Enhanced Management Method of Storage Area Network (SAN) Server with Random Remote Backups

Transcription

1 ELSEVIER Available online at MATHEMATICAL AND SCIENCE ~ DIRECT" COMPUTER MODELLING Mathematical Computer Modelling 42 (2005) Enhanced Management Method of Storage Area Network (SAN) Server with Rom Remote Backups SONG-KYOO KIM Mobile Communication Division, Samsung Electronics 94-1 Imsudong, Oumi, Gyeongbook , South Korea amang, kim~s amsung, com (Received December 2003; revised accepted June 2005) Abstract--A SAN (storage area network) is a mass storage solution, that is, designed to provide a high-reliability high-speed network over fiber optic link. This paper is dealing with the stochastic SAN server management method which is focused on reliability. The (remote) backup servers are hooked up by the long-haul network replace broken main severs immediately. If the SAN servers are represented as "machines", this system can be solved by using the stochastic maintenance model with main unreliable rom auxiliary spare (remote backup) machines, subject to rom breakdowns, repairs, two replacement policies: one for busy another for idle or vacation periods. When the repair facility is not available because of the given conditions, auxiliary machines are being used for backups. Unlike existing models, the availability of auxiliary machines is changed for each activation of the system. Analytically tractable results are obtained by using a duality principle (which enables us to treat a more rudimentary system), semiregenerative analysis, multivariate marked renewal processes. The results are demonstrated in the framework of optimized SAN server allocation problems with unreliable backup 2005 Elsevier Ltd. All rights reserved. Keywords--Stochastic maintenance, SAN management, Availability, Stochastic optimization, Closed queues, Duality principle. 1. INTRODUCTION Optical networks provide enormous capacities in the networks a common infrastructure over which a variety of services can be delivered. Storage area network (SAN) is one of optical network applications to interconnect computer systems with other computer systems peripheral equipments. A SAN architecture is identified by fibre channel consists of one or more servers connected to storage devices via switching devices such as hubs, switches, bridges. Connections between clients storage devices are controlled by SAN servers that should be very reliable can support high-speed connections between storage clients. SANs operate at bit rates ranging from 200 Mbps to 1 Gbps over fiber optic link. While the bit rate itself is relatively modest, the SAN is attractive because of the optical layer that can support a huge number of such connections between two data centers. Optical fibre offers much higher bwidth than copper cables is less susceptible to various kinds of electromagnetic interference other /05/$ - see front matter (~) 2005 Elsevier Ltd. All rights reserved. Typeset by Ajk/~S-TEX doi: /j.mcm

2 948 STORAGE AREA NETWORK (SAN) SERVER undesirable effects [1]. SAN server for resilience against disaster is a major issue of reliable network management. We assume that the remote backup SAN servers are geometrically separated with main SAN servers hooked up by the long-haul networks over fiber optic links. The stochastic model in this paper is considered to improve the availability of SAN servers. Let us assume that we have SAN spare servers which are linked with long-haul networks. Once all m + 1 servers become intact, the system renders a checkup. Patching applications or upgrading the operating system is a very typical pattern of maintenance. During this period, the system borrows servers with the latest version of the operating system applications from the total quantity of from an external provider. Since the usage of external resources are rom because these are also unreliable. Our model provides decision factors such as the optimal control level, the number of spare servers, failure rates, so on, that are beneficial for the SAN architecture. The mechanism that is mentioned above can considered as the stochastic model that consists of machines a repair facility which fixes broken machines goes on vacations when all machines are working properly. Unlike other existing models, the availability of auxiliary machines is changed for each activation it considers the rom number of external backups for maintenance purposes beyond M/G/1 closed queueing system. If each SAN server is considered as "machines", the model forms a class of closed queueing systems with the initial quantity of m + 1 main unreliable machines, S rom auxiliary reserve machines, also called "super-reserve" machines. The concept of super-reserve machines is intoduced in [2]. Main machines represent the SAN servers super-reserve machines means their remote backups. Main machines are subject to "exponential failures" their repairs are rendered (in the FIFO order) by a single repair facility (also known as the "repairman") with generally distributed repair times. The rom super-reserve facility is "activated" the repairman is idle or vacationing whenever the number of main machines is restored to its original quantity. Even though the number of super-reserve machines is rom, we assume that the number is fixed when the super-reserve machines are activated. During this time, dropped machines line up in the %vaiting room" until the whole super-reserve facility is exhausted one more machine breaks down. Then, a repairman's busy period resumes its cycle continues. The classical model with m unreliable machines no backup [3] is also described in terms of closed queueing systems, i.e., those with so-called "finite sources." the rom availability of super-reserve machines makes our model different. The supplementary variable method semi-regenerative analysis are used with additional results for semi-markov processes analytic solution is very appealing in optimization. The current methodology (without super-reserve facility) stems from [4-6] further explored with super-reserve facility [2]. 2. DUALITY PRINCIPLE: GENERAL 2.1. Stochastically Congruent Models The duality principle we would like to begin with includes another model, which is more simple than the main one (Model 1) to which we will refer as to Model 2. Model 2 is similar to Model 1, except that it does not have the backup facility idle periods. We rather associate it with repairman's vacations, which are distributed as regular repairs. However, upon his return, the repairman brings a br new machine, which replaces any one that breaks down during his vacation trip if any such available. Otherwise, the new machine he brings in substitutes any other machine in both cases the old machine is disposed. Model 2 is directly connected with yet another model, which we will call Model 3. Model 3 is a regular multichannel queueing system, in notation, (GIo, GI)/M/m/O (see Appendix A.1). Model 2 Model 3 are best observed in a parallel fashion [5]. The number Zt of intact machines in Model 2 corresponds to the total number of customers in Model 3. m - Zt gives the number of defective machines waiting for repair in Model 2 it also gives the number

3 S.-K. Kim 949 of available channel in Model 3. Whenever a machine of Model 2 breaks down, a customer of Model 3 is processed leaves the system. A customer in Model 3 is called for service to occupy a channel just vacant, while in Model 2, the latter joins the queue of defective machines. Whenever a machine is completely repaired, it joins the main facility while this event corresponds to the arrival of a new customer from an outside source. If the channel is full, this customer gets lost. Understably, the channel must have been filled up upon the previous arrival there has been no service completion since then. Correspondingly, the repairman leaves on vacation when all machines become intact upon his return, the repairman attempts to replace a defective machine by a new one he carries out. If there is none available, he may return the new machine for refund. In any event, the status of the reliability system, in this case, is not altered. We call Models 2 3 "stochastically congruent." This idea has been previously utilized in [5,6], in connection with simpler models Notations In this section, we will describe all the above models more formally. Denote by Z 1 the total number of intact main machines at time t in Model 1. If a repairman fixes all machines completely, the total number of main machines is restored to m + 1. Then, he goes on vacation until S + 1 machines break down, after which the repairman resumes his work. In other words, a busy period begins. As we already mentioned, backup machines are replaced when main machines fail during repairman's idle period. Denote by S, the rom number of external backups that are used during idle periods. The PMF (probability mass function' of S (the number of super-reserve backups) is s(n):=p{s=n}, n=0,1,..,n... (2.1) with the mean r - E[S] Nmax is the maximum number of externall backups with the control level N(< N... ). Let TO(= 0),71,... be the successive moments of repair completions. The rom variable 7n+1 -~-~ is supposed to have a PDF (probability distribution function), A(x) := P{~+l - ~n _< x}, x > 0, (2.2) with mean a = E[7-,~+1 - T~] < oo. If upon the service completion, the totm number of intact machine is greater than m, the PMF of next service completion period is Ao(x) with the mean ao = f~+ xao(dx). Each of the main machines breaks down independently of each other of repairs, according to the exponential distribution with parameter 0 < # < oo. In Model 2, there is a maximum of m main served by a single repairman. The total number of main machines at time t (> 0) is denoted by Zt. Unlike Model 1, there are no idle periods even when all main machines become intact. Let T~ be the n th repair completion. If the line of defective machines is nonempty, the repairman continues to repair a next machine immediately. When the cumulative number of intact machines becomes m at Tn, the repairman leaves the system returns at T~+I with a br new machine. We assume that his vacation is distributed as his regular service time. The new machine replaces a defective one, if by then available, or otherwiserefund the new machine. We suppose that the last action does not affect the status of the system in this particular problem setting. The rom variable T,~+I - Tn is stochastically equivalent to Tn+l --T~ of Model 1. The assumption about the failures distribution is the same as of Model 1 (i.e., exponential distribution with parameter #). 3. DUALITY PRINCIPLE: CONNECTION BETWEEN THE MODELS In case of the given number of external backups S, it can be shown that 7-1,72... T1, T2,... are sequences of stopping times of the processes, x, Z I' 0)) E1 {0,1,..,re+l} (3.1) ( t,t> =

4 950 STORAGE AREA NETWORK (SAN) SERVER (,t2,(p )xee,(zt;t>o)) E {0,1,.,m}, (3.1a) respectively, that these processes are semiregenerative relative to these sequences [5, p. 246]. Let ~ := Z 1 Xn := ZT,~-, n 0, 1, 2,...,rn_ Consequently, (~1,"~[1, (P X)xc E, (~n;?~ = 1, 2,... )) --* E (3.2) (f~,ll,(p~)xce,(xn;n=l,2,...)) ~E (3.2a) are embedded Markov chains (MC). Both Markov chains are stochastically equivalent ergodic. Their limiting probabilities are expressed through the common invariant probability measure P = (P0, P1... Pc). Since ((n) (X,~) are stochastically equivalent, only one of them will be treated. Let (ft1, 11, (P~)x~E, (Ytl;t > 0)) ~ E (ft,tl, (P~)~S, (Yt;t > 0)) --~ E be the minimal semi-markov processes associated with the above sequences of stopping times, respectively. In the case of the Model 1, we need to find the limiting probabilities, (cf., [7, p. 342]), where y~ := lim P{Yt' = k} = PkMk k e E, (3.3) x--,~ P M ' { a,[ ] k=0,1,...,m-1, (3.4) Mk := Ek[T1] = E ao(s + 1) + (S 1)., k = m, m# j PM is the scalar product of P M = (M0, M1,..., Mm) T, which equals PM=fi+Pm.E aos + m------fi-- ' where O~ = (1 - Pc) a + Pmao. It is readily seen that the process Zt Zt 1 on their busy periods are stochastically equivalent, or formally, P~{Zt = k} = P~{Zlt = k I t E Busy Period} (k E E). Furthermore, Therefore, P~ {Zt = k} = Pz {Zlt = k l t c Busy Period} _ P~{Zlt = k,z 1 C {0, 1,...,rn}} Px{Zl t C {O, 1,...,m}} 0, k=m+l, = = k} 1_ p~{ztl = m+ 1}, k<m. P~{ZI=k}=(1-Px{Zlt =m+l})p~{zt=k}, k=0,1,...,m. (3.5) On the other h, breaking a period when Yt 1 = m in two intervals, of which the first one is an idle period the second one is the repair period of all backup machines, we have lira px {t E Idle Period I S = n, Yt ] = m} = (n + 1)/m# = 1. t--~ (n + 1)~rap + (n + 1)a0 1 + aom#" (3.6)

5 S.-K. Kim 951 From (3.3), P,,,(n + i)(i + a0m#) lim P* {Yt 1 = m ] S = n} = m#(a + P,,aon) + Pm(n + 1)" t (3.r) Let ~k 1 := limt +~ P~{Z 1 : k },~k(n):= 1 limt-~ px {Z~ = k I s = n}, k = 0,1,...,m, be the limiting probabilities conditional probabilities of the process Zt 1. These probabilities exist, under the same conditions as those for the embedded process. Then, from (3.6),(3.7), it follows that ~m+l (7/) : lim Px{z1 t = rn + ] I S = n} Pm(n + 1) :. (3.8) t---~oz m~(a + Pmaon) + Pm(n + 1) From (3.7),(3.5) yield = -- ~m+] (n)) 7rk, k = O, 1,...,m, (3.9) where 7rk := limt~ Px{Zt = k} is subject to our consideration in Section 4. conditional expectations, we have, i [ Pm(S 1) ] E = [~m+l (S)] = ~ mp(~t + PmaoS) + Pm(S + 1) 7rm+ l Because of (3.10) From (3.1), (3.10), (3.11) yield 1 l~, [(1 1 71"k = -- ~rn+l (S))] 71"k, k = 0,1,...,m. (3.11) i fi s(n)p.,(n + 1) X,n+l = ~#(0, Pmaon) + P.~(n + 1) rt=o 1 ~ S(~1,)TD~#(Ct -~ Pmaon) ~k ~k n=o mp(a + Pmaon) + P.z(n + 1)' k = 0,1,...,m. 4. STOCHASTIC PROCESS WITH CONTINUOUS TIME PARAMETER We now turn to the continuous time parameter queueing process of Model 3. The below treatment is similar to that of Dshalalow [5] (see also [2]). We bring some details, for the sake of consistency. Let (Nt) be the counting process associated with the point process (Tn) Vt := TN,+I --t, t >_ O. {Vt; t _> 0} gives the residual time from time t to the next repair completion TN,+I. The process ( a,u,( v L~,(z~,E)L>0 ) ~(~,;~(~)), ~:=Ex~+, is weak Markov. Let (Pi i) be the Markov semigroup associated with (Zt, Vt) assumed to be absolutely continuous, i.e., with P](k [0, y]):=pi{z t =k, Vt C [0, y]}, (4.1) 7r~(t,u)du = Pi{Z t = k, Vt E [u,u+du)}, i,k C E, A(x), j =O,...,m-1, Aj (x) = Ao (x), j = m. ~, t c R+, (4.2)

6 952 STORAGE AREA NETWORK (SAN) SERVER Since we are dealing with equilibrium, we do not need to consider the initial state. So, we are dropping the index (i) from 7r. To find the limiting distribution 7r of the process (Zt), we will start with the Kolmogorov differential equations, assuming that Aj(x) (of (3.2)) has the Radon- Nikodym density aj(x), which is pointwise continuous. From (4.2), the Kolmogorov equations arc as follows, (0 0) 0u 7rk(t, u) = -kptrk(t, u) + (k + 1)~zTrk+l (t, u) (4.3) (o0) +Trk_l(t,O)a(u), k = 1,...,m- 1; Ot O-u 7rk(t,u) = --mptrrn(t,u) + ptrm(t,o)a (u) + 7r,~_l(t,O)a(u), k = m. (4.3a) The process (Zt, Vt) is aiso semiregenerative relative to the sequence To, T1,..., its limiting distribution exists. Let 7rk(u) := t~eclim 7rk(t,u),Crk(O):= jf 7rk(u)e-O~du, Re(0) _> 0, k C E. + Letting t --~ ec in (4.3) then applying the Laplace transform, we have (0 - ] #)Trk(O) = 7rk(0) -- (]; + 1)#~k+1(0) -- OL(0)Tl'k--l(0), k ,..., 7)2 -- 1; (4.4) With in (4.4), (0 -- TFt~t)~rn(O ) = 7rrn(0 ) -- 71"m_1(0 ) O~(0) -- 71"m(0 ) Oz0(0). -~u~ =~(o) (k+l)u~k+~ ~k_~(0). Summation over k = 1,..., n - 1(< m - 1) in (4.6) then yields From (4.6), Using (Tr, 1) = 1, we get 7rn-1 (0) 7On , ~t ~- 1~...,~7~. n# ~m_l(o) m~ Tr0 = 1 - ), 7rn. n=l To find the unknowns 7rk(0) when k = 0, 1,..., m, we use the approach in [3,4]. Denote (4.5) (4.6) (4.7) (4.8) (4.9) Note that pjk(t) :- PJ {zt = k I r~ > t}. ~ pjk(t) A(dt) = Pjk, + j,k c E, (4.10) are the transition probabilities of the embedded MC (Xn). We use the natural assumption that PJ{Z~=klTl>t}=PJ{Zt=klTl>t+u}, Let K3t(k [0, y]) := PJ{Zt = k, Vt c [0, y], T1 > t}. theorem, lim P{(k [O,y])= (~)-lj~epj /~ KJt(k [O,y])dt, t~oo + Vu>O. Then, from the main convergence kse, yc~+, (4.11)

7 S.-K. Kim 953 where a= (1 -Pm)a+Pmao. By elementary probability arguments, then, K~ (k x [0, y]) : Pjk (t) [A (t + y) - A (t)] lim P[(k x [O,y]) = lim Pi{Zt = k, Vt C [O,y]} t ----* O0 t--+ O~ : (~,)--1 E PJ ~ KJ(k x [0, y])dt dee + = (gt)-i E PJ pjk(t)[aj(t + y) - Aj(t)] dt, j@e + = (a)-i E PJ p;k(t) aj (t + x) dx dt j6e + =0 j, kee, yen+ (by Fubini's theorem), = (~)--1 E PJ pjk(t)aj(t + x) dt dx. =0 jee + (4.12) From (4.2) (4.12), ~(x) = (~)-~ ~ Pj jee + pj~(t)~j(t + ~)dt. (4.13) When x -+ 0 in (4.13), due to (4.10) Pk = Y-]~jeE PjPjk, Pk ~k(o)=--, k=o,...,,~ (4.14) a where Pk from (A.1). From (4.7)-(4.9) (4.14), we finally arrive at the limiting distribution rr, m 7to = 1 - Errn' k = 0, n~l Pkl 7r k -- k#a ' k = 1,...,m, (4.15) where P = {P0,..., P,z} from (A.1). For the process Z 1, the corresponding formulas yield (from Section 3), 7rm+ 1 : E m#(g + Prnao n) 4- Pm(n + 1)' ~ ~(~).~(~ + Pmao~) ~k = ~k,~(a + P~a0~) + P~(~ + 1)' 7z=O (4.16) k = 0,1,...,m, (4.17) along with (4.15).

8 954 STORAGE AREA NETWORK (SAN) SERVER 5. OPTIMALITY OF THE SAN SERVER MANAGEMENT In this section, we will deal with a class of optimization problems that arise in reliability. Let us formalize a pertinent optimization problem. Let a strategy, say Z, specify, ahead of the time, a set of acts we impose on the system, such as the choice of repair distribution, the number of main external backups, statistics of failure rates dependent on the nmnber of backup machines that enables us to spend more or less time on the maintenance so on. On the other h, a system can be subject to a set, say C, of cost functions. Denote by (Z, C, t) the expected costs within [0, t], due to the strategy Z costs C define the expected cumulative cost rate over an infinite horizon, (E, C) := t +o~ 7 1im 1 (Z, C, t). (5.1) We formulate our problem for a special case. Let g(u) denote the cost function associated with the system spending u units of time for the super-reserve machine usage during all idle periods that fall into the period of time [0, t]. If {Yt~; t > 0} is the semi-markov process (from Section 2.3) g(u) is a linear function, i.e., g(u) = cu where c is cost rate of idle periods, then f0 t 1{,~+~} (Y~)du gives the time spent by the system during all repairman's idle periods within [0, t], where 1A is the indicator function of a set A, Ug(t) = ]Ei [g ( fot l{~} (Y~) du) ] = ce~ [/ot l{rn} (Y~) du] is the corresponding expected cost. By Fubini's theorem, ~0 t Ug(t) = c pi {y~ = m} du. (5.2) Let fl(n) be the reward rate for n intact main machines. Then, the expected reward for all working machines in the interval [0, t] is [f0t ] [r~l /t ] Ufl(t) = IE i fl(z 1) ds = E i fl(k) lik}(z 1) ds kk=0 =0 (5.3) (which by Pubini's theorem is) m+l t = Zsl(k)f P {zl=k}as. k=o =0 If Lt denotes the number of failed machines waiting for repair, during a busy period, at time t, then it is t = m + 1 Zt 1. Analogous to the above calculations, the expected cost due to failures during all busy periods within the time period [0, t] is [jf0 t ] rn+l /~ Uf2(i,t) =E i f2(l,)ds = E f2(m+l-k) Pi{z~ =k} ds. k=0 0 (5.4) Since the Markov renewal function Ri(t) = Ei[Nt] ]Ei[~n~_-0 l[0,t](t~)] represents the total expected number of machines refurbished in the time interval [0, t], the functional ldr i (t) = rr i (t) gives the total reward for all machines completely processed in [0, t], if r is the common reward for each repaired machine.

9 S.-K. Kim 955 The cumulative cost of the entire procedures involved in the reliability system of main borrowed super-reserve machines, in the interval [0, t] is (Z, C, t) = Ug (t) + Ufl (t) -- U f2 (t) + LIR ~ (t) m+l =~ {z: =~} du+ ~ fl(k) {z~ =k} ds k=o =0 m+ 1 t + Zf~(.~+l-k)f, P~{zJ=k} d~+~r*(t) k=0 =0 t ~0 t m+l ftpi =~ P~{Z3=.~} d~+~ (fi(k)+f~(m+l-k)). {Z l=k} d~+~r~(t) k=0 Js=0 Among various constraints, it is reasonable to include that the quantity of backup machines must be large enough as to provide some maintenance (or upgrade or retraining) of m + 1 main machines this can be done as t~oo t =0 1{re+l} (Z 1) du > 1"(re+l), (5.5) where F is an appropriate "reconciliation" function of m. The value on the left of (5.5) is the portion of a maintenance period within a unit time (we can call it shortly the maintenance rate). On the right-h side, we set the minimal maintenance rate as a function F of internal t machines. In terms of our system, f =0 l{m+l}(z~)du is an "idle period" within [0, t], during which we assign the repairman to the above mentioned maintenance. In thc case of computer networks, an upgrade can be anything from installing a newer version of operating system virus checking to patching the network server applications. Now, we turn to convergence theorems for semiregenerative, semi-markov, Markov renewal processes, (i) (ii) lim ~Ri(t)- 1 t-~ PM' lira -/o 1 t pi {Z 1 k} ds 7rk, t---* oo t 1 (iii) lira -1 ft pi {y~ = k} du PkMk t~o~ t Jo PM ' to arrive at the objective function (Z, C) which gives the total expected rate of all processes over an infinite horizon. In light of equations (i)-(iii), we have m+ i PmMm 1 6 (Z, C) = t--,~olim 6 (Z, C, t) = c p~ + r~-~ + E (fl (k) + f2 (m k)) 7r~. (5.6) k=0 With fi being linear functions, we have rn+l P~Mm 1 (Z,C) =c PM +r~--m+ E [kh+(m+l)l]tc~, (5.7) k=0

10 956 STORAGE AREA NETWORK (SAN) SERVER where h, l are relevant (cost) constant coefficients. Recall that from (4.15)-(4.17), we have 1 ~ S (n) P,,~ (n -F 1) 71-m+1 = 2_~?T~t (a -}- Pma0?%) Jr- Prn (~ q- 1)' tz=0 rr k = rrk E s (n) rnp (5 + Pmaon) m 7r0 : 1 - ~ rr~, k = 0, k = 0, l,...,m, :rk - Pk-1 k~t> ' k= 1,...,rn. With these attachments (A.1)-(A.5), formula (5.7) for the objective function is complete. 6. OPTIMIZATION EXAMPLES In this section, we intend to demonstrate the tractability of our results in Sections 2-6 on a special case of super-reserve machines. We allow an exponential distribution of repair times. Since our method involves operating with an embedded Markov process in a multichannel open queue with a finite buffer, we get back to Section 3 for some particular formulas M/M/m/0 Queueing System The pertinent special formulas of Section 3 under these assumptions are as follows, A(x) -- Ao (x) = i - e -(1/a)x, (6.1) JfR ~ o(1/a)e-( +l/a)z dx a(o) = + e - x da(x) = = _ 1 ~-ao' 1 1 oe (rap (1 - z)) = 1 + am#(1 - z)' (6.2) Now, we can calculate the remaining parameter such as a~ from (3.3), B, from (3.2) to get Pk (from (3.1)). Once we get Pk, the probabilities rrk can be found by using formula (4.15) SAN Server Allocation Optimization: Unreliable External Backups Servers Let us consider the number of external backup servers has a binomial distribution. In this case, the system borrows N external backup servers but these backups are also unreliable at the moment of borrowing. If the probability of the intact backup servers is p (i.e., Bernoulli rom variable), the sum of unreliable backups yields the binomial rom valuables. So, (3.1) yields s(n) = (N)pn(1- p) N-n, (6.3) N 7rm+ 1 1 =E [qol+l(s)] = ~ (pl+l(n)8(n)" r~--o We also specify the remaining of the five primary cost functions are as follows, fl(n) = G.n, f2(n) = B.n, (6.4) where G is the reward for one SAN server, B is the penalty for one broken server during all busy periods all per unit time. From (5.7) (6.4), we have rn+l PmM,~ 1 (Z(N),C):t_~olim (Z(N),C,t):c.--+r.pM ~+~-~(fl(k)+f2(m+l-k))rr~ k=0 m+l PmM,~ 1 1 =c. p~ r.~--~+ ~ (Gk+B(m+l-k))rr k k=o

11 S.-K. Kim 957 Optimal Soultion of Servers o i... i E O , The Number of Ser'cers with 7 channels (m=6) Figure 1. Optimization of SAN sever backups. where m+l -1 1 k--0 Finally, we arrive at the following expression for the objective function, P,~Mm 1 (Z(N),C)=c. p~+r.~+(g-b)2~+b(m+l) (6.6) Here, we use formulas (3.7), (4.15)-(4.17), (6.6). Notice that we have the parameter N vary. Even though we cannot see the maximum number of backups N in (6.6), it is included in all ~ (see (3.10),(3.11)). Comparing (6.6) with (5.7), we identify h = G - B l -- B. We restrict the initial strategy of this model to one, which includes only the control level N of super-reserve machines. In other words, we need to find an N, such that (6.5) (Z(N),C) = min { (Z (N),C): N = 1,2,... }. (6.7) As an illustration, we take c = 2, G = 2, r = 4, B = 1. Repair time distribution is exponential with mean a = 1.2 the parameter # = 0.2. Take the total number of SAN servers as five. Now, wc calculate (Z(N), C) No that gives a minimum for (&(N), C). In other words, the control level No sts for the available number of backup servers (super-reserve machines) which minimizes the total cost of this system. Below is a plot of (Z(N), C) for N C {1, 2,..., 20}. Our calculation yields that No = 5 for which the minimal cost equals It means that we allocate our internal resources to seven main (m + 1) machines obtain the decision value No = 5 which is the number of external backups under availability p (= 0.2) for a backup server. The maximum number of external servers is needed to minimize the cost of this SAN model. 7. CONCLUSIONS This paper shows the analytical approach of the SAN management method by using the closed queueing system with flexible conditions. This approaches theoretical, but feasible to apply realworld applications such as networked server allocation management [8] design of enhanced secure network infrastructure [9]. Analytically tractable results are obtained by using a duality principle, semiregenerative analysis multivariate marked renewal processes. Implementation of this model comparison between the model the actual data will be the further research that is related with this paper.

12 958 STORAGE AREA NETWORK (SAN) SERVER APPENDIX A.I. Embedded Process of Model 3 Model 3, as mentioned, is the conventional (GIo, GI)/M/m/O (nlultichannel) queue. Recall that such a system is characterized by the general-independent input (i.e., a renewal process), m parallel channels without any buffer. A customer enters a free channel available with his service dem distributed exponentially with parameter #. Inter-renewal times are distributed in accordance with the PMF A(x) Ao(x). Model 2, as we see it, is congruent to Model 3, while Model 1 is dual with Model 2, so also with Model 3. The latter is a classical system investigated by Takacs [3]. The stationary probabilities P = (P0, P~,..., Pro) for the embedded process are known to satisfy the following formulas, B,(k) r (-1) ~-k r~k Pk rr~ 1 k = O,...,m- I, (A.1) where Bn = r=n (A.2) O~0 ~ (7)/ Ozr ' I, r=0, ~ := ~ ai (A.3) l-'ii=0 --, 1 - a~ r > I, a (0) = e- ~A(du), a (0) e- ~Ao(du), (A.4) /0 /0 o c~o at = a (r#), a r = (r#). (A.5) REFERENCES 1. R. Ramaswami K.N. Sivarajan, Optical Networks: A Practical Perspective, American Press, San Diego, CA, (2002). 2. s. Kim J.H. Dshalalow, A versatile stochastic maintenance model with res. super-reserve machines, Methodology Computing in App. Probability 5 (1), 59-84, (2003). 3. L. Takacs, Introduction to the Theory of Queues, Oxford University Press, New York, NY, (1962). 4. J.H. Dshalalow, On the multiserver queue with finite waiting room controlled input, Adv. Appl. Prob. 17, , (1985). 5. J.H. Dshalalow, On single-server closed queues with priorities state dependent parameters, Queueing Systems 8, 23~254, (1991). 6. J.H. DshMalow, On a duality principle in processes of servicing machines with double control, J. of Appl. Math. Sire. 1 (3), , (1988). 7. E. Cinlar~ Introduction to Stochastic Processes, Prentice Hall, Englewood Cliffs, N J, (1975). 8. S. Kim, Enhanced networked server management with rom remote backups, SPIE Proc. on Performance Control of N. G. Comm. Networks 5244, , (2003). 9. T. Kawno, S. Matsui, T. Yasue C. Konno, Secure communiction infrastructure for mobile, Hitaeh Review 48 (1), 15-20, (1999).