1 Introduction
Modern data centers consume tremendous amounts of energy to supply networking, computing, and storage services to global IT companies. Concerns about energy consumption have prompted researchers to explore operational methods that maximize energy efficiency and satisfy a certain level of quality of service (QoS), (Anselmi and Verloop, 2011; Ko and Cho, 2014; Liao et al., 2015)
. QoS can be achieved by adding constraints that impose upper bounds for response timerelated metrics, e.g., the mean virtual response time and the tail probability of the response time. In general, these constraints are binding, because of the conflict between the QoSrelated metrics and energy consumption. Binding the QoSrelated constraints implies that the metrics are maintained as a constant value, and suggests the need to investigate the stabilization of response times. Although some proposed methodologies
(Anselmi and Verloop, 2011; Ko and Cho, 2014; Liao et al., 2015) assume the stationarity of data traffic arrival processes, nonstationary properties, such as timevarying arrival rates from real data (CAIDA, 2016), make it difficult to analyze queueing system performance.In this paper, therefore, we study the service rate controls that stabilize the mean virtual response time to a certain target value in a single server PS queue representing a computer server in a data center under timevarying arrival rates and controllable service rates, i.e., a queue. Our approach is similar to Whitt (2015) and Ma and Whitt (2016), who considered three different service rate controls, two of which were designed to stabilize the mean (virtual) waiting time in a queue. The slowly timevarying traffic patterns of internet services (CAIDA, 2016) justify our use of pointwise stationary approximation (PSA) Green and Kolesar (1991). We adopt different heavytraffic approximation results (HTA), because our objective is to stabilize the mean virtual response time, which is one of our target performance measures. We propose two service rate control schemes:
(1)  
(2) 
where is the desired response time, is the mean job size, is the arrival rate function, , and , with and are the squared coefficient of variations (SCV) of the base interarrival and service time distributions. Equation (1) is a modification of the wellknown squareroot control (SR) suggested in Whitt (2015), and Equation (2) is a new control scheme, which we call the differencematching (DM) control, because it maintains the difference between and as a constant . The DM control is easy to implement thanks to its simplicity.
Figure 4 shows the mean queue length process, (green line), and the mean virtual response time process, (red line), of the simulated queues with an Erlang base arrival distribution and a lognormal job size distribution with the SR control as in Equation (1
) and three different timevarying arrival rates. The dotted black lines are 95% confidence intervals and the dotted blue line plots the arrival rate function; its dedicated yaxis is on the right. The plots show that the response time is almost perfectly stabilized by the SR control under the lighttraffic condition.
Figure 7 depicts the performance measures when the target response time is relatively long. While the stabilization looks poor for both controls, their relative amplitude – one of our performance measures described in Section 4.2 – is under 10%. Figure ((b))(b) depicts that the DM control achieves the target response time, which implies that using the DM control shows better accuracy – the other performance measure in Section 4.2 – under the heavy traffic condition (long response time).
This paper contributes to the published literature on queueing systems by studying the response time stabilizing controls for a queue; proposing a new control scheme, i.e., DM control, for heavytraffic conditions; undertaking extensive simulations of the proposed control schemes; and gaining insights into their effectiveness for data centers.
The remainder of this paper is organized as follows. Section 2 introduces a singleserver PS queueing model with a timevarying arrival rate and a controllable service rate. We explain some details for simulating a queue, which is not straightforward, unlike its stationary counterpart. Section 3 explains the procedure to derive the two service rate controls, and some simple characteristics of the controls. Section 4 reports the results of the simulations including the interesting phenomena we find. Section 5 concludes and suggests some future research directions.
2 The model
Section 2.1 introduces a singleserver queueing model with nonstationary nonPoisson arrivals under the PS discipline and the service rate control. Section 2.2 explains the procedures to simulate such queueing systems. Throughout this paper, we use the following notations:

: arbitrary periodic function with a period

: spatial scale average of ; for any

: arrival rate function

: service rate function

: base interarrival times between and
job; i.i.d. random variables having a general distribution function
with a mean and an SCV 
: service requirement that the job brings; i.i.d. random variables having a general distribution function with a mean and an SCV

: instantaneous traffic intensity;

: time when the job arrives

: time when the job departs

: arrival process; number of job arrivals during interval

: departure process; number of job departures during interval

: queue length process; number of jobs in the system at time

: virtual response time process; sojourn time that a virtual customer arriving at time spends in the system
2.1 The queue
We consider a single server processor sharing queueing system where arrivals follow an NSNP. We assume that the timedependent arrival rate function is continuous and bounded finitely both below and above. Under the assumption, the cumulative arrival function is welldefined for and so is the inverse .
Each job has its own service requirement, e.g., job size, to be processed by a server. Assume that the job size is determined upon arrival in ICT service systems, e.g., packet size or file size. Let be the service requirement that the job brings, and assume that ’s are independent and identically distributed. Appropriate control schemes dynamically determine service rate function . Assume that function is continuous and bounded so that it can be integrate on compact intervals to obtain a cumulative service function . The amount of service processed by the server during time interval is .
The PS policy is a workconserving service discipline which is commonly used to describe computer systems (especially CPUs) Gautam (2012). All jobs in the system evenly share the server or processor at any given time, e.g., if the processor runs at a processing speed of bits/s and there are jobs, then each job is processed by bits/s.
2.2 Simulating the queue
Simulating a queue is difficult and computationally expensive because of nonPoisson arrivals, time nonhomogeneity, processor sharing, and other factors. Therefore, we combine two algorithms (Gerhardt and Nelson, 2009; Ma and Whitt, 2016) for simulation. The first algorithm by Gerhardt and Nelson (2009) provides the supporting theory for generating an NSNP from its stationary counterpart, and the second algorithm by Ma and Whitt (2016) gives a numerical approximation method to relieve the computational burden when the rate function is periodic.
2.2.1 The arrival process
Let be the NSNP arrival process we want to simulate. Construct the process by applying the change of time to a stationary renewal process. Let be the stationary renewal process with i.i.d. interrenewal times . Then,
(3)  
(4) 
where In particular, we call the standard equilibrium renewal process (SERP) when and is a random variable having the stationary excess distribution given by
(5) 
By defining to be the composition of and , i.e., with , we construct an NSNP. We generate samples from the arrival process using the inversion method described in Gerhardt and Nelson (2009). Algorithm 1 describes the procedure. An NSNP generated by Algorithm 1 has the following property:
Constructing the arrival process prompts the following remark.
Remark 1 (Gerhardt and Nelson, 2009).
, for all , and , for large .
We note that NSNP is a generalization of the simple nonstationary Poisson process (NSPP), where
is exponentially distributed. It can be verified easily this by plugging 1 into
.2.2.2 The service times
The service completion time is determined as soon as a job arrives when the FCFS discipline applies. Under the PS policy, however, it is not determined upon arrival, because future arrivals will affect the service times of of the jobs already existing in the system. Express the service completion time or the departure time of the job that brings a random amount of service requirement as:
(6) 
where is the arrival time of the job and is the number of customers in the system at time .
2.2.3 The response time process
Let denote the entire time that a job spends in the system if it arrives at time and brings a amount of service requirement. Since has a whatif characteristic, this is often called virtual response time (or virtual sojourn time) at time . When we use omitting , we still assume a random service requirement. Our primary interest in the queue is the mean virtual response time process for . Note that the stochastic nature of in Equation 6 means that cannot be obtained conveniently as its FCFS counterpart where the Lindely’s recursion is applicable.
To obtain the virtual response time process in a queue, we store the path
of the queue for every replication of the simulation. The path contains the status of the system at each recording epoch. After a replication is terminated, rerun the simulations from each recording epoch during a replication length (say
), given the stored status at time , with a newly inserted job which is the virtual job. Each rerun of the simulation terminates when the virtual job is finished and results in a realization of a virtual response time . We obtain the expected process by averaging at 10,000 replications.3 Methods
As mentioned in Section 1, we combine the pointwise stationary approximation (PSA) and the heavytraffic approximation, which were used by Whitt (2015) and Ma and Whitt (2015) to stabilize the waiting times (excluding service times) in queues, and adjust the combined approximations to stabilize the response times (waiting time + service time) in queues. Below, we explain our methods.
3.1 Pointwise stationary approximation with heavytraffic limits
We briefly visit the pointwise stationary approximation (PSA) (Green and Kolesar, 1991; Whitt, 1991), which is known to be an appropriate approximation when the arrival rate changes slowly relative to the average service time (Whitt, 1991, 2015). Thus, we consider that the performance at different times is similar to the performance of the stationary counterpart with the instantaneous model parameters.
3.2 Two service rate controls
Whitt (2015) derived the PSAbased service rate control to stabilize the waiting time. We take a similar approach, but our service rate control stabilizes the response time. We derive two service rate controls based on and heavytraffic approximations. Hereinafter, we use the subscripts and to indicate the discipline from which the result derives, e.g., variability factor and .
3.2.1 The squareroot (SR) control
In queueing systems, the workload processes are identical under any workconserving disciplines. Thus, we derive a control based on a queue as an experimental trial, which we later discover to be appropriate for queues under lighttraffic conditions (see Section 4.3.3 for the details).
The heavytraffic approximation for the expected steady state response time in a queue is (Chen and Yao, 2001):
(8) 
where is the service rate, is the mean job size, is the traffic intensity, and is the variability parameter, given the SCVs for the arrival base and job size distributions. Approximate the expected response time at time in a queue based on the PSA:
(9) 
where is the instantaneous traffic intensity at time . Fixing the LHS by a target response time and adjusting the terms gives:
(10) 
Finally, obtain the solution to the quadratic equation above:
(11) 
We call Equation (11) the squareroot (SR) control, which is the naming convention used by Whitt (2015).
3.2.2 The differencematching (DM) control
Recall that Equation (7) is the heavytraffic approximation for the steadystate mean virtual response time () in a queue:
where , , and are defined as in Equation (8), and is the variability parameter for the PS queue. Approximate the expected response time process based on the PSA:
(12) 
Fixing the LHS by a certain constant and adjusting the terms gives a service rate control that is much simpler than :
(13) 
As mentioned in Section 1, we call Equation (13) the differencematching (DM) control because is a constant .
3.2.3 Simple analysis on the two service rate controls
The two controls derived above result in different service rate functions except when both base distributions have SCVs 1. The most representative example is the queue. Applying , the SR control (11) reduces to , which is the same as the DM control (13) with . It prompts the following remark.
Remark 2.
For the timevarying queues having both the base distributions (arrival base and job size) of SCV 1, the two controls coincide.
Another simple but interesting phenomenon is that both controls become identical as we decrease or increase the target response time .
Proposition 1.
The two controls coincide as (heavytraffic) or (lighttraffic).
4 Simulation experiments
We investigate the performance of the two service rate controls through simulation experiments. Table 1 summarizes the simulation parameters.
4.1 Simulation setting
We use the sinusoidal arrival rate function with constants , , and . Therefore, we have three functions of the same amplitude but of different periods. Two are the slowly timevarying functions () and the third one () is not. We include the third, however, to observe how the controls work when the arrival rate is a quickly timevarying function.
To observe the asymptotic behavior, we set the replication length to at least three cycles of the periods, e.g., we conduct simulations for period on a 20,000 unit time and periods on a 2,000 unit time considering the length of periods. For the target response time , we use two different values: 0.1 for the short and 10.0 for the long response times. Because the service rate controls are inversely proportional to , each value of results in lighttraffic and heavytraffic, respectively. For each independent system, we conduct 10,000 replications to obtain the ensemble average of the performance measures.
We consider three different distributions for arrival base and job size distribution: Erlang distribution (ER); exponential distribution (EXP); and lognormal distribution (LN). The distributions have mean 1 and different SCVs. Specifically, we use ER with and LN with . The SCV of EXP is always 1 by definition. We make five pairs of base arrival/job size distributions: EXP/EXP, ER/ER, LN/LN, ER/LN, and LN/ER. Note that the combination EXP/EXP corresponds to a queueing system with NSPP arrival process and exponential service requirement, i.e., . Table 2 summarizes the variability factors, and , associated with each pair of distributions.
System  
Arrival rate function  
Periodic coefficient  
Replication length  
Service rate function  , 
Target response time  0.1 (lighttraffic), 10.0 (heavytraffic) 
Number of replication  10000 
Exponential (SCV=1.0)  
Distributions  Erlang (SCV=0.5) 
Lognormal (SCV=2.0) 
Distribution pair (arrival base/job size)  
Exponential/Exponential  1  1 
Erlang/Erlang  0.5  0.6667 
Lognormal/Lognormal  2  1.3333 
Erlang/Lognormal  1.25  0.8333 
Lognormal/Erlang  1.25  1.6666 
4.2 Two metrics to evaluate the effectiveness of the controls
We use two metrics to measure the performance of the two controls. First, we define the relative amplitude (RA) by
as a measure of stabilization. Second, we define the relative gap (RG) by
as a measure of accuracy. We obtain the two metrics by numerically calculating:
(14)  
(15) 
where is the period of the expected response time process , is an arbitrary long time after the process has been stabilized, and is the target response time. For the values of , we use the same values as the periods of the arrival rate functions since we observe that the periods are the same for both and .
The two measures above are favorable as they become closer to 0%. For RA, there is no negative value since the amplitude is a positive amount. Note, however, that RG allows a negative value such that the control overestimates the service rate which gives a smaller spatial average than our original target.
4.3 Results
report both the absolute values (amplitude and spatial average) and the relative values (RA and RG). For the performance of the controls, we heuristically call them
good if they control the response time with and , and poor otherwise.In the following plots, the green line corresponds to the mean queue length and the red line to the mean virtual response time of the simulated queues under the various combinations of control and distribution. The dotted black lines are the 95% confidence intervals, and the dotted blue line plots the arrival rate function and has its dedicated yaxis on the right.
In the following subsections, we summarize the results of Tables 48 by their traffic intensity. We obtain each traffic intensity by targeting the response time (short or long) we desire according to Proposition 1. Specifically, the instantaneous traffic intensity is approximately when and when , for the distribution pairs.
4.3.1 Control performances in lighttraffic systems (s=0.1)
Figures 11 and 15 depict the two expected processes and in lighttraffic systems under the two controls where the base distribution pair is Erlang/Erlang. The figures show universally good stabilizing performances (), even under the quickly timevarying arrival rate (), but, the accuracy of the DM control is poor. Specifically, the expected response time process stabilizes around 0.15 although the target is 0.1, which corresponds to about 0.5% of RG (Figure 15). Intuitively, this poor performance stems from the inaccuracy of the heavytraffic approximation in lighttraffic systems. Meanwhile, the SR control results in only about 0.05% of RG (Figure 11). Throughout the simulation experiments, we observe this tendency consistently from all of the distribution pairs (see Section 4.3.3 for the details).
4.3.2 Control performances in heavytraffic systems (s=10)
Figures 25 and 35 depict the heavytraffic systems under the two controls and three pairs of base distributions (EXP/EXP, ER/ER, LN/LN). Compared to the lighttraffic systems, we do not observe perfectly controlled results. For quickly timevarying arrival rate (), the poor control performance is obvious since the PSA is not appropriate. We observe positive results for the slowest timevarying arrival rate () despite the imperfect stabilization. The consistently better accuracy of DM control justifies its use in heavytraffic systems.
4.3.3 Why does DM fail to meet the target response time in lighttraffic?
In lighttraffic systems, the probability that two or more jobs will present simultaneously becomes smaller, i.e., it is rare that multiple jobs will share the same processor. We recall the following heavytraffic based PSAs of the expected virtual response time processes (Equations (9) and (12) in Section 3).
Letting , the two approximations above converge, respectively, to
(16)  
(17) 
and then the two controls reduce to the constants:
(18)  
(19) 
Throughout the simulation experiments, we use the base distributions having mean and the target mean virtual response time for lighttraffic so regardless of the distributions. In comparison, varies depending on both the base arrival and job size distributions because of variability factor .
In a queue with a service rate function , the response time of a job with random size and arrival time denoted by , is expressed as
(20) 
Approximating under the lighttraffic condition and letting , the above expression reduces to the analytic form:
(21) 
Replacing by the mean job size obtains the numerical values shown in Table 3. We observe that the simulation results and the approximately calculated values coincide, e.g., in Figure ((c))(c) and for Erlang/Erlang in Table 3 are 0.15. In comparison, we observe that is consistently 0.1 regardless of the distributions based on the reasoning we use to obtain the numerical values in Table 3.
We gain two insights into lighttraffic systems. First, the PS queue exhibits behavior similar to the FCFS queue. Second, the two service rate controls do not require time dependency. Thus, we conclude that the SR control is appropriate for stabilizing response times in queues as the target response time shortens.
Distribution pair  
Exponential/Exponential  10  0.1 
Erlang/Erlang  6.667  0.15 
Lognormal/Lognormal  13.333  0.07 
Erlang/Lognormal  8.333  0.12 
Lognormal/Erlang  16.666  0.06 
4.3.4 Heavytraffic behavior of the two controls
Figure 40 plots the the result of two controls under the different distribution pairs. As we calculated in Section 4.3.3, the two controls are significantly different when the target response time is short where the target response time is around zero. The difference between them diminishes as the target response time becomes longer. However, the difference in convergence speeds causes the two controls to perform differently in nonasymptotic heavytraffic systems, e.g., traffic intensities are around 0.9 throughout the heavytraffic systems.
5 Conclusion
This paper studied the service rate functions that control the mean virtual response time required to obtain stabilization in queues with slowly timevarying arrival rates. Modifying Whitt (2015)’s method for analyzing PS queues resulted in a modified squareroot (SR) service rate control and we introduced a new differencematching (DM) service rate control that appears practically advantageous due to its ease of use and simplicity. Extensive simulation experiments were performed to investigate the performance of two controls. The SR control was effective under a lighttraffic condition with a short target response time relative to the interarrival times. Neither control, however, perfectly stabilized the response time under a heavytraffic condition. The DM control outperformed the SR control in terms of meeting the target mean virtual response time.
We suggest several research directions based on the results presented in this paper. Limit theorems, e.g., fluid and diffusion limits, can be derived for
queues with periodically timevarying arrival rate functions. We believe that such supporting theories should provide important clues to achieving perfect stabilization of the response time process. Lighttraffic behaviors in queueing situations also deserve more analysis, since studies of timevarying queues are scarse to the best of our knowledge. Conceivably, interpolating the two controls could extend the coverage of the target response time beyond short and long. Of course, practical applications in ICT infrastructures should be accompanied.
Acknowledgment
This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF2016R1D1A1B04933453).
References
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Appendix A Performances of the suggested controls
a.1 Numerical data
Amplitude (RA)  Spatial Average (TG)  Amplitude (RA)  Spatial Average (TG)  
0.1  0.001  0.0044 (4.37%)  0.1 (0.0%)  0.0037 (3.7%)  0.1001 (0.0%) 
0.01  0.0035 (3.45%)  0.1001 (0.0%)  0.0039 (3.9%)  0.1001 (0.0%)  
0.1  0.003 (2.95%)  0.1015 (0.02%)  0.0025 (2.5%)  0.1012 (0.01%)  
10.0  0.001  0.7555 (7.44%)  10.1493 (0.01%)  0.6808 (6.72%)  10.1321 (0.01%) 
0.01  1.7516 (17.07%)  10.264 (0.03%)  1.8011 (17.58%)  10.2452 (0.02%)  
0.1  1.1104 (10.82%)  10.2665 (0.03%)  1.0656 (10.57%)  10.0773 (0.01%) 
Amplitude (RA)  Spatial Average (TG)  Amplitude (RA)  Spatial Average (TG)  
0.1  0.001  0.0037 (3.52%)  0.1047 (0.05%)  0.005 (3.37%)  0.1487 (0.49%) 
0.01  0.0039 (3.72%)  0.1045 (0.04%)  0.0044 (2.93%)  0.1485 (0.49%)  
0.1  0.0026 (2.46%)  0.1064 (0.06%)  0.0027 (1.78%)  0.1513 (0.51%)  
10.0  0.001  1.1872 (8.64%)  13.7467 (0.37%)  0.7828 (7.20%)  10.8678 (0.09%) 
0.01  2.5553 (18.56%)  13.765 (0.38%)  1.8884 (17.19%)  10.9824 (0.10%)  
0.1  1.4838 (10.74%)  13.822 (0.38%)  1.3446 (12.04%)  11.1719 (0.12%) 
Amplitude (RA)  Spatial Average (TG)  Amplitude (RA)  Spatial Average (TG)  
0.1  0.001  0.0049 (5.27%)  0.0922 (0.08%)  0.004 (5.28%)  0.0751 (0.25%) 
0.01  0.0057 (6.14%)  0.0921 (0.08%)  0.0044 (5.88%)  0.075 (0.25%)  
0.1  0.0041 (4.38%)  0.0936 (0.06%)  0.0041 (5.42%)  0.0761 (0.24%)  
10.0  0.001  0.4776 (7.34%)  6.5104 (0.35%)  0.8366 (9.05%)  9.2406 (0.08%) 
0.01  0.9325 (14.33%)  6.5061 (0.35%)  1.6268 (17.64%)  9.2206 (0.08%)  
0.1  0.8908 (13.47%)  6.6132 (0.34%)  0.9665 (10.3%)  9.3818 (0.06%) 
Amplitude (RA)  Spatial Average (TG)  Amplitude (RA)  Spatial Average (TG)  
0.1  0.001  0.0049 (5.07%)  0.0976 (0.02%)  0.0055 (4.56%)  0.1197 (0.2%) 
0.01  0.0047 (4.81%)  0.0974 (0.03%)  0.005 (4.22%)  0.1193 (0.19%)  
0.1  0.0031 (3.08%)  0.0995 (0.01%)  0.0046 (3.8%)  0.1209 (0.21%)  
10.0  0.001  0.5891 (8.26%)  7.1291 (0.29%)  1.0496 (10.06%)  10.4293 (0.04%) 
0.01  1.2382 (17.11%)  7.2366 (0.28%)  2.0119 (19.39%)  10.375 (0.04%)  
0.1  0.8216 (11.29%)  7.2763 (0.27%)  1.2382 (11.84%)  10.456 (0.05%) 
Amplitude (RA)  Spatial Average (TG)  Amplitude (RA)  Spatial Average (TG)  
0.1  0.001  0.0034 (3.51%)  0.0978 (0.02%)  0.003 (4.91%)  0.0601 (0.4%) 
0.01  0.0039 (3.99%)  0.0977 (0.02%)  0.003 (4.97%)  0.0602 (0.4%)  
0.1  0.0026 (2.64%)  0.0997 (0.0%)  0.0023 (3.82%)  0.0609 (0.39%)  
10.0  0.001  0.6518 (5.63%)  11.5748 (0.16%)  0.4418 (5.43%)  8.1343 (0.19%) 
0.01  1.8191 (15.4%)  11.8151 (0.18%)  1.0204 (12.49%)  8.1706 (0.18%)  
0.1  1.223 (10.41%)  11.7491 (0.17%)  0.9637 (11.56%)  8.3389 (0.17%) 
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