=Paper=
{{Paper
|id=Vol-2332/paper-03-005
|storemode=property
|title=
On Optimization of Energy Consumption in Cloud Computing System
|pdfUrl=https://ceur-ws.org/Vol-2332/paper-03-005.pdf
|volume=Vol-2332
|authors=Anastasia V. Daraseliya,Eduard S. Sopin,Vladimir V. Rykov
}}
==
On Optimization of Energy Consumption in Cloud Computing System
==
https://ceur-ws.org/Vol-2332/paper-03-005.pdf
23
UDC 519.218.31
On Optimization of Energy Consumption in Cloud Computing
System
Anastasia V. Daraseliya* , Eduard S. Sopin*† , Vladimir V. Rykov‡
*
Department of Applied Probability and Informatics
Peoples’ Friendship University of Russia
Miklukho-Maklaya str. 6, Moscow, 117198, Russia
†
Institute of Informatics Problems, FRC CSC RAS
44-2 Vavilov Str., Moscow 119333, Russia
‡
Departmen of Applied Mathematics and Computer Modeling
Gubkin Russian State University of Oil and Gas
65 Leninsky Prospekt, Moscow, 119991, Russia
Email: nastyadar6@gmail.com, sopin_es@rudn.university, vladimir_rykov@mail.ru
We constructed mathematical models of cloud computing systems taking into account
various mechanisms for increasing energy efficiency in terms of queuing theory, and analytical
expressions for the main characteristics of energy consumption and server performance metrics
were obtained. We investigated and compered three different energy efficiency improving
mechanisms of cloud computing systems: the shutdown delay mechanism, the switch on
delay mechanism and the threshold-based switch on mechanism. The general principle of
functioning mechanisms for energy efficiency improving is that all mechanisms try to find a
middle ground between continuous operation without shutdowns and with switching on as
soon as it remains empty. We formulated the energy consumption optimization problem of
the cloud computing system for each parameter used in this energy efficiency mechanisms.
We conducted a numerical analysis of the formulas for solving the optimization problem of
energy consumption in cloud computing system based on the initial data close to the real
ones.
Key words and phrases: cloud computing, energy efficiency, queuing system, optimiza-
tion.
Copyright © 2018 for the individual papers by the papers’ authors. Use permitted under the CC-BY license —
https://creativecommons.org/licenses/by/4.0/. This volume is published and copyrighted by its editors.
In: K. E. Samouylov, L. A. Sevastianov, D. S. Kulyabov (eds.): Selected Papers of the 12th International Workshop on
Applied Problems in Theory of Probabilities and Mathematical Statistics (Summer Session) in the framework of the
Conference “Information and Telecommunication Technologies and Mathematical Modeling of High-Tech Systems”,
Lisbon, Portugal, October 22–27, 2018, published at http://ceur-ws.org
24 APTP+MS’2018
1. Introduction
Recently, the concept of energy efficiency improving in cloud computing systems is
becoming popular. There are various methods to implement this. One way to improve
energy efficiency is scheduling and load balancing the servers, VMs, and applications [3].
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 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.
Moreover since the service-level agreement (SLA) must not be violated, the provider
needs to maintain the required level of energy consumption. However, while maintaining
the SLA, one of the parameters is the response time, so here we consider the optimization
problem of energy consumption with a constraints on the response time.
2. Modeling of energy efficiency improvement mechanisms
We consider a baseline model [7] as a single-server queuing system described by
the Markov process with Processor Sharing policy where the maximum numbers of
the customers is 𝐶. We do not consider distribution of processing volume of a task
in the paper, however, it can be done by means of queuing systems with limited
resources [6]. Customers arrive according to the Poisson law with rate 𝜆. Service times,
switch on and switch off durations are exponentially distributed with the parameters
𝜇, 𝛼 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 cause change of the system state to the switch on mode. After exponentially
distributed time with rate 𝛼, the system switches to the operating mode, in which
serving of customers is started. When the system remains empty in the operating mode,
it switches off immediately. Fig. 1 shows the transition intensities diagram for the
baseline model. For the base model, the set of states 𝑆1 is represented in the following
form: 𝑆1 = {(𝑠, 𝑘)|𝑠 = 1, 2, 1 ≤ 𝑘 ≤ 𝐶} ∪ {(𝑠, 𝑘)|𝑠 = 3, 0 ≤ 𝑘 ≤ 𝐶} ∪ (0, 0) [4, 5].
In [9] we derive the system of equilibrium equations, based on the transition intensity
diagram [8], [7], which makes it possible to obtain stationary probabilities 𝑝𝑠,𝑘 that the
system is in (s,k) state.
Due to the high energy consumption for shutting down the cloud server, in some cases
it’s more beneficial to leave it in operating mode pending the arrival of new customers.
In [8] we consider the model with server shutdown delay mechanism. In contrast to the
base model, where it was assumed that the server shuts down as soon as it remains
empty, in this model the system 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. Fig. 2 shows the transition intensities diagram for the model with
the shutdown delay mechanism. The set of states for this model is represented in the
following form: 𝑆2 = {(𝑠, 𝑘)|𝑠 = 1, 1 ≤ 𝑘 ≤ 𝐶} ∪ {(𝑠, 𝑘)|𝑠 = 2, 3, 0 ≤ 𝑘 ≤ 𝐶} ∪ (0, 0).
In [8] we derive and solve the system of equilibrium equations for the model with
shutdown delay mechanism.
Also we consider the model with server switch on delay, as well as in base model,
system passes in switch off mode at once after it remains empty. But it does not switch on
immediately on arrival of a new customer, and waits exponentially distributed time with
rate 𝜃. Fig. 3 shows the transition intensities diagram for this model. For this system,
the set of states 𝑆3 is represented in the following form: 𝑆3 = {(𝑠, 𝑘)|𝑠 = 0, 3, 0 ≤ 𝑘 ≤
Daraseliya A. V., Sopin E. S., Rykov V. V. 25
Figure 1. Transition intensities diagram. Baseline mathematical model
Figure 2. Transition intensities diagram. Mathematical model with the
shutdown delay mechanism
Figure 3. Transition intensities diagram. Mathematical model with the switch
on delay mechanism
26 APTP+MS’2018
Figure 4. Transition intensities diagram. Mathematical model with the
threshold-based switch mechanism
𝐶} ∪ {(𝑠, 𝑘)|𝑠 = 1, 2, 1 ≤ 𝑘 ≤ 𝐶}. In [9] we derive the system of equilibrium equations
for the model with the switch on delay mechanism.
Then we consider the mode with the threshold-based switch mechanism, in which
system passes from standby mode in switch on mode only after arrived of a certain
number 𝜅 of customers. Fig. 4 shows the transition intensities diagram for this
model. For this system, the set of states 𝑆4 is represented in the following form:
𝑆4 = {(𝑠, 𝑘)|𝑠 = 0, 0 ≤ 𝑘 ≤ 𝜅 − 1} ∪ {(𝑠, 𝑘)|𝑠 = 1, 𝜅 ≤ 𝑘 ≤ 𝐶} ∪ {(𝑠, 𝑘)|𝑠 = 2, 1 ≤ 𝑘 ≤
𝐶} ∪ {(𝑠, 𝑘)|𝑠 = 3, 0 ≤ 𝑘 ≤ 𝐶}. In [9] we derive the system of equilibrium equations for
this model with the threshold-based switch mechanism.
We derived the system of equilibrium equations for each model, based on the transition
intensity diagrams, which makes it possible to obtain stationary probability distribution
of the system. Taking into account the normalization condition and using matrix methods,
the system of equilibrium equations can be solved numerically, but we represent the
analytical solution in [8].
3. Energy consumption indicators and the performance characteristics of
cloud systems
With the system stationary distribution, we calculate the energy consumption in-
dicators. We will assume that in the switch on / off mode, the power consumption
is constant and equal to the average values 𝑃1 and 𝑃3 , respectively. In the operating
mode, the power consumption 𝑃2,𝑘 depends on the server occupancy. Through 𝑃2,𝑚𝑎𝑥
we denoted the maximum value of the server’s power consumption in the operating
mode, and through 𝑃2,𝑚𝑖𝑛 we denoted the power consumption in idle mode. The energy
consumption in the standby mode will be calculated by 𝑃0 . By analogy with the formula
given in [2], we derive the formula for the average server power consumption:
𝐶
∑︁ 𝐶
∑︁ 𝐶
∑︁ 𝐶
∑︁
𝑃 = 𝑃0 𝑝0,𝑖 + 𝑃1 𝑝1,𝑖 + 𝑃3 𝑝3,𝑖 + 𝑃2,𝑖 𝑝2,𝑖
𝑖=0 𝑖=0 𝑖=0 𝑖=0
where
𝑃2,𝑚𝑎𝑥 − 𝑃2,𝑚𝑖𝑛
𝑃2,𝑘 = 𝑃2,𝑚𝑖𝑛 + 𝑘
𝐶
According to Little’s law, 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 ,
where blocking probability 𝜋 is
Daraseliya A. V., Sopin E. S., Rykov V. V. 27
𝜋 = 𝑝0,𝐶 + 𝑝1,𝐶 + 𝑝2,𝐶 + 𝑝3,𝐶
The average number 𝑁 of customers is given by
3 ∑︁
∑︁ 𝐶
𝑁 = 𝑖𝑝𝑘,𝑖
𝑘=0 𝑖=1
The average response time 𝑇 follows directly from Little’s law and formula (3):
∑︀3 ∑︀𝐶
𝑘=0 𝑖=1 𝑖𝑝𝑘,𝑖
𝑇 =
𝜆(1 − 𝜋)
4. Optimization problem
In order 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, it is necessary to formulate and solve the energy consump-
tion optimization problem for the each parameters of the energy efficiency increasing
mechanisms.
The optimization problem can be formulated as
⎧
⎨ 𝑃 → min,
⎪
⎪ 𝑅1 : 𝑇 ≤ 𝑇0 ,
𝑅2 : 𝑃 ≥ 0,
⎩
where the energy consumption 𝑃 of the cloud system is minimized under constraint
𝑇0 on the average response time threshold.
For a model with the shutdown delay mechanism minimizing the energy consumption
𝑃 by the parameter 𝛾 can be written as follows:
⎧
⎨ 𝑃 (𝛾) → min,
⎪
⎪ 𝑅1 : 𝑇 ≤ 𝑇0 ,
𝑅2 : 𝑃 ≥ 0,
⎩
By analogy, we can write down the minimization problem for models with the switch
on delay and the threshold-based switch mechanisms through the parameters 𝜃 and 𝜅
⎧
⎨ 𝑃 (𝜃) → min,
⎪
⎪ 𝑅1 : 𝑇 ≤ 𝑇0 ,
𝑅2 : 𝑃 ≥ 0,
⎩
⎧
⎨ 𝑃 (𝜅) → min,
⎪
⎪ 𝑅1 : 𝑇 ≤ 𝑇0 ,
𝑅2 : 𝑃 ≥ 0.
⎩
For each of these three mechanisms, the optimization problem was considered sepa-
rately.
28 APTP+MS’2018
Figure 5. The dependence of the power consumption 𝑃 on the rate 𝛾. Model
with the shutdown delay mechanism
Figure 6. The dependence of the average response time 𝑇 on the rate 𝛾. Model
with the shutdown delay mechanism
5. Numerical analysis
In this section we present results a numerical analysis of the formulas to solved the
optimization problem.
On the energy profile of the cloud system installed at the University of Cardiff [1], it
can be seen [1] that the inclusion of the server lasts 150 seconds, and the shutdown is 30
seconds. Further, for convenience, it was represented in minutes. The values of 𝑃𝑖 were
taken from [1], according to which 𝑃0 = 10 W, 𝑃1 = 170 W, 𝑃3 = 120 W, 𝑃2,𝑚𝑖𝑛 = 105
W and 𝑃2,𝑚𝑎𝑥 = 268 W.
The results of numerical analysis for the values 𝐶=20, 𝜇=20, 𝛼=1, 𝛽=2 and 𝑇0 = 2.5
are presented in Fig. 5–10.
The plots of the server’s power consumption for the model with the shutdown delay
mechanism (Fig. 5) and for the model with the switch on delay mechanism (Fig. 7) show
that the consumed power increases very fast for small values of the arrival flow intensity
𝜆.
The plots of the average response time 𝑇 (Fig. 6) for the model with the shutdown
delay mechanism show that the greatest dependence of the average sojourn time 𝑇 on
the arrival flow intensity 𝜆 is observed at values of 𝛾 from 1 to 7. Also note, that for the
Daraseliya A. V., Sopin E. S., Rykov V. V. 29
Figure 7. The dependence of the power consumption 𝑃 on the rate 𝜃. Model
with the switch on delay mechanism
Figure 8. The dependence of the average response time 𝑇 on the rate 𝜃. Model
with the switch on delay mechanism
arrival flow intensity 𝜆 = 1, the condition 𝑅1 of the optimization problem is performed
with 𝛾 less than 5. For 𝜆 = 5, the condition 𝑅1 of the optimization problem is satisfied
on the whole segment of the function.
In Fig. 8 note that for small values of 𝜃, the difference in the average sojourn time 𝑇
is the greatest. For the arrival flow intensity 𝜆 = 5, the condition 𝑅1 of the optimization
problem is fulfilled when the value of 𝜃 is greater than 3.
The plots of the average response time 𝑇 (Fig. 10) for the model with the threshold-
based switch mechanism show that for the large values of 𝜅, the difference in the average
sojourn time 𝑇 is the greatest, and vice versa, the small 𝜅 values have almost no effect
on the average sojourn time. For the arrival flow intensity 𝜆 = 5, the condition 𝑅1 of
the optimization problem is fulfilled when the value of 𝜅 is less than 6.
6. Conclusions
We consider a cloud computing system with three different energy efficiency improving
mechanisms as a system with Processor Sharing policy. We investigate how a waiting
time before a server goes to switch on / off mode and threshold-based switch affects
30 APTP+MS’2018
Figure 9. The dependence of the power consumption 𝑃 on the parameter 𝜅.
Model with the threshold-based switch mechanism
Figure 10. The dependence of the average response time 𝑇 on the parameter 𝜅.
Model with the threshold-based switch mechanism
the energy efficiency of a cloud system. We carried out a numerical analysis of the
formulas for solving the energy consumption optimization problem. Numerical analysis
showed that that the server switch on mechanism is most efficient in terms of power
consumption, but the server shutdown delay mechanism allows the system to work at a
lower system load and is more effective in terms of response time. The mechanism with
server threshold-based switch on gives an improvement for power, but deterioration in
time.
Acknowledgments
The publication has been prepared with the support of the “RUDN University
Program 5-100” and funded by RFBR according to the research projects No. 18-07-00576
and No. 19-07-00933.
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