MULTISERVER RETRIAL QUEUE WITH SETUP TIME AND ITS APPLICATION TO DATA CENTERS

. This paper considers a multiserver retrial queue with setup time which is motivated from application in data centers with the ON-OFF policy, where an idle server is immediately turned oﬀ. The ON-OFF policy is designed to save energy consumption of idle servers because an idle server still consumes about 60% of its peak consumption processing jobs. Upon arrival, a job is allocated to one of available oﬀ-servers and that server is started up. Otherwise, if all the servers are not available upon arrival, the job is blocked and retries in a random time. A server needs some setup time during which the server cannot process a job but consumes energy. We formulate this model using a three-dimensional continuous-time Markov chain obtaining the stability condition via Foster–Lyapunov criteria. Interestingly, the stability condition is diﬀerent from that of the corresponding non-retrial queue. Furthermore, exploiting the special structure of the Markov chain together with a heuristic technique, we develop an eﬃcient algorithm for computing the stationary distribution. Numerical results reveal that under the ON-OFF policy, allowing retrials is more power-saving than buﬀering jobs. Furthermore, we obtain a new insight that if the setup time is relatively long, setting an appropriate retrial time could reduce both power consumption and the mean response time of jobs.

1. Introduction. Data centers are the main infrastructure for cloud computing services. In data centers, there are a huge number of computer servers (server farm) which consume a large amount of energy. Therefore, power-saving in data centers is one of the most important issues because it has a big impact on the operational cost and the environment. It is reported that power consumption of an idle server is about 60% of its peak [2]. A natural way for saving power consumption is to turn off physical servers once they become idle. However, this policy, sometimes called as ON-OFF policy, has some side effects. First, arriving jobs are not processed immediately upon arrivals because off-servers are required to restart thus increasing the delay. Second, during the restart period (i.e., setup time hereafter), the server cannot process jobs but consumes energy. In this paper, we consider the applications where one physical server processes one job at a time as in [9].
Investigating the delay-performance and energy consumption trade-off by a queueing model is useful because it can provide some insights that help us to design optimal system saving the power consumption while keeping an acceptable service level for users. Gandhi et al. [9] analyze a multiserver queue with setup times for data centers using the recursive renewal reward approach. The authors investigate the tradeoff between power-consumption and delay-performance. Phung-Duc [22] analyzes the same model and obtain exact closed form expressions of the stationary distribution using the generating function approach and the matrix analytic methods. Furthermore, in [22], the switching cost between ON and OFF is taken into account in the power consumption.
An important feature missing in [9,22] is the retrial phenomenon of jobs. It is common in computer systems that requests are resubmitted after some random time if they are blocked. For many applications such as Gmail, etc., a request is repeated after some waiting time. It is even more important to take into account the retrial phenomenon in systems with power-saving controls because these policies clearly increase the dropping (blocking) probability and the delay for jobs, thus accelerating retrials. On one hand, allowing retrials may increase the possibility that the server can be in OFF state thus reducing power consumption. On the other hand, a job may suffer from blocking leading to a long delay.
The aim of our paper is to quantify the tradeoff between the power consumption and delay-performance taking into account the presence of retrials. To this end, we consider a multiserver retrial queue with setup time which is a natural retrial version of the multiserver queue in [9,22]. We assume that the multiserver queue is buffer-less, i.e., no waiting position is available. Instead, an arriving job that cannot find any available servers is blocked and it retries in a random time independently of other jobs in the orbit and the service of servers. During retrial intervals, the job is said to stay in the orbit. We formulate the multiserver retrial queue using a threedimensional continuous-time Markov chain and establish the stability condition which is the theoretical contribution of our paper. We show that the stability condition of the multiserver retrial queue with setup time is stronger than that of the corresponding non-retrial queue in [9,22]. From a practical point of view, this result suggests that under power-saving controls such as the ON-OFF policy, more servers are needed in order to stabilize the system with retrials in comparison with that without retrials (with an infinite buffer).
It has been widely known in fairly general settings that the stability condition for a retrial model is the same with that of the corresponding model without retrials [13,14]. The stability condition of our multiserver retrial queue is a new finding in the sense that it is different from that of the corresponding queue without retrials. In this line, we recently became aware of Shin [24] in which a stability condition is derived for MAP/PH/c/K retrial queues with vacation. It should be noted that vacation model is essentially different from setup model because in the former a vacation is independent of waiting customers while in the latter OFF servers are activated by waiting jobs.
In order to compute the stationary distribution, we treat the Markov chain as a level-dependent quasi birth-and-death process with infinitely many levels. We show that our Markov chain possesses a similar structure to the one of [19], allowing us to compute the stationary distribution efficiently. We also propose a heuristic technique to determine the truncation point for the level-dependent quasi-birthand-death process. Finally, we provide some numerical examples to show impacts of retrials and setup time on the power-saving policy in data centers. Furthermore, we show the power consumption and delay performance trade-off and new insights into the performance of data centers. A brief version of our paper was presented in [21].
We shortly summarize the contributions of this paper for clarity. First, we give the stability condition of the multiserver retrial queue with setup time. Second, we develop a heuristic technique to determine the truncation point of the leveldependent quasi-birth-and-death process. Third, we provide performance evaluation of the retrial queue in terms of the power consumption and delay.
The rest of this paper is organized as follows. Section 2 presents the multiserver retrial queue with setup time and its formulation based on a three-dimensional continuous-time Markov chain. Section 3 provides the stability condition of the queueing model. Computational algorithms for the stationary distribution are summarized in Section 4. Section 5 presents performance evaluation via numerical experiments. Concluding remarks are presented in Section 6.

System model and Markov chain.
2.1. Queueing model. We consider an M/M/c/c retrial queue with setup time where a server in the queueing system corresponds to one physical server in data centers. We assume that a physical server can process one job at a time as in [9]. In this system, the server is immediately turned off if there is no job to process. An arriving job is allocated to an unoccupied off-server and that the server is started up. The server needs some setup time to be active so as to process the waiting jobs. Upon the completion of a service, the server picks one waiting job (waiting at another setup server) if any and the server in setup for that job is turned off. Thus, in our model a server has one of three states: active (serving a job), off or setup. We assume that the setup time of the server follows the exponential distribution with mean 1/α and the service time is exponentially distributed with mean 1/ν. The arrival process of jobs is the Poisson process with rate λ and is independent of the setup and service times. Blocked jobs that see all the servers occupied (active or setup) join the orbit from which they retry independently with rate µ. A job in the orbit retries until being served. Regardless of a newly arriving job or a retrial job from the orbit, the process of serving a job is the same. Although our model is buffer-less, jobs still wait for available servers from setup or service completions.
Remark 1. Our model with c = 1 is different from the single server retrial queue with setup time in [20,23] and its variant [4], where the server is not turned off if there are some jobs in the orbit. In contrast, this paper analyzes the case where the servers are not aware of jobs in the orbit. Therefore, the server is turned off immediately upon the completion of a service. As a result, our model with c = 1 is identical to the M/G/1 retrial queue with the same arrival rate and retrial rate. Further, the service time of this M/G/1 queue is the sum of the setup time and the service time of our model.

2.2.
Continuous-time Markov chain. Let us denote by S 1 (t) the number of jobs in the service facility at time t. It should be noted that S 1 (t) is also the sum of the number of active servers (the number of jobs in service) and the number of servers in setup (the number of jobs waiting for an active and available server) at 18 TUAN PHUNG-DUC AND KEN'ICHI KAWANISHI time t. Let us denote by S 2 (t) the number of active servers (serving a job) at time t. Furthermore, let N (t) denote the number of jobs in the orbit at time t. Then, it is easy to see that the stochastic process forms a continuous-time Markov chain on the state space S defined by Remark 2. There are several ways to construct equivalent Markov chains for this system. For example, one may consider the process (S 1 (t) − S 2 (t), S 2 (t), N (t)). However, as we will show, our Markov chain possesses a special structure for which the framework in [19] could be adopted to develop an efficient algorithm.

2.3.
Level-dependent QBD process. We formulate {X(t); t ≥ 0} as a leveldependent quasi birth-and-death (QBD) process [16] by taking N (t) as the level variable and (S 1 (t), S 2 (t)) as the phase variable. Then it is easy to see that the infinitesimal generator Q is given by where Q where I is an identity matrix of appropriate dimension. For i = 1, . . . , c, 3. Stability analysis. We consider a sufficient condition for the positive recurrence of {X(t); t ≥ 0}. We define the diagonal matrix ∆ as ∆ = diag{q c,0 , q c,1 , . . . , q c,c } and we define square matrices It holds that A 0 , A 1 and A 2 are non-negative matrices and are independent of k.
We consider a (discrete-time) level-independent quasi birth-and-death process with block matrices A 0 , A 1 and A 2 , respectively. Furthermore, A 0 , A 1 and A 2 record the transition probabilities increasing the level by 1, keeping in the same level and decreasing the level by 1, respectively. Letting It is clear that A is the transition probability matrix of a discrete-time Markov chain on the state space {0, 1, . . . , c}. We define A(z) by It should be noted that A in our model is reducible. It is clear that A(z) is also reducible and has the same structure of A. In fact, by a direct calculation, we can see that A(z) has the block structure given by . . , λ c (z) are the other eigenvalues of A(z). Furthermore, we can choose an eigenvector v(z) corresponding to χ(z) such that v(z) ≥ 0. In Lemmas 3.1 and 3.2, we prove that under a certain condition (the stability condition presented in Theorem 3.3), there exists some z ∈ (0, 1) such that v(z) > 0, based on which we can construct a Lyapunov function for guaranteeing the stability of our Markov chain.
Remark 3. Neuts [15] mainly treated the case A(z) is irreducible. It was pointed out in Neuts [15] (pages 14-15) that when A(z) is reducible ad hoc arguments are needed, which is our case. Let Proof. Assume that χ c−1 (z) > a j,j (z) for z 0 < z < 1 and j = 0, 1, . . . , c − 2. Then it is obvious that and v c−2 (z) are positive, we reach a contradiction again. Therefore we have v c−3 (z) > 0. Repeating the same argument, we conclude that v 0 (z), . . . , v c−2 (z) are all positive.
Let us denote by ν c−1 the invariant probability vector of A c−1,c−1 (1), and by β c−1 the vector given by is the first derivative of the function of A c−1,c−1 (z) with respect to z. It is well-known [15] that the equation Because it holds that χ c−1 (0) > 0 in our model, there exists such a solution, if ν c−1 β c−1 > 1.

TUAN PHUNG-DUC AND KEN'ICHI KAWANISHI
Proof. The proof is immediately followed from the convexity of χ c−1 (z) and Lemma 3.1.
If it holds that then the stochastic process {X(t); t ≥ 0} is positive recurrent.
Proof. The proof is essentially based on the technique by Diamond and Alfa [5,6], which makes use of the Foster criterion. We use the approach by Tweedie [25] or Statement 8, p. 97 in Falin and Templeton [7], which states that a Markov process with infinitesimal generator Q = [q s,p ] on state space S is regular and ergodic, if there exist a lower bounded function ϕ(·) on S, some finite subset S 0 ⊂ S and some ǫ > 0 such that y(s) = p∈S q s,p ϕ(p) ≤ −ǫ for s / ∈ S 0 , and y(s) < ∞ for s ∈ S 0 . For z ∈ (max{η, z 0 }, 1), let ϕ (k) be a column vector of size (c + 1)(c + 2)/2 given by . . .
We can obtain ν c−1 and β c−1 as It is easy to see that (1) follows from ν c−1 β c−1 > 1.

Remark 5.
Using a similar sample path analysis as in [10], we can also show that (1) holds if the stochastic process {X(t); t ≥ 0} is positive recurrent.
Remark 6. It is interesting that the stability condition of multiserver queues with setup time is more stringent in the presence of retrials. Indeed, the stability condition of the multiserver queues with setup time and without retrials [9,22] is ρ < 1. We note that a similar stability condition is also obtained for MAP/PH/c/c retrial queue with vacation.

TUAN PHUNG-DUC AND KEN'ICHI KAWANISHI
Remark 7. The same Lyapunov function can be directly used to derive the sufficient stability condition for M/M/c/c + K retrial queues with setup time. We confirmed that in the case K > 0, the stability condition is ρ < 1. The stability conditions of the models with and without buffer are different. It is well known that the stability condition for a retrial queue without setup time and with classical (linear) retrial rate is the same as that of the corresponding model without retrials [13,14]. This is evident because in the saturated situation where there are many customers in the orbit and all the servers are active. In this situation, the time for a customer to access from the orbit to the server tends to zero and the orbit behaves like the buffer. However, in the model with setup time and without buffer (M/M/c/c), when all the servers are active and a job is completed, the server is turned off regardless the presence of jobs in the orbit. Thus, in this sense the service capacity is loosen. In this saturated situation, there likely exists the case that one server is in setup process and the rest of c − 1 servers are active. In other words, it seems that one server may lose its service capacity in the saturated situation. This explains why the effect is maximized when c = 2. This can be discussed later in formula (3) in detail. However, the effect disappears in the model with buffer because in the saturated situation there are always customers waiting in the buffer and the server is not turned off upon service completions.

Algorithm for the stationary distribution.
In what follows, we assume that {X(t); t ≥ 0} is positive recurrent. If we assume that {X(t); t ≥ 0} is positive recurrent, then we can define π It holds that π is the unique solution of the linear systems given by πQ = 0, πe = 1, where 0 and e denote a row vector and a column vector of zeros and ones with an appropriate size, respectively.
The matrix analytic method [12] tells us that π (k) has a matrix-product form solution given by where {R (k) ; k = 1, 2, . . .} are the minimal nonnegative solution to The boundary vector π (0) is uniquely determined by 2 ) = 0, Computing the stationary distribution {π (k) ; k = 0, 1, . . .} in level-dependent quasi birth-and-death processes essentially reduces to computing the rate matrices {R (k) ; k = 1, 2, . . .}. Bright and Taylor [3] propose an algorithm to compute the rate matrices efficiently by extending the logarithmic reduction algorithm [11]. It is reported that a simple algorithm of Phung-Duc et al. [16], based on a matrix continued fraction representation, is easier to implement and less memory-consuming than Bright and Taylor's algorithm.

A heuristic technique for truncation point.
In order to apply the algorithm in [16], it is necessary to decide where to truncate the level in advance. We propose a heuristic technique to determine the truncation point N . Let us consider a single server M/M/1/1 retrial queue (without setup time) with arrival ratē λ, service rateν, and retrial rate µ. We assume the retrial queue is stable. We denote by L 1 the stationary number of jobs in the orbit for this single server retrial queue. Furthermore, let us consider the number of jobs L 2 that arrive during an exponentially distributed setup time with mean 1/α. Our heuristic technique is to regard L = L 1 + L 2 as an approximation of the number of jobs in the orbit of our multiserver retrial queue with setup time. We will chooseλ andν such that the stability condition of the M/M/1/1 queue (without setup time) coincides with our multiserver model with setup time. Instead of multiserver retrial queues, we make use of the single server M/M/1/1 retrial queue because it allows us to obtain an explicit expression of the generating function of the orbit size distribution. We selectλ andν asλ = λ/c,ν = rν.
Under the stability condition of our model ρ < r,ρ =λ/ν < 1 and thus the generating function Π(z) of L 1 is given by [7] as: where a =λ/µ. It is easy to see that the generating function Q(z) of the number of jobs L 2 arriving during the setup time is given by Q(z) = (1 − θ)/(1 − θz), where θ =λ/(λ + α).
The generating function H(z) of L = L 1 +L 2 is given by H(z) = Π(z)Q(z), under the assumption that L 1 and L 2 are independent. There are two rationales behind the use of L for the estimation of the truncation point of our multiserver retrial queue with setup time. First, for a non-retrial multiserver queue with staggered setup, i.e., only one server can be in setup mode at time, the queue length is decomposed to that of the model without setup time and the number of Poisson arrivals during the setup time [18]. Although our model is not staggered setup, we expect that a similar relation, i.e., the orbit size of our multiserver queue with setup time is approximately decomposed into that of the model without setup and the number of Poisson arrivals during the setup time. The second rationale is that it is reasonable to estimate the truncation point of a multiserver retrial queue (without setup time) by a single server retrial model with the same traffic intensity [17]. Since there are two heuristic approximations, our method is rather rough and thus needs to be carefully verified.
Let us denote by [z n ]f (z) the coefficient of z n in the series expansion of an arbitrary analytic function f (z). We use the notation g(n) ∼ h(n) defined by lim n→∞ g(n)/h(n) = 1. By investigating the dominant singularity of Π(z)Q(z), we and Γ(·) is the gamma function. Using the asymptotic result, we propose a heuristic technique to determine the truncation point N by where ǫ 1 ∈ (0, 1) is an arbitrary small value. It should be noted that the explicit asymptotic expression m n (ρ, θ, a) of [z n ](Π(z)Q(z)) gives us the method to compute N with less effort than those involving matrix computations and thus is practically useful. One of the drawbacks of our heuristic technique is its reliability whose validity will be presented later.

4.2.
Approximate stationary distribution. For a given truncation point N , we obtain an approximation R (N ) to R (N ) using the algorithm in [16]. Then we can obtain an approximation { π (k) ; k = 0, 1, . . . , N } to the stationary distribution {π (k) ; k = 0, 1, . . .} by where R (k) = R (k) ( R (k+1) ) for k = 1, 2, . . . , N − 1, and π (0) is the unique solution of the linear systems given by 2 ) = 0, Because of the special structure of Q (k) 0 , we observe that R (k) (X) for a square matrix X of dimension (c + 1)(c + 2)/2 must have the form c,i (X) is a (c+1)×(i+1) matrix for i = 0, 1, . . . , c. Exploiting the property of the sparse structure, it is clear that the approximate stationary distribution of our queueing model can be efficiently computed. In fact, we observe that the structure of the Markov chain of our queueing model is the same as the one of [19] and thus the stationary distribution can be computed by Algorithm 3 in [19].
Assuming that the mean power E[P ] is proportional to the number of servers in setup as well as the number of active servers, we propose to evaluate E[P ] by where C a and C s are the energy per time unit for an active server and a server in setup, respectively. If we assume that a server in setup consume the same power as in its peak, it holds that C s = C a . As a reference model for comparison, we also consider the corresponding M/M/c queue without setup time, where servers become idle if there are no jobs to serve. We refer to the policy of the model as ON-IDLE policy. It is easy to see that the mean power consumption E[P M/M/c ] for this model is given by where C i is the energy per time unit for an idle server. It should be noted that E[P M/M/c ] does not depend on the retrial rate µ.

TUAN PHUNG-DUC AND KEN'ICHI KAWANISHI
Furthermore, let E[W ] denote the mean response time of a job which is the time interval from arrival to the acceptance of the job to a server including retrial times. E[W ] is obtained from the stationary distribution as follows due to Little's law.

5.
Numerical examples. We show some numerical examples to illustrate impacts of retrials and setup time on the performance in terms of the power consumption. For all numerical examples, we set the mean service time as a unit and thus we fix ν = 1. We also fix C a = C s = 1. If we assume that the power consumption of an idle server is about 60% of its peak [2], it implies that C i = 0.6C a . The job arrival rate λ is adjusted so that we obtain a specific value of ρ. It is reported [2] that the utilization of servers is typically between 10 and 50 percents. Unless otherwise stated, we select the value of ρ = 0.5 to evaluate the performance in the worst-case scenario.    (2). We choose the retrial rate µ = 1/10. We observe that our heuristic technique gives us sufficiently small value of the stationary probability at the truncation point. It should be noted that the mean setup time to activate an off-server is typically larger than the mean service time of a job. As far as these numerical examples, our heuristic technique is practically useful enough to discuss the performance of data centers. Numerical experiments show that our heuristic technique works well for the case with relatively large retrial rate. Thus, in all of our numerical results, we consider µ ≥ 1/10.

5.2.
Effect of retrials on power consumption. Figure 1 shows the power consumption against the retrial rate. We observe that the power consumption decreases with the decrease in the retrial rate. Thus, the ON-OFF policy works more effectively for long interval of the retrial time. This is because servers are kept off much more time as decreasing the retrial rate. It should be also noted that the multiserver retrial queue with setup time approximately approaches to the corresponding nonretrial queue as increasing the retrial rate. Therefore Figure 1 suggests that from a power-saving point of view, it is better to reject blocking jobs and allow retrials instead of buffering.
We also observe that the power consumption for α = 1/10 is almost insensitive to the retrial rate and is less than that of the M/M/c without setup time. In contrast, the power consumption for α = 1/100 is more sensitive to the retrial rate and is higher than that of M/M/c without setup time. This suggests that there exists a threshold α * , which may depend on µ and possibly on ρ as well, exceeding α * the power-saving (ON-OFF) policy is more effective than the ON-IDLE policy. Numerical examples of the power consumption against the setup rate in Figure 2 show the evidence supporting that suggestion.
Next we investigate the impact of retrials on the power consumption from the viewpoint of the number of servers. We show the ratio E[P ]/c versus the retrial rate in Figure 3. We observe that the ON-OFF policy works effectively as the number of servers increases. This advantage can be viewed as the economies of scale in terms of power-consumption per server.

5.3.
Effect of retrials on response time. In this subsection, we investigate the effects of retrials on the mean response time in Figure 4 for ρ = 0.5 and in Figure 5 for ρ = 0.8. Here the response time of a job is the time from the arrival until the the job is accepted to the system. We observe that if the setup time is relatively fast, α = 1/10, the mean response time monotonically decreases with the retrial rate as is expected because the response time is likely dominated by the retrial times when the setup time is short enough. This together with the observations in Section 5.2 show the trade-off between power consumption and delay-performance.  Interestingly, we observe in the case α = 1/100 that the mean response time first decreases and then increases with the increase in µ. The explanation for the first part (i.e. µ is small) is the same as for the case α = 1/10. On the other hand, the reason for the monotonic increase of the mean response time in µ could be explained as follows. If a job frequently retries, the job is likely blocked again because all the servers are either busy or in setup. As a result, the number of retrials increases leading to the increase in the response time. Our result suggests that if the mean setup time is relatively large (about 100 times of the service time), setting an appropriate retrial time could reduce both power consumption and response time.   Figure 6 shows the power consumption versus the traffic intensity. Table 3 shows the truncation point N and π (N ) e corresponding to Figure 6 by setting ǫ 1 = 1.0 × 10 −12 in (2). The numerical values are obtained for α = 1/10 and µ = 1. For ρ ≥ 0.6, we observe that our heuristic technique for the truncation point provides larger values of π (N ) e than those of ρ ≤ 0.5. Even for ρ ≥ 0.6, the heuristic technique could provide large values  of truncation point N by adjusting ǫ 1 so that we obtain sufficiently small values of π (N ) e comparing with those of ρ ≤ 0.5. However, we observed that selecting smaller ǫ 1 , e.g. ǫ 1 = 1.0 × 10 −15 , gives us almost the same power consumption computed numerically by setting ǫ 1 = 1.0 × 10 −12 . Figure 6 suggests that the ON-OFF policy consumes less energy than the ON-IDLE policy up to a moderate value of ρ. Furthermore, the ON-OFF policy is more effective as increasing the number of servers. This is also a consequence of the economies of scale. We adopt the same truncation point presented in Section 4.1 for the M/M/c/c + K model. Figure 7 shows the effect of the buffer on the mean power consumption for K = 0, 10, 20 and 30. We observe that the power consumption for the cases with buffer is greater than that for the case without buffer (K = 0). In the case with buffer, jobs have more chance to wait. As a reason, the servers have less change to be in the IDLE state and are more likely to be in setup than the case without buffer. 6. Concluding remarks. In this paper, we have analyzed a multiserver retrial queue with setup time for power-saving data centers. We have analyzed the queueing model by a level-dependent quasi birth-and-death process. Based on the Foster-Lyapunov criteria, we have established a sufficient condition of the stability for the multiserver retrial queue. We have shown that the level-dependent quasi birth-anddeath process has the same structure as in [19] allowing us to compute the stationary distribution efficiently. We have proposed a heuristic technique for determination of the truncation point which has shown to be practically useful. Numerical results have revealed that allowing retrials is less power-consuming than buffering jobs. Furthermore, the longer the retrial interval is, the less data centers consume power. We have also revealed that the power-saving (ON-OFF) policy is more effective compared with the ON-IDLE policy, if the mean setup time is sufficiently short in comparison with the mean service time of a job. Furthermore, we have found a new insight that if the setup time is relatively long, setting an appropriate retrial interval could lead to reducing both power consumption and response time.
Before concluding this paper, we remark on some side effects of retrials in the multiserver queue with setup time. As in Remark 6, retrials of jobs adversely affect the stability of the multiserver queues with setup time. Indeed, let us rewrite the stability condition (1) in Theorem 3.3 as In case α/ν ≪ 1, (3) implies that we have to reduce the incoming jobs by almost half to stabilize the system with c = 2. Fortunately, this side effect decreases as the number of servers c increases. From a managerial point of view, providing servers with shorter setup time is more beneficial for stable operation of data centers under ON-OFF policy in our model.