ANALYSIS OF AN APPROACH TO INCREASE ENERGY EFFICIENCY OF A CLOUD COMPUTING SYSTEM

1. Introduction

Energy efficiency of cloud system is very important issue for cloud providers. The servers can be put into standby state in order to improve the energy efficiency of a cloud system in case of light load. On the one hand, the switching to standby mode allows us to reduce power consumption, and on the other hand, it leads to extra power usage to turn on/off the server. Therefore, it is important to understand under what conditions it will be advantageous to put the server in standby state, and under what conditions it is more profitable to leave it in the operating mode.

In [1] we have considered a cloud system taking into account switch on and switch off periods of servers, and it was assumed that the server switches off immediately as it remains empty. In this paper, we consider a model, in which the server does not switch off immediately after it is empty, but waits for an exponentially distributed time. For simplicity, we consider only one server with a number of virtual machines working on it.

2. Mathematical model of a cloud system

We consider a multi server queuing system with C servers. Customers arrive according to the Poisson law with rateλ. Service times, switch on and switch off durations are exponentially distributed with the parametersμ, ɑ and β, respectively. The system state is described by the vector (s, k), where k is the number of customers in the system, s is the server state. Here s = 0 means that the system is in the standby mode, s=1 reflects switch-on mode and s=2 and s=3 represent operating and switch off modes, respectively. Arrival of a customer in an empty system causes change of the system state to the switch on mode. After exponentially distributed time with rate ɑ, the syste enters the operating mode, in which serving of customers is started. When the system remains empty in the operating mode, it does not switch off immediately, but waits exponentially distributed time with rate γ. If a customer arrives during that waiting period, then the system starts serving. Otherwise, the state is changed to the switch off mode. If a customer arrives during the switch off mode, then the system turns to the switch on mode immediately after the completion of the switch off. Otherwise, the system falls to the stand by mode. Figure 1 shows the transition intensities diagram. p1,С   p1,С1   p3,С ;

   p2,0   p3,0;

  p2,1   p2,0; 3 V1   pi, j  1.

i0 j0

Taking into account the normalization condition (12) and using matrix methods, the system of equations of equilibrium ( 1 ) – (11) can be solved numerically, but below we represent the analytical solution of ( 1 ) – (11).

The expression for p3,0 follows directly from formula ( 1 ):

p3,0   p0,0.

The stationary probabilities p3,k follow from formula ( 2 ) taking into account formula ( 1 ):

  k  k1 p3,k     p3,k1       p3,0     k p0,0, 1  k  С 1.

The equation for stationary probabilityp3,V can be represented by formula ( 3 ) and expression (14):

    C1    C1    C1 p3,C   p3,C1        p3,0         p0,0       p0,0.

The expression for p1,1 follows from formula ( 4 ) and the expression for p3,1, is obtained from (14): p3,1       p3,0     

     p0,0 p1,1     p0,0     p3,1   1  p0,0  2 p0,0    2     2           p0,0.

The stationary probabilities p1,k follow from ( 5 ) by substituting expression (14) and the simple algebraic transformations:

  p1,k     p1,k1     p3,k , 2  k  C 1 

    p1,k1    2kk1  p0,0      2k2   2k1       p1,k2     k1   p0,0      k    p0,0     k1

       k1    

k1  2k1 p1,1   i1    k1i   i p0,0   2k1 k1    i1 p1,1     k    p0,0 

i1   i1     k1  2     2    k     p0,0  k12     2    k     p0,0      k1 p1,1    2kk1  p0,0 ki11 11k 

      2k1   k     k     k       k       

   2 k1       k     k 

 p0,0.    k   k p0,0 

The expression for p1,V follows from (6) by substituting expression (15) and the simple algebraic transformations:

  p1,C   p1,C1   p3,C  (10) (11) (12) (13) (14) (15) (16) (17)    k1  2     2   2 k1      k     k      C1 (18)

    k         k   k   p0,0        p0,0.

The expression for calculating stationary probabilities for the operating mode p2,0 follows from formulas (7) and ( 1 ):

       p2,0   p3,0   p0,0.

The equation of stationary probability p2,1 can be represented by the formula (8) and the expression (19):          p2,1   p2,0   p0,0.

The expression for p2,2 is obtained by substituting expressions (16), (19) and (20) into formula (9): p2,2      p2,1   p2,0   p1,1          p0,0     p0,0 22 p0,0             p0,0  22 p0,0 

     2       2     2    2 p0,0         p0,0. (19) (20) (21) The stationary probabilities p2,k follow from (10) and the simple algebraic transformations:     p2,k  p1,k   p2,k1  p2,k1, 2  k  C 1;  p2,k1      p2,k  p2,k1  p1,k , 2  k  C 1;  p2,k      p2,k1  p2,k2  p1,k1, 3  k  C; p2,k   p2,k1   p2,k2  p1,k1, 3  k  C;

    p2,k        p2,k2   p2,k3  p1,k2    p2,k2  p1,k1      2 p2,k2  2   p2,k3    2   p1,k2   p2,k2   p1,k1 

      2 2    p2,k2  2   p2,k3    p1,k1  2 p1,k2      22  p2,k2  2   p2,k3    p1,k1  2 p1,k2    2     2     2  

p2,k3   p2,k4   p1,k3   2   p2,k3    p1,k1  2 p1,k2    3  2 3 2   3 p2,k3  2  3  2 

p2,k4   3  2   2   3     3      p1,k1  2  p1,k2  2  2   2 

 p1,k3  

    2     2  p2,k4   p2,k5   p1,k4    3

p2,k4      p1,k1  2  p1,k2  2  2   2  3  2   2   3     2     2   4

 p1,k3   p2,k4  3  2   2   3   4

p2,k5    p1,k1  2  p1,k2  2  2   2  p1,k3  3  2   2   3 

 3   4  3 24 2  3   4 p2,k4   3  2 4 2   3 

p2,k5   p1,k1  2  p1,k2   2    2   2 p1,k3   3  2  2   3   3

 p1,k4  

 p1,k4 .

  We represent the resulting expression (22) as sum of sequences: 4  i 4i III   ki03 il0 i i p1,kl1   ki03 il0  i p1,kl1   kl03 p1,kl1 il0  i  In turn, we represent the third part (26) of expression (23) in the form of four parts:

III1  k3 kl2 k2 k3   l  k2 l0   kl1    k1 

l0      k1   k1 1    k2

         

    k2      1     ; III2  k3 1 1 k3    l 1

   l0  l1   kl1    k1  l0      k1 

  k2 1     

    1     

   k1 1    ;

   1  (23) (24) (25) (26) (27) (28)    1 k1 11 k2    1 k1 11 k2 .

 Then we substitute the obtained expressions (27) – (30) into expression (26):

III3  kl03 kl2 klk1l1 kl1kl1  kl03  klk2l1  k3 kl2  k2 k3    l  k2 k3   ll0  kl1 

   k1 l0      k1 l0    

1   k1 1  

 1   k1 

   2     k1    k2 1       

  k2  k1     1             2    k1     k1     1    k2    1    

k1   k1  1   k2 

  1    

     p0,0 1 2  2    1   k2    k1 11k1      2     k1      k2 1 1    k2 

   1          k1      k2 1 1    k2 

   1    .

Then we substitute the expressions (31), (24) and (25) into the formula (23): p2,k  1  k1      2       22 p0,0   1  k2        

 1    2  1   

 p0,0  (29) (30) (31) 

  p0,0 1  2  2    1    k2         k1 1    k1 

    2        k1      k2 1  11 k2      k1     k2 1  11k2 

3. Energy consumption indicators

After receiving the system stationary distribution, we calculate the energy consumption indicators. We will assume that in the switch on / off mode, the power consumption is constant and equal to the average value. In the operating mode, the power consumption depends on the server occupancy. By analogy with the formula given in [3,4,5], we derive the formula for the average server power consumption:

P  P0 C p0,i  P1 iC0 p1,i  P3 iC0 p3,i  iC0 P2,i p2,i , (33)

i0 where

P2,k  P2,min  P2,max  P2,min k. (34)

V1

The values of Pi were taken from [2], according to which P0 = 10 W, P1 = 170 W, P3 = 120 W,P2, min = 105 W and P2, max = 268 W.

The average number N of customers in the system is equal to the average effective arrival rate λ(1-π) multiplied by the average sojourn time T. Expressed algebraically the law is

N   (1 )T, where blocking probability π is

  p1,C  p2,C  p3,C.

The average number N of customers is given by

The results of numerical analysis for the values C=7, =µ20, ɑ=1, β=2 are presented in figures 2 – 4. (32) (35) (36) (37) (38) The plots of the server’s power consumption (fig. 2) for thoautrthemcoodneslumsehdopwower increases very fast for small values of the arrival flow intensity λ, also note that with the increase of which the system doesn’t go into standby mode, the power consumption also increases.

In Fig. 3, we note that the largest drop for power consumption occurs at small values of γ that co large values of waiting time before the system goes to the standby mode, and that with the increase of the load intensity λ, energy consumption also increases.

In Fig. 4, we note that at small values of γ difference in sojourn time is not big. sojourn time also increases. It means, the larger the waiting time before the system goes to the standby mode, the greater sojourn time.

5. Conclusion

In the paper, we considered a cloud computing system in which the server switches off after a random time after it was left empty. The intensity γ monotonically affects the average consumed power, so investigate its effect on performance measures also.

With inc we nee

Благодарности Acknowledgement

Публикация подготовлена РФФИ в рамках научных

при поддержке проекто-в07-№0305115 и Программы №-071-053608.

Р-1У0Д0Н» и«5 при финансовой поддержке

The publication was prepared with the support of the “RUDN according to the research projects No. 15-07-03051 and No. 15-07-03608.

Uni-v1e0r0si”ty anPdrogfruanmded5 by

RFBR

References

Note on the authors:

Sopin Eduard S., professor at the Applied Probability and Informatics departmentof Peoples’ University of Russia; PhD, senior researcher at the Institute of Informatics Problems, Federal Research Center Computer Science and Control of the Russian Academy of Sciences, [e-mail] Daraseliya Anastasia V., student of Applied Probability and Informatics department, Peoples’ University of Russia, [e-mail]

Friendsh

Friend Об авторах: Сопин Эдуард Сергеевич, кандидат физик-оматематических нау,к доцент кафедры прикладной информатики и теории вероятно, стРеойссийский университет дружбы народ;освтарший научный сотрудник Института прикладной информ, аФтиедкиеральный исследовательский центр «Информатика и упрлаевние» Российской академии ,нsаoуpк[e-mail] Дараселия Анастасия Валерьевна, студент кафедры прикладной информатики и теории Российский университет дружбы народnоaвs,[e-mail]

References

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