of occupied eoies. Theean job esponse tie T n of job class =,,..., f + o ) when the vecto of nube of jobs is n = n, n,..., n ) is equal to the ean tie while the eoy is occupied. ince the sevice ate of job class fo the cental seve odel n is denoted as n, n depends on T n the nube of jobs in the cental seve odel n. The state tansition of the // queuing odel with two job classes one is finite input souce and the othe is open input souce) is shown in Figue 3, whee the sevice ates fo the cental seve odel change depending on the nube of jobs in the cental seve odel. This is a two diensional bith-death pocess. The equilibiu equations with the steady-state pobability Q n) =Q n, n ), when the nube of axiu nube of jobs in the cental seve odel is and the nube of jobs in the cental seve odel is n =n, n ), ae as follows siila to the case with highe diensions). ) n =0, n =0 K + )Q 0, 0) = 0 Q, 0) + 0 Q 0, ) ) n =,,...,, n =0 {K n ) + + n }Q n 0 n, 0) = K n +) Q n, 0) + n +) Q n +, 0) n 0 + Q n n, ) 3) n =, +,..., K, n =0 {K n ) + + }Q 0 n, 0) = K n +) Q n, 0) + Q 0 n +, 0) + Q n n, ) 4) n =0, n =,,..., {K + + n }Q 0n 0, n ) = Q 0, n ) + Q n, n ) + n +) Q 0n 0, n +) 5) n =0, n =, +, +,... {K + + }Q 0 0, n ) = Q 0, n ) + Q n, n ) + Q 0 0, n +) 6) n +n, n =,,,, n =,,, {K n ) + + n + n n n }Q n n n, n ) = K n +) Q n, n ) + Q n, n ) + n +) Q n n n +, n ) + n +) Q nn n, n +) 7) n +n =, n =,,...,, n =,,..., {K n ) + + n + n n n }Q n n n, n ) = K n +) Q n, n ) + Q n, n ) + n Q n +, n ) + n n n n n Q n, n +) 8) n +n >, n =,,..., K, n =,, 3,... When the lattice point, ) such as + = is on the shotest oute fo 0, 0) to n, n ), and Q n, n) is the steady-state pobability along with the oute. IBN: -603-45-9, CEA Pess
Int'l Conf. cientific Coputing CC'7 7 {K n ) + + + } Q n, n ) = K n +) Q n, n ) + Q n, n ) + Q n, ) + Q n, n ) n 8-) n +n >, n =,,...,, n =n +, n +,..., Q n n, n) Q n, n) n 8-) n +n >, n =+, +,..., K, n =,,..., Q n, n) Q n, n) n 8-3) n +n >, n =,,...,, n =+, +, +3,... Q n n, n) Q n, n) 0 8-4) n +n >, n = +, +,..., K, n =+, +, +3,... Q n, n ) Q n, n) 0 Fo the state n, n ) of the akov chain, when n n, all jobs ae in the cental seve odel and executing CPU and I/O pocessing, and when n +n, n +n jobs ae in the syste waiting queue and waiting fo a pat of the eoy esouces to be eleased. The tansition diaga of the two diensional bith-death pocess is shown in Figue 3. Howeve, the equilibiu equation does not have the poduct fo solution. Theefoe, soe appoxiation is equied to solve it. When the odel has a single job class, it can be descibed with a one diensional bith-death pocess. Its tansition diaga is shown in Figue 4, and the equilibiu equation is as follows: a) Finite input souce i) n =0 K Q 0) Q ) ii) n =,,, {K n ) + n n }Q n ) = K n +) Q n )+n +) Q n +) n a) Finite input souce b) Open input souce Figue 4: tate tansition diaga single job class) iii) n =, +,, K {K n ) + }Q n ) = K n +) Q n ) + Q n +) iv) n =K Q K ) = Q K ) b) Open input souce i) n =0: Q 0) Q )\ ii) n =,,, + n n )Q n ) = Q n ) + n +) n Q n +) iii) n =, +, +, + )Q n ) = Q n )+ Q n +) olutions fo the equilibiu equation ae in the following poduct fo. a) Finite input souce n K Q 0) i Q n ) K Q 0) i n,,, K) b) Open input souce n Q 0) i i i Q n ) Q 0) i i i n i ) i i i ) i i i n i n n,,, ) K i ) n,,, ),,, ) In finite input souce, fo the state tansition at i =,,...,, ultiply by facto K i ), while fo the state i i tansition at i =, +,..., K, ultiply by facto K i ), and in open input souce, fo the state tansition at i =,,...,, ultiply by facto, i i while fo the state tansition at i =, +, +,... ultiply by facto. Fo two diension case, we conside a oute fo lattice point 0, 0) to n, n ) shown in Figue 5, and fo the hoizontal state tansition at the lattice such as i +i on the oute, ultiply by facto IBN: -603-45-9, CEA Pess
8 Int'l Conf. cientific Coputing CC'7 K i ) i i ii i ii ii i ii, and ultiply by facto fo the vetical state tansition. When the lattice point i, i ) such as i +i, fo the state tansition outside of the lattice point, ) such as + = on the oute between, ) and i, i )), ultiply by facto K ) o. Thus, the coefficient of Q n, n ) elated to Q 0, 0) is epesented as the suation of the ultiplication based on all the outes fo 0, 0) to n, n ). Fo exaple, fo the oute fo 0, 0) to, ) when =3, and K =5, which is the case of n +n, the ultiplication along the oute of boken line i) in Figue 5 is 5 Q 0,0). Fo 0 0 the oute fo 0, 0) to 4,), which is the case of n +n, the ultiplication along the oute ii) is 5 4 Q 0,0) 0 3. ince thee ae ultiple outes fo 0, 0) to n, n ), the coefficient of Q n, n ) elated to Q 0, 0) is appoxiately epesented as the total of the ultiplication based on all outes. iilaly to the case above, we can appoxiately calculate the state pobability of a queuing netwok with ultiple job classes when f > o o >. 4. Nueical expeients We evaluated the poposed appoxiation technique though nueical expeients. We used the following paaetes.. Nube of teinals: K = 3 ~ 0 o K = 5 ~ 0. Nube of eoy esouces: = 3 o 5 3. Think-tie: t = 0 4. Aival ate: =0.0 ~ 0.4 5. Nube of I/O nodes: = 6. Total sevice tie at each node 0 =.0, = =0.5 0 =.0, = =.0, whee is the total sevice tie of job class at node. Figues 6 ~ 9 show the ean syste esponse ties of job classes and, when is fixed at 3 o 5. Figue 6, 7 show the case of =0., K =3, 4,, 0 o K =5, 6,, 0, and Figue 8, 9 show the case of K =6, =0.0, 0.04,, 0.4. The ean syste esponse tie is the ean tie fo job aival to depatue fo the inne level, that is the ean tie fo depatue fo the teinal to coing back to the teinal in the finite input souce and that is the ean tie fo aival to depatue to outside of the syste in the open input souce. iilaly to the case of a single job class, the ean syste esponse tie fo both job class onotonically inceases and the ean syste esponse tie fo the finite input souce daws a convex cuve. When the nube of teinals K of job class is inceased and the aival ate of job class is fixed the only taffic of job class is inceased), the ean esponse tie of job class and job class inceases. This is because of the entie cental seve odel is oe cowded by inceasing taffic of the job class. iilaly, when the aival ate of job class is inceased and the nube of teinals is fixed, both esponse tie onotonically inceases. We can see that the ean esponse tie of job class incease oe apidly than job class in heavie taffic ange. This eason can be pesued that the behavio of the ean syste esponse tie of job class in the heavie taffic ange is appoxiately linea to the nube of jobs in the cental seve odel. IBN: -603-45-9, CEA Pess
Int'l Conf. cientific Coputing CC'7 9 Figue 6: ean syste esponse tie =3, t =0, =0.) Figue 8: ean syste esponse tie =3, t =0, K =6) 5. Conclusion Figue 7: ean syste esponse tie =5, t =0, =0.) We poposed an appoxiation technique fo evaluating the pefoance of copute systes in hybid input souce using a queuing netwok and analyzed its pefoance easues though nueical expeients. The concept of the appoxiation is based on sepaately analyzing the inne level CPU, I/O equipent, and eoy) and oute level teinals, outside of the syste, and counication lines). The nueical expeients claified the chaacteistics of the syste esponse tie. In the futue we plan to exaine the accuacy of the poposed appoxiation technique by copaing it with exact solutions o siulation esults. 6. efeences [] F. Baskett, K.. Chandy,.. untz and F. G. Palacious, Open, Closed, and ixed Netwoks of Queues with Diffeent Classes of Custoes,'' J. AC, Vol., No., pp.48--60, Apil 975. Figue 9: ean syste esponse tie =5, t =0, K =6) [] H. Kobayashi, odeling and Analysis,'' Addison- Wesley Publishing Copany, Inc. 978. [3] T. Kuasugi and I. Kino, Appoxiation ethod fo Two-laye Queueing odels,'' Pefoance Evaluation 36--37, pp.55--70, 999. [4] J. A. olia and K. C. evcik, The ethod of Layes,'' IEEE Tans. on oftwae Engineeing, Vol., No.8, pp.689--700, Aug. 995. [5] A. Gandhi,. Dooudi,. Hachol-Balte and A. chelle-wolf, Exact Analysis of the //k/setup Class of akov Chains via ecusive enewal ewad, IGETIC 3, pp.53--66, June 03 [6]. Takaya,. Ogiwaa, N. atazali, C. Itaba, I. Koike, T. Kinoshita, Queuing Netwok Appoxiation Technique fo Evaluating Pefoance of Copute ystes with eoy esouce used by ultiple job types, CC04, pp.4--46, July 04 [7] ayuko Hiose, adoka hiatoi, atazali Nooafiza, yo Tsuboi, I. Koike, T. Kinoshita, Queuing Netwok Appoxiation Technique fo Evaluating Pefoance of Copute ystes with Finite Input ouce, CC05, pp.9--5, July 05 IBN: -603-45-9, CEA Pess