We constructed mathematical models of cloud computing systems taking into account various mechanisms for increasing energy eficiency 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 diferent energy eficiency 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 eficiency 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 eficiency 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.
Recently, the concept of energy eficiency improving in cloud computing systems is
becoming popular. There are various methods to implement this. One way to improve
energy eficiency is scheduling and load balancing the servers, VMs, and applications [
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.
We consider a baseline model [
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 [
Also we consider the model with server switch on delay, as well as in base model,
system passes in switch of 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 . Fig. 3 shows the transition intensities diagram for this model. For this system,
the set of states 3 is represented in the following form: 3 = {(, )| = 0, 3, 0 ≤ ≤
} ∪ {(, )| = 1, 2, 1 ≤ ≤ }. In [
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 of customers. Fig. 4 shows the transition intensities diagram for this
model. For this system, the set of states 4 is represented in the following form:
4 = {(, )| = 0, 0 ≤ ≤ − 1} ∪ {(, )| = 1, ≤ ≤ } ∪ {(, )| = 2, 1 ≤ ≤
} ∪ {(, )| = 3, 0 ≤ ≤ }. In [
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 [
With the system stationary distribution, we calculate the energy consumption
indicators. We will assume that in the switch on / of mode, the power consumption
is constant and equal to the average values 1 and 3, respectively. In the operating
mode, the power consumption 2, depends on the server occupancy. Through 2,
we denoted the maximum value of the server’s power consumption in the operating
mode, and through 2, we denoted the power consumption in idle mode. The energy
consumption in the standby mode will be calculated by 0. By analogy with the formula
given in [
=0 =0
According to Little’s law, the average number of customers in the system is equal to the average efective arrival rate (1 − ) multiplied by the average sojourn time , where blocking probability is
The average response time follows directly from Little’s law and formula (3): 3 = ∑︁ ∑︁ , where the energy consumption of the cloud system is minimized under constraint 0 on the average response time threshold.
For a model with the shutdown delay mechanism minimizing the energy consumption by the parameter can be written as follows:
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 and ⎧ ⎪ → min, ⎨
1 : ≤ 0, ⎪⎩ 2 : ≥ 0, ⎧ ⎪ ( ) → min, ⎨
1 : ≤ 0, ⎪⎩ 2 : ≥ 0, ⎧ ⎪ ( ) → min, ⎨
1 : ≤ 0, ⎪⎩ 2 : ≥ 0, ⎧ ⎪ ( ) → min, ⎨
1 : ≤ 0, ⎪⎩ 2 : ≥ 0.
For each of these three mechanisms, the optimization problem was considered separately.
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 Cardif [
The results of numerical analysis for the values =20, =20, =1, =2 and 0 = 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 .
The plots of the average response time (Fig. 6) for the model with the shutdown delay mechanism show that the greatest dependence of the average sojourn time on the arrival flow intensity is observed at values of from 1 to 7. Also note, that for the arrival flow intensity = 1, the condition 1 of the optimization problem is performed
= 5, the condition 1 of the optimization problem is satisfied on the whole segment of the function.
In Fig. 8 note that for small values of , the diference in the average sojourn time is the greatest. For the arrival flow intensity = 5, the condition 1 of the optimization problem is fulfilled when the value of is greater than 3.
The plots of the average response time (Fig. 10) for the model with the thresholdbased switch mechanism show that for the large values of , the diference in the average sojourn time is the greatest, and vice versa, the small on the average sojourn time. For the arrival flow intensity values have almost no efect = 5, the condition 1 of the optimization problem is fulfilled when the value of is less than 6. 6.
We consider a cloud computing system with three diferent energy eficiency improving mechanisms as a system with Processor Sharing policy.
the energy eficiency 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 eficient in terms of power consumption, but the server shutdown delay mechanism allows the system to work at a lower system load and is more efective in terms of response time. The mechanism with server threshold-based switch on gives an improvement for power, but deterioration in time.
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.