УДК 519.218.31
Дараселия А.В.1, Сопин Э.С.1,2
1 Российский университет дружбы народов, г. Москва, Россия
2 Федеральный исследовательский центр «Информатика и управление» РАН, г. Москва, Россия
АНАЛИЗ ОДНОГО ПОДХОДА ДЛЯ СНИЖЕНИЯ ЭНЕРГОПОТРЕБЛЕНИЯ ОБЛАЧНОЙ
СИСТЕМЫ*
Аннотация
Повышение энергоэффективности систем облачных вычислений является одной из
важнейших задач провайдеров облачных услуг. Одним из наиболее очевидных решений для
снижения энергопотребления является динамическое управление количеством работающих
серверов в зависимости от нагрузки. Однако такой подход приводит к дополнительных
временным и энергозатратам на подключение и отключение вычислительных мощностей.
Исследования показывают, что эти дополнительные затраты могут существенно
снижать выгоду от использования динамического управления. В связи с этим облачные
провайдеры применяют механизмы, снижающие количество включений/выключений
серверов. Один из таких механизмов исследуется в данной статье. Рассматривается
облачная система, в которой сервер, не имеющий активных виртуальных машин,
выключается не сразу, а по прошествии некоторого времени. Построена математическая
модель системы облачных вычислений с учетом времени включения и выключения для
анализа показателей энергопотребления. Проводится анализ условий, при которых имеет
смысл с точки зрения энергопотребления не переводить сервер в режим ожидания сразу же
как опустела очередь запросов. Были получены аналитические выражения для
стационарного распределения и основных показателей производительности и
энергопотребления.
Ключевые слова
Облачные вычисления; энергоэффективность; теория массового обслуживания.
Daraseliya A.V.1, Sopin E.S.1,2
1 Peoples’ Friendship University of Russia, Moscow, Russia
2 Federal Research Center Computer Science and Control of the Russian Academy of Sciences, Moscow, Russia
ANALYSIS OF AN APPROACH TO INCREASE ENERGY EFFICIENCY OF A CLOUD
COMPUTING SYSTEM
Abstract
Enhancing energy efficiency of cloud computing systems is one of the key challenges for cloud
providers. One of the most obvious ways to decrease energy consumption is dynamic control of
switched on servers according to the system load. However, this approach leads to additional energy
and time loss due to switching on/off of computing resources. Recent research shows that the
additional energy and time loss may significantly decrease positive effect of dynamic control. In this
regard, cloud providers employ various mechanisms that decrease server switching number. One of
the mechanisms is considered in the paper. We consider a cloud computing system, in which a server
does not switch off immediately, as it remains empty but after a certain time. We develop a
mathematical model of a cloud computing system with switch on/off periods and analyze its energy
efficiency metrics. We investigate how energy efficiency of a cloud system is affected by a waiting time
before a server goes to standby mode.
Keywords
Cloud computing; cloud computing; energy efficiency; queuing system.
* Труды II Международной научной конференции «Конвергентные когнитивно-
информационные технологии» (Convergent’2017), Москва, 24-26 ноября, 2017
Proceedings of the II International scientific conference "Convergent cognitive information
technologies" (Convergent’2017), Moscow, Russia, November 24-26, 2017
79
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 system
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.
Figure. 1.Transition intensities diagram
We derive the system of equilibrium equations, based on the transition intensity diagram (Fig. 1), which makes
it possible to obtain stationary probability distribution of the system:
(1)
p p ; 3,0
0,0
(2)
p3,k p , 1 k С 1;
3,k 1
(3)
p3,V1 p ;
3,С 1
(4)
p1,1 p p ;
0,0 3,1
(5)
p1,k p1,k 1 p , 2 k С 1;
3,k
p1,С
p
p ;
(6)
1,С 1 3,С
(7)
p2,0 p3,0 ;
(8)
p2,1 p ;
2,0
80
p2,2
p p
p ; (9)
2,1
2,0
1,1
p2,k 1
p
p
p , 3 k С 1; (10)
2,k 1 1, k
2, k
(11)
p2,С p1,С p2,С 1 ;
3 V1
(12)
pi 0 j 0
i, j 1.
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):
(13)
p3,0 p .
0,0
The stationary probabilities p3,k follow from formula (2) taking into account formula (1):
(14)
k
k 1
p3,k p3,k 1 p3,0 p0,0 , 1 k С 1.
k
The equation for stationary probabilityp3,V can be represented by formula (3) and expression (14):
C 1 C 1 C 1
(15)
p3,C p3,0 p
3,C 1 0,0
p 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
1 2 2 2 (16)
p1,1 p0,0 p3,1 p0,0 p0,0 p0,0 .
The stationary probabilities p1,k follow from (5) by substituting expression (14) and the simple algebraic
transformations:
p1,k p1,k 1 p , 2 k C 1
3,k
2 k 1
p1,k 1
p
k 0,0
2 k 2 2 k 1
p1,k 2 p 0,0 p
k 0,0
k 1
k 1
k 1
2 k 1
p1,1 p
i 1 i 0,0
k 1i
k 1
2 k 1
p1,1
k 1
p 0, 0
i 1
i 1 i 1
k
k
k 1 1
2 k 1 k 1
p1,1 p 0, 0
i 1 1
k
2 k 1 k k
k 1
k
k
p
2 2
p
k 0,0 0, 0
k 1 2 2 2 k 1
k k
p . (17)
p0,0
k k k 0,0
The expression for p1,V follows from (6) by substituting expression (15) and the simple algebraic
transformations:
p1,C p p
1,C 1 3,C
81
k 1 2 2 k 1
k k C 1
(18)
2
p p0,0 .
k
k k
0,0
The expression for calculating stationary probabilities for the operating mode p2,0 follows from formulas (7)
and (1):
p
p
p . (19)
2,0
3,0
0,0
The equation of stationary probability p2,1 can be represented by the formula (8) and the expression (19):
p2,1
p
p . (20)
2,0
0,0
The expression for p2,2 is obtained by substituting expressions (16), (19) and (20) into formula (9):
p 2, 2
p
p 2, 0 p1,1
p p 2 2 p
0,0
2 ,1 0, 0 0, 0
p p
2 2
0,0 0,0
2
2 2 (21)
p p .
2 0,0
0,0
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
p 2,k 2 p 2 , k 3 p1,k 2 p 2,k 2 p1,k 1
2 2
2
p 2,k 2 p 2,k 3 p1,k 1 2 p1,k 2
2
2
2
p 2,k 2
p 2,k 3 p1,k 1 2 p1,k 2
2 2
2 2
p 2,k 3 p 2,k 4 p1,k 3 2 p 2,k 3 p1,k 1 2 p1,k 2
2
3 2 2 3
p 2 , k 3
2 2 p
2,k 4
3
3
p 2 2 p
p1,k 1 1, k 2 1, k 3
2 2
3 2 2 3 2 2
p 2,k 4 p 2,k 5 p1,k 4 p 2,k 4
3
3
p 2 2 p
p1,k 1 1, k 2 1, k 3
2 2
p
3 2 2 3 2 2
p
3 2 2 3
2,k 4 2 , k 5
4
4
p 2 2 p 3 2 2 3 p
p1,k 1 1, k 2 1, k 3 1, k 4
2 2 3
4 3 2 2 3 4
p2,k 4
p3 2 2 3
4 4 2, k 5 (22)
p1,k 1
p p p .
2 2 3 2 2 3
1, k 2 1, k 3 1, k 4
2 2 3
We represent the resulting expression (22) as sum of sequences:
82
4 3
i 4 i
i 3 i
p2,k i 0 p2,k 4 i 0
p2,k 5
4 4
1 2 3
i 1i i 2 i i 3 i
p1,k 1 i 0
p1,k 2 i 0
p1, k 3 i 0
p1, k 4
2
2
3
4 3 l
i 4 i
i 3 i
4 1 i l i
i 0
p 2,k 4 i 0
p 2 , k 5 i 0 p1,k l 1
4 4 l 0 l
g g 1 l
i g i
i g 1i
g 1 i l i
i 0
p 2, k g p 2,k g 1 i 0
i 0
p1,k l 1
g g l 0 l
k g 2 g k 2
k 2 k 3 l
i k 2 i
i k 3 i
k 3 i l i
i 0
p 2, 2 i 0
p 2,1 i 0 p1,k l 1
k 2 k 2 l 0 l
k 2 k 3
i k 2 i
2 2 2 i k 3 i
i 0
p i 0
p0,0
k 2 2 0,0 k 2
l
k 3
i l i k l 2 2 2 2 k l 2 k l 1 k l 1
i 0 p . (23)
l 0 l k l 1
k l 1 k l 1
0,0
The resulting expression (23) can be represented in three parts:
k 1
k 2
i k 2 i
k 2
1
k 2
i
1 k 1 (24)
I i 0 i i ;
k 2
i 0
1
i 0
1
k 2
1 (25)
i 1 i
k 3 k 3
k 3 k 3
1 k 2
II i k 3 i
k 2 i 0 i 0
i 1 i 1
i 0
i 0 1
1
;
k 3 l i i k 3 l k 3 l
III p1,k l 1 p1,k l 1 p1,k l 1 i
i
i 0 i 0 i 0 i 0 l 0 i 0
1 k 2 k 3
1 p1,k l 1
k 3
p1,k l 1 l 1
l 0 1 1 l 0
p0,0
k 3
k l 2 2 2 2k l 2 k l 1 k l 1
1 l 1
1 l 0 k l 1 k l 1
k l 1
k 3 k l 2
2 2 k 3 k 1 2 2
p0 , 0
1 l 0 k l 1 l 0 l 1 k l 1
k 3 2 k l 2
k l 1
k l 1
k 3 2 k l k l 1 k l 1 (26)
.
l 1
k l 1 k l 1 k l 1 k l 1
l 0 l 0
In turn, we represent the third part (26) of expression (23) in the form of four parts:
k 2
1
k l 2
k 2
k 2
k 3 k 3 l
III1 k 1
l 0 l 0 1
k l 1 k 1
k 2
1
k 1
k 2
(27)
1 ;
k 1
1 l
(28)
k 3 k 3
1 1 1
III 2 l 1 ;
l 0 1
k l 1 k 1 k 1
l 0
83
III 3
k 3
k l 2 k l 1 k l 1 k 3 k l 2
k 3
k l 2
l 0
k l 1 k l 1
l 0
k l 1
l 0
k l 1
k 2 k 3 k 2 k 3
l l
l 0 k 1
k 1
l 0
k 2 k 2
1 1
k 2 k 2
k 1
1
k 1
1
k 2 k 2
1 1
k 1 k 1
(29)
;
k 3
III 4
k l 1
k l 1
l 0 k l 1
l 1 k l 1
k 3
k l 1 k 3
l 1
k l 1
l 0
l 1k l 1 k l 1
l 0 k l 1 k l 1
l l
1 k 3
1 k 3
k 1 k 1
l 0 l 0
k 2 k 2
1 1 (30)
1 1
.
k 1 k 1
1 1
Then we substitute the obtained expressions (27) – (30) into expression (26):
2 2 k 2
III p0,0 1
1
k 1
1
k 1 2 2 1
k 1 1
k 2
k 2
k 1 1 k 1 1
2
k 2
k 2
1 1
2 k 1
k 1
k 1 k 1
1 1
k 1
2
2
1
k 2 k 1
p0,0 1
1
1
k 2
k 2
1 1
k 1
2
1
k 2
k 2
k 1 1 1 (31)
.
1
Then we substitute the expressions (31), (24) and (25) into the formula (23):
p 2,k
1 k 1 2 2 2
p 0,0
1 k 2
p 0, 0
1
2
1
84
k 1
2 1
p0,0
2
1
k 2
k 1
1 1
k 2
k 2
k 2
k 2
1 1 1 1 (32)
k 1 k 1
2
1
1
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:
C C C C
(33)
PP p P p P
0 p P p ,
i 0
0,i 1
i 0
1,i 3
i 0
3,i i 0
2,i 2,i
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 , (35)
where blocking probability π is
p1,C p2,C p3,C . (36)
The average number N of customers is given by
3
N ipk ,i
C
(37)
k 0 i 1
The average sojourn time T follows directly from formulas (35) and (37):
3 C
ip . k ,i
(38)
T k 0 i 1
(1 )
4. Numeral analysis
The results of numerical analysis for the values C=7, µ=20, ɑ=1, β=2 are presented in figures 2 – 4.
Figure. 2. The dependence of the power consumption P on the arrival flow intensity λ
85
Figure. 3. The dependence of the power consumption P on the waiting time before the system goes to the standby mode
The plots of the server’s power consumption (fig. 2) for our model show that the consumed power increases
very fast for small values of the arrival flow intensity λ, also note that with the increase of waiting time, during
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 corresponds to
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.
Figure. 4. The dependence of the average time T on the waiting time before the system goes to the standby mode
In Fig. 4, we note that at small values of γ difference in sojourn time is not big. With increase of γ, the average
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 we need to
investigate its effect on performance measures also.
Благодарности
Публикация подготовлена при поддержке Программы РУДН «5-100» и при финансовой поддержке
РФФИ в рамках научных проектов № 15-07-03051 и № 15-07-03608.
Acknowledgement
The publication was prepared with the support of the “RUDN University Program 5-100” and funded by RFBR
according to the research projects No. 15-07-03051 and No. 15-07-03608.
References
1. Daraseliya A.V., Sopin E.S., Energy efficiency analysis of Cloud computing system with setup and vacation periods of server //
ITTMM-2017. — 2017. — S. 119-121.
2. Javier Conejero, Omer Rana, Peter Burnap, Jeffrey Morgan, Blanca Caminero, Carmen Carrion, Analysing Hadoop Power
Consumption and Impact on Application QoS // Future Generation Computer Systems. — Vol.55 Issue C, — Feb. 2016, — S. 213-
223. W3C Markup Validation Service. URL: http://validator.w3.org/.
3. Anton Beloglazov, Jemal Abawajy, Rajkumar Buyya, «Energy-aware resource allocation heuristics for efficient management of data
86
centers for Cloud computing » // Future Generation Computer Systems. — vol.28, — 2012. —S. 755 – 768.
4. DaraseliyaA.V., SopinE.S., O zadache optimizacii jenergopotreblenija oblachnoj infrastruktury // Information Technologies and
Mathematical Modeling names after A.F. Terpugov (ITMM – 2017). —2017.
5. Sopin E.S., Daraseliya A.V., Yarkina N.V., On the virtual machines migration effectiveness in cloud systems // 19-th International
Conference on «Distributed computer and communication networks: control, computation, communications – DCCN-2016». —2016.
—S. 408-411.
Литература
1. Дараселия А.В., Сопин Э.С., Анализ энергопотребления системы облачных вычислений с учетом разогрева и выключения
серверов // Информационно-телекоммуникационные технологии и математическое моделирование
высокотехнологичных систем. — 2012. — С. 119-121.
2. Javier Conejero, Omer Rana, Peter Burnap, Jeffrey Morgan, Blanca Caminero, Carmen Carrion, Analysing Hadoop Power
Consumption and Impact on Application QoS // Future Generation Computer Systems. — Vol.55 Issue C. — Feb. 2016. — C. 213-
223.
3. Anton Beloglazov, Jemal Abawajy, Rajkumar Buyya, «Energy-aware resource allocation heuristics for efficient management of data
centers for Cloud computing » // Future Generation Computer Systems. — vol.28. — 2012. — C. 755 – 768.
4. Дараселия А.В., Сопин Э.С., О задаче оптимизации энергопотребления облачной инфраструктуры // XVI Международная
конференция имени А.Ф. Терпугова «Информационные технологии и математическое моделирование» (ИТММ – 2017). —
2007.
5. Э.С. Сопин, А.В. Дараселия, Н.В. Яркина, Об эффективности миграции виртуальных машин в облачных системах // 19-я
межд.конф. «Распределенные компьютерные и коммуникационные сети: управление, вычисление, связь». — 2016. — Т.3—
С. 408-411.
Note on the authors:
Sopin Eduard S., professor at the Applied Probability and Informatics departmentof Peoples’ Friendship
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, sopin_es@rudn.university
Daraseliya Anastasia V., student of Applied Probability and Informatics department, Peoples’ Friendship
University of Russia, nastyadar6@gmail.com
Об авторах:
Сопин Эдуард Сергеевич, кандидат физико-математических наук, доцент кафедры прикладной
информатики и теории вероятностей, Российский университет дружбы народов; старший
научный сотрудник Института прикладной информатики, Федеральный исследовательский
центр «Информатика и управление» Российской академии наук, sopin_es@rudn.university
Дараселия Анастасия Валерьевна, студент кафедры прикладной информатики и теории вероятностей,
Российский университет дружбы народов, nastyadar6@gmail.com
87