A Capacity Supply Model for Virtualized Servers

Transcription

1 96 Iformatia Eoomiă vol. 3, o. 3/009 A apaity upply Model for Virtualized ervers Alexader PINNOW, tefa OTERBURG Otto-vo-Guerike-Uiversity, Magdeburg, Germay {alexader.piow stefa.osterburg}@iti.s.ui-magdeburg.de This paper deals with determiig the apaity supply for virtualized servers. First, a server is modeled as a queue based o a Markov hai. The, the effet of server virtualizatio o the apaity supply will be aalyzed with the distributio futio of the server load. Keywords: Markov hai, Weibull distributio, server, apaity supply Motivatio Nowadays, data eters are beig ru iidet-drive. Quite ofte, further hardware systems are istalled as a reatio o ew ustomer eeds. Despite aquisitio osts for further hardware systems beig osidered uritial, the extesio of the iformatio ifrastruture leads to higher admiistratio, maiteae, ad fially persoel osts. For a effiiet usage of the existig iformatio ifrastruture, there are oepts suh as Virtual ad Adaptive omputig to logially separate hard- ad software. Realizig these oepts allows abadoig iidet-drive busiess strutures. The paper disusses a model to pla the apaity supply of physial ad virtual servers. I the first step, the effet of the server load o the respose time will be desribed. Therefore, a server is modeled as a queue based o a Markov hai. The effet of server virtualizatio o the apaity supply will be aalyzed with the distributio futio of the server load. Determiig the ritial erver Load The operator of a data etre provides the ustomer with physial ad immaterial resoures. The maximum output of a resoure is defied as its apaity [6. Whe desribig omputer systems as hardware uits, they a be osidered as a set osistig of proessor, storage ad iput/output devies. The operator of a data etre aot use the omplete apaity supply of a omputer system beause this will yield a uaeptable respose time. The utilizatio of the apaity supply is alled server load. The level of server load that does ot satisfy the ustomer eeds a be alulated usig the queuig theory. Therefore, the server will be modeled as a time otiuous Markov hai. The server reeives requests represeted as a evet. These evets are Poisso distributed. The arrival rate λ is defied as the umber of requests to the server i a give time period. With the arrival of a ew request the state i of a server is trasitioig to state i+. The state probability π i (t represets the probability of i requests i a server at time t. The server is proessig these requests. The termiatio of proessig is represeted by a evet. These evets are also Poisso distributed. The proessig rate ν defies the umber of request a server a hadle i a give time period. Arrival ad proessig rates λ ad ν are o-varyig ad idepedet from the umber of requests i this model. Figure [ illustrates this otext. The ratio of arrival ad proessig rates desribes the server load. The proessig rate eeds to be larger tha the arrival rate i order to avoid a ifiite growth of the uproessed requests: λ < ( ν The mea umber of requests proessed by a server depedig o the server load is [: K ( Little's theorem [7 determies the mea umber of requests proessed by a server depedig o the arrival rate λ ad the mea respose time K : K λ T (3 By substitutig Little's theorem i equatio the server load a be determied i depedee of the mea respose time T ad the

2 Iformatia Eoomiă vol. 3, o. 3/ arrival rate λ: Tλ λt ; (4 + Tλ For a give mea respose time ad arrival rate the ritial server load a be alulated by equatio 4. Fig.. Queue as Markov hai 3 Determiig the apaity upply A o-varyig arrival rate λ was assumed for the determiatio of the ritial server load. I reality, the arrival rate is varyig over the time. It will be assumed that the arrival rate i differet time periods is o-varyig at differet levels (see figure. Fig.. Arrival rate λ over time ( µ F (6 D ( exp( π I suh a time period a steady state is reahed. The arrival rate λ used to determie the ritial server load is the maximum arrival rate a server must be able to hadle to esure a oritial respose time. The levels of the arrival rate λ are subjet to a distributio futio. I the model, a ormal distributio will be assumed. From equatio follows that the server load is also ormally distributed. The server apaity supply, desribes the maximum output of the resoure. The requests to a server ause the a- D P( D The itegral of the distributio futio for paity demad D. The server load a be desribed as a ratio of apaity supply ad apaity demad D : D (5 Beause of the distributio futio of the server load the apaity demad D is also ormally distributed. The quota of server requests that do ot exeed a defied apaity demad a be determied by the distributio futio F of the ormal distributio: D the apaity demad D aot be alulated

3 98 Iformatia Eoomiă vol. 3, o. 3/009 aalytially ad desribed by a kow futio i a losed form [3. Therefore, the ormal distributio of the apaity demad will F x; Θ exp The parameters Θ {,, } of the distributio futio are defied as sale, shape ad be approximated by the Weibull distributio. The distributio futio F of the Weibull distributio is [8: for x loatio parameters. The desity futio is give as [8: x ( (7 x x f ( x exp for x (8 a The expetatio of the three parameter Weibull distributio is determied by the sale, shape ad loatio parameters [8: µ E(X (9 + Γ + ale ad shape parameters defie the variae of the three parameter Weibull distributio [8: Γ + Γ + (0 The shape parameter speifies the skewess of the Weibull distributio. The ormal For the shape parameter approximatig the ( distributio a be approximated by the ormal distributio the followig values for Weibull distributio with a speial shape parameter the gamma futio a be alulated [: [4: Γ + Γ + Γ + Γ + The sale parameter depeds o the variae for a shape parameter approximatig the ormal distributio: ( The loatio parameter depeds o the variae ad the expetatio value µ for a shape parameter approximatig the ormal distributio: Therefore, the distributio futio desribig the apaity demad by the three parame- (3 µ (4 ter Weibull distributio is:

4 Iformatia Eoomiă vol. 3, o. 3/ exp exp ( µ D D D F (5 The quota of server requests exeedig a defied apaity demad D is the differee betwee the total probability ad the distributio futio of the apaity demad D. For a give apaity demad with a ormal distributio approximated by the three parameter Weibull distributio, the quota of ritial server requests к exeedig the apaity demad D is: > D F P exp ( ( D D (6 Aordig to equatio 5, the apaity demad D a be substituted by the ritial server load. exp (7 From equatio 7, the apaity supply a be determied depedig o the expetatio value μ ad the variae of the apaity demad D ad the ritial server load к: [ [ l( l( l( exp + (8 Replaig the shape, sale ad loatio parameters with values for the approximated ormal distributio, the eeded apaity supply of a server with a quota of к allowed ritial server requests is:

5 00 Iformatia Eoomiă vol. 3, o. 3/009 [ l( µ (9 [ l( Hee, the apaity supply is determied by the expetatio value ad the variae of the apaity demad, the umber of server requests proessed i a give time period, ad the mea respose time of a server request µ 4 Determiig the apaity upply i ase of Virtualizatio I the ext step, the effet of virtualizatio o the apaity supply will be aalyzed. A physial server a be split ito differet virtual servers. I the model, a virtualizatio tehique with a dyami alloatio of the apaity supply of a physial server to the virtual servers is assumed. The apaity supply a be used ompletely by the virtual servers. Thus, virtualizatio overhead will be igored i the model. The apaity demads of the virtual servers are radom variables. These radom variables are orrelated if a statistial depedey betwee them exists. The apaity demads of the virtual servers should be uorrelated. The ovariae matrix [5 is: 0, i j ov( x with ij (0 i, i j The expetatio value of the physial server E( is the sum of the expetatio values of the virtual servers E( V Di [3: E( E ( V D V D V D i E( V Di ( i µ V Di The variae of the apaity demad of the physial server Var( for two virtual Var( Var Var ( + V D servers is i geeral [3: ( + Var( + ov(, V D V D The distributio of the apaity demad should be uorrelated. I this ase, the variae of the apaity demad of a physial server for virtual servers is: V D V D V D (

6 Iformatia Eoomiă vol. 3, o. 3/009 0 Var( Var i i ( V D Var( V Di V Di V D V D (3 Igorig the virtualizatio overhead, the apaity supply P of a physial server is the sum of the apaity supplies Vi of the virtual servers: P Vi i (4 server load, the quota of allowed ritial server requests к, the expetatio value of the apaity demad of a virtual server μ ad the variae of the apaity demad of a vir- I the model, the physial server is operatig [ l( µ P (7 With equatio 5, the expetatio value ad variae of the physial server a be substituted by the expetatio value ad variae of the virtual servers. The required apaity supply P o a physial server with idetial virtual servers depedig o the tual server : ritial idetial virtual servers. The expetatio values ad the stadard deviatios are equal for all virtual servers. The expetatio value E( ad the variae Var( of the apaity demad of the physial server are: E( µ µ (5 Var( For idetial servers operated by a physial server the required apaity supply P of the physial server is: P V (6 Equatio 9 desribes the quota of ritial server requests to a physial server without virtualizatio: [ l( µ P (8 Beause the physial server operates idetial virtual serves, the required apaity buted to the virtual servers: supply of the physial server a be distri- P V [ l( µ (9 [ l( laim: If less tha half of the server requests ause a ritial apaity demad (к < 0.5, µ the required apaity supply per virtual server dereases with ireasig umber of vir-

7 0 Iformatia Eoomiă vol. 3, o. 3/009 tual servers. Proof: If this is true, equatio 9 must be a d d V slopig futio for к < 0.5: 3 Term Term [ l( [ l( < > 0 For a positive umber of virtual servers ad a positive server load term is always egative. Hee, term is: (30 [ l( > l( > (3 l( < 0,6865 < e From the apaity plaig poit of view, it is useful to operate as may as possible virtual servers o a physial server. The required apaity supply of a ifiite large physial server operatig a ifiite umber of virtual servers would be the ratio of the expetatio value of the apaity demad ad the ritial server load: P + ov(, < I ase of orrelatio, the required apaity supply should be less tha i the uorrelated ase. Therefore, two idetial physial servers operatig two idetial virtual servers [ l( q. e. d P µ lim (3 p Beause of the limitatio of the apability of a physial server suh a mahie aot be implemeted. If a physial server operates two idetial virtual servers, the impat of the orrelatio o the apaity supply a be show. The equatio to alulate the apaity supply of two orrelated virtual servers is: [ l( µ (33 will be ompared. The two virtual servers o oe physial server are orrelated the others are ot. The followig iequatio desribes this ase: µ (34 > + ov(, [ l( µ

8 Iformatia Eoomiă vol. 3, o. 3/ Less tha half server requests should ause a ritial apaity demad. Applyig equatio 3 for к < 0.5 results to: [ l( > 0 (35 > The server load is always positive. Hee, the iequatio of the apaity supplies a be writte as follows: + ov(, (36 > + ov(, This iequatio is always true for egative ovariae betwee the apaity demads of the virtual servers. ov (, < 0 (37 I the uorrelated ase, for к < 0.5, a larger apaity supply is required tha i the ase of egative orrelatio. The radom variables ad represet the apaity demad of the virtual servers. Negative orrelatio meas that ad have a iverse liear oheree [3. Positive orrelatio will raise the required apaity supply. Thus, it is reasoable to aggregate virtual servers with egative orrelatio o a physial server to miimize the required apaity supply. 5 Example A data etre is plaig the apaity supply of four idetial servers. A server should be able to proess i mea 00 requests per seod. The arrival ad the proess rates are Poisso distributed. The mea respose time eeds to be 0.03 seods. This requiremet should be satisfied for 95% of the requests. The utilizatio of the apaity supply is ormally distributed. The ormal distributio will be approximated by the Weibull distributio. The expetatio value of the apaity demad is 000 MIP with a stadard deviatio of 00 MIP. Figure 3 illustrates the distributio ad desity futios of the apaity demad of a server. The apaity demads of the servers are uorrelated. First, the apaity supply is determied for the four servers realized as physial servers. Furthermore, the apaity savig will be determied if four virtual servers are operated by oe physial server with a dyami alloatio of the apaity supply. The arrival rate λ of the Poisso distributed Fig. 3. Distributio ad desity futios server requests is 00 requests per seod: λ 00 (38 The mea respose time T to proess a request is 0.03 seods: T 0,03 (39 From equatio 4, the ritial server load

9 04 Iformatia Eoomiă vol. 3, o. 3/009 a be determied by the arrival rate λ ad the mea respose time T. The ritial server load must ot be exeeded to omply with the mea respose time speifiatio: Tλ + Tλ ( To ahieve a mea respose time of 0.03 seods with 00 requests per seod, a server load of 75% must ot be exeeded. 00 [ l( P [ l( The required apaity supply is 60 MIP. The apaity savig by virtualizatio is the differee betwee the apaity supply of the four physial servers without virtualizatio ad the physial server operatig four virtual servers: (46 The apaity savig for operatig four virtual servers o a physial server is 87 MIP. 6 olusio The apaity supply desribes the maximum output of a server. The utilizatio of the apaity supply is the apaity demad. The server load is the ratio of the apaity demad ad the apaity supply. The appliatio of a Markov hai demostrated that it is impossible to utilize the full apaity supply. For a give arrival rate λ of server requests the ritial server load to ahieve a required respose time T a be determied. The expetatio value μ of the apaity demad D of a server is 000 MIP: µ E( 000 (4 The stadard deviatio of the apaity demad D of a server is 00 MIP: 00 (4 I 95% of the ases a additioal request should ot exeed a server load of 75%. The quota of ritial server requests к is: 0.05 (43 From equatio 9 the apaity supply omplyig with these requiremets a be determied: ( ( The apaity supply of a server is 768 MIP. I the ext step, the apaity supply of a physial server operatig four idetial virtual servers will be determied. The apaity supply of the physial server a be determied by equatio 8: The level of the arrival rate λ is ot fixed i the ourse of time but is statistially varyig. For a kow distributio futio of the levels of the arrival rate the apaity demad ot exeeded by a defied quota of server requests a be determied by the expetatio value ad the variae. The apaity demad a be used for determiig the apaity supply. Assumig a ormal distributio approximated by the Weibull distributio for the level of the arrival rate λ demostrates that the use of virtualizatio saves apaity. Negative orrelatio of the apaity demad of two virtual servers leads to more apaity savigs tha i the uorrelated ase. Referees [ G. Bolh, Queueig etworks ad Markov hais: modellig ad performae evaluatio with omputer siee appli-

10 Iformatia Eoomiă vol. 3, o. 3/ atios, Wiley, New York, 998. [ I. N. Broshtei, K. A. emedyayev, G. Musiol ad H. Muehlig, Hadbook of Mathematis, priger, Berli, 007. [3 F. M. Dekkig,. Kraaikamp ad H. Paul, A Moder Itrodutio to Probability ad tatistis, priger, Lodo, 005. [4. Y. D. Dubey, Normal ad Weibull distributios, Naval Researh Logistis Quarterly 4 (967, pp [5 W. Härdle ad Z. Hlávka, Multivariate tatistis: Exerises ad olutios, priger, New York, 007. [6 W. Ker, Idustrielle Produktioswirtshaft, Poeshel, [7 tuttgart, J. D. Little, 99. A proof for the queuig fo rmu a: l L λw, Operatios Researh: The joural of the Operatios Researh oiety of Ameria 9 (96, pp [8 D. N. P. Murthy, M. Xie ad R. Jiag, Weibull models, Wiley-Itersiee, Hoboke, NJ, 003. Alexader PINNOW studied Busiess ad omputer iee util 003. The he worked as software egieer speialized i fiaial markets. ie 006 he is sietifi assistat at the Very Large Busiess Appliatios Lab of the Otto-vo-Guerike-Uiversity i Magdeburg, Germay. His area of researh is the data etre as a produtio faility for IT-servies with fous o apaity maagemet i adaptive ad virtualized data eters. He published several papers i this field of study. The researh i this area is a ooperatio of the VLBA-Lab ad the Germay based IT-servie provider T-ystems. tefa OTERBURG studied omputer iee util 00 ad passed his diploma with distitio. He had worked as a software egieer ad osultat for Mirosoft ERP software (Dyamis AX for several years, before i 006 he beame sietifi assistat at the Very Large Busiess Appliatios (VLBA Lab of the Otto-vo-Guerike-Uiversity i Magdeburg. His area of researh is the data eter as a produtio faility for IT-servies with fous o availability maagemet i adaptive ad virtualized data eters. The researh i this area is a ooperatio of the VLBA-Lab ad the Germay based IT-servie provider T-ystems.