Detection Performance of Packet Arrival under Downclocking for Mobile Edge Computing

Mobile edge computing (MEC) enables battery-powered mobile nodes to acquire information technology services at the network edge. These nodes desire to enjoy their service under power saving. The sampling rate invariant detection (SRID) is the first downclocking WiFi technique that can achieve this objective. With SRID, a node detects one packet arrival at a downclocked rate. Upon a successful detection, the node reverts to a full-clocked rate to receive the packet immediately. To ensure that a node acquires its service immediately, the detection performance (namely, themiss-detection probability and the false-alarm probability) of SRID is of importance. This paper is the first one to theoretically study the crucial impact of SRID attributes (e.g., tolerance threshold, correlation threshold, and energy ratio threshold) on the packet detection performance. Extensive Monte Carlo experiments show that our theoretical model is very accurate. This study can help system developers set reasonable system parameters for WiFi downclocking.


Introduction
Mobile edge computing (MEC) [1] aims to provide computing resources and information technology services at the network edge.In MEC, various battery-powered mobile nodes (such as smartphone) will access these resources and services via MEC application servers such as LTE base station and wireless access point (AP).These battery-powered nodes desire to enjoy their service under power saving.
In this paper, we assume that a number of batterypowered nodes access an AP (acting as an MEC application server) via a WiFi network.These devices adopt a novel algorithm called sampling rate invariant detection (SRID) [2] for power saving.SRID is the first downclocking mechanism (adopted in WiFi).With SRID, a node detects one packet arrival at a downclocked rate.Upon a successful detection, the node reverts to a full-clocked rate to receive the packet immediately.For each detection, there are two types of typical errors: miss-detection (i.e., the AP sends a packet but the node does not detect it) and false-alarm (i.e., the AP sends nothing but the node detects a packet mistakenly).To ensure that a node acquires its service immediately, the detection performance (namely, the miss-detection probability and the false-alarm probability) of SRID is of importance.This paper is concerned with the detection performance.Our contributions are summarized as follows: (i) To the best of our knowledge, this paper is the first one to theoretically analyze the detection performance of WiFi downclocking.Our theoretical model characterizes the crucial impact of SRID attributes (e.g., tolerance threshold, correlation threshold, and energy ratio threshold) on the packet detection performance (i.e., the miss-detection probability and the falsealarm probability).(ii) We run extensive Monte Carlo experiments to verify that our theoretical model is very accurate.We show that as the downclocked rate decreases, the falsealarm probability increases significantly, which will lead to a serious adverse impact on packet detection.
This study can help system developers set reasonable system parameters for WiFi downclocking.So far, downclocking has received great attention [2][3][4][5][6][7][8][9][10][11][12][13].Among the most relevant works, [2] is the first paper that brought downclocking to low-power WiFi networks and proposed the SRID algorithm, which was considered as one of the most classical amendments on power saving of 802.11 protocols [12,13].In WiFi, the dominant source of energy consumption is the idle listening operation [7,8], where a node needs to frequently detect unpredictably arriving packets or assess a clear channel with high power.Therefore, SRID reduced the power consumption by allowing a WiFi node to downclock its sample rate in idle listening mode.SASD [3] was proposed to reduce the power consumption in SRID further by allowing nondestination nodes in idle listening to enter a doze state.AS-MAC [4] was proposed to avoid contention and reduce delay by asynchronously scheduling the wake-up time of neighboring nodes via a downclocking mechanism for wireless sensor networks.SloMo [5] was proposed to allow WiFi nodes to operate their radios at lower clock rates when receiving and transmitting at low bit rates.Sampleless [6] allowed energy-constrained devices to scale down their sampling rates regardless of channel conditions.The above works mainly focused on the hardware implementation of the downclocking mechanism or evaluated its performance via simulation.In contrast, this paper is the first one to model the impact of downclocking on the packet detection performance theoretically.
The rest of this paper is organized as follows.Section 2 gives an overview of SRID.Section 3 theoretically analyzes the detection performance of SRID.Section 4 presents Monte Carlo results that reveal the crucial impact of SRID attributes on the detection performance.Section 5 concludes this paper.

Overview of SRID
In the downclocking mechanism, one basic problem is how to detect unpredictably arriving packets at a downclocked rate, so that the node can revert to a full-clocked mode to receive the arriving packets.
The SRID that adopts the downclocking mechanism is designed for WiFi networks.With the help of Figure 1, we specify how SRID works.Assume a WiFi network consisting of one access point (AP) and a number of nodes.The AP is always in the active mode, while each node is in the downclocked mode by default.When the AP has a packet to transmit toward a node, the operations of the AP and the node are as follows.
(i) The AP first transmits an additional preamble called -preamble, and then a sequence of dummy bits, and finally a conventional 802.11 packet.Here, the preamble is used to notify the node of the arrival of an expected packet.The dummy bits are used to provide a guard interval that allows the node to revert to the full-clocked mode from the downclocked mode.(ii) The node continuously detects its -preamble via self-correlation and then reverts to the full-clocked mode upon a successful detection.
In the next two subsections, we detail the construction and the detection of an -preamble.

Construction of 𝑀-Preamble.
In SRID, an -preamble consists of  ( ≥ 2) duplicated versions of a complex gold sequence (CGS), the length of each CGS sequence being   +   .Figure 1(a) shows an example of -preamble, where  = 3.Thus the total length of -preamble  can be expressed as where   represents the minimum length of the CGS (used for -preamble).The integer  represents the address of the node in SRID, which is assigned by the AP.1/  is the minimum downclocking factor of radio hardware.For example, assume that the full-clocked frequency is 20 MHz.Then the minimum downclocked frequency is 20 * (1/  ) MHz.

Detection of 𝑀-Preamble.
In SRID, a node continuously performs self-correlation to detect its -preamble.Assume that a node operates with a downclocking factor of 1/ ∈ [1/  , 1].
Let () denote the sampling value of the node at the sampling point .
Wireless Communications and Mobile Computing 3 Let () denote the self-correlation result of the node at the sampling point .To detect its -preamble, at each sampling point , the node with address  performs the selfcorrelation between the latest  1 samples and the previous  1 samples (offset by ).Therefore, () can be calculated by where  1 =   / is the size of the self-correlation window (in sampling points) and  = (  +   )/ is the number of sampling points of a CGS when the downclocking factor is 1/.Note that  1 and  are shown in Figure 2.
Let () denote the energy level at sampling point , which can be calculated by ( We say that an -preamble is successfully detected if the total number of successfully detected points,   , is greater than  1  2 ; namely, where  1 is the tolerance threshold and  2 = ( − 1) is the total number of sampling points (from the 2nd CGS to the -th CGS), as shown in Figure 2. We say that a sampling point is detected successfully if the following two conditions are satisfied.
Condition 2. At sampling point , the energy ratio (in dB) of   () and   ( − ) exceeds a threshold   ; namely, where   () =  −1 1 () + (1 −  −1 1 )  ( − 1) represents a moving average of energy level, with a window size equal to  1 .The reason of introducing Condition 2 is to reduce the probability that Condition 1 is satisfied but no -preambles are transmitted.

Detection Performance Analysis
In this section, focusing on the downlink traffic from the AP to nodes, we theoretically analyze the crucial impact of SRID attributes (namely, tolerance threshold  1 , correlation threshold , and energy ratio threshold   ) on the detection performance.
Due to the downclocked rate and the noise, each SRID detection result is associated with four mutually exclusive minievents: (a) successful detection: AP sends an preamble and the node detects it successfully, (b) missdetection: AP sends an -preamble but the node does not detect it, (c) false-alarm: AP does not send an -preamble but the node detects it mistakenly, and (d) Null: AP does not send an -preamble and the node detects nothing.To study the detection performance, we only need to calculate the successful detection probability   = Prob(successful detection) and the false-alarm probability  fa = Prob(falsealarm), because Prob(miss-detection) = 1 − Prob(successful detection) and Prob(Null) = 1 − Prob(false-alarm).
We note that each detection result is determined depending on whether the AP sends an -preamble.Below, we introduce two competing hypotheses: H 0 :  () =  () ,  = 0, 1, . . .,  H 1 :  () = ℎ () +  () ,  = 0, 1, . . ., , where H 0 is referred to as the null hypothesis (i.e., AP does not send an -preamble to a node) and H 1 as the alternative hypothesis (i.e., AP sends an -preamble to a node).Under hypothesis H 0 , at the sampling point , the node receives the noise, and therefore its sample value is () = (), where () is the Gaussian white noise.Under hypothesis H 1 , at the sampling point , the node receives the -preamble signal and the noise, and therefore its sampling value is () = ℎ() + (), where () represents the sampling value on the -preamble and ℎ represents the channel coefficient.

Expression of 𝑃 𝑑
. We now express   .According to (4), we have The sampling process is a Bernoulli process, where a sampling point is marked success if Conditions 1 and 2 (specified in Section 2.2) are satisfied.Therefore, the number of successfully detected points in  2 trials,   , follows a binomial distribution.Thus   is expressed by where  1 is the probability of Condition 1 being satisfied under H 1 and  ER1 is the probability of Condition 2 being satisfied under H 1 .
Note that ( − ) = ∑ , because AP transmits one -preamble (which consists of  sampling points only) for each packet, and thereby the node only receives the noise before the -preamble.Similar to  1 , we can calculate  ER1 via the Monte Carlo method.

Expression of 𝑃 fa
. We now express  fa .Similar to   ,  fa can be expressed as follows: where  terms of   , which is given by (9).Similarly, we can calculate  fa .
In Algorithm 1, we input the SRID parameters and output the value of   .In the algorithm, we run Monte Carlo experiment for 100000 times.We now detail each experiment.
(i) In lines (3) to (4), we generate an -preamble of  samples points, which simulates the AP's -preamble transmission.
(ii) In line (5), we invoke the Matlab function, awgn(⋅), to simulate the additive white Gaussian noise (AWGN) channel and then the node's received signal is the result that the AP's -preamble signal passes through the AWGN channel.
Finally, after we finish 100000 runs, we first calculate the average of  1 , avg( 1 ), and the average of  ER1 , avg( ER1 ), and then calculate   , as shown in line (14).

Model Verification
In this section, we present the Monte Carlo results to illustrate the crucial impact of SRID attributes and SNR on the detection performance (namely, the successful detection probability   and the false-alarm probability  fa ).The default parameter settings are set by [2] and are shown in Table 1.In Figures 3 and 4, each Monte Carlo result is on average over 100000 runs.In addition, we use "SRID(1/)" to denote the SRID detection with the downclocking factor of 1/.In all figures, the labels "ana" and "sim", respectively, denote the theoretical and simulation results.Figures 3(a) and 3(b), respectively, plot   and  fa as the SNR varies, when 1/ = 1/2, 1/4, 1/8, 1/16.From Figure 3, we have the following observations.(i) Given 1/,   increases and  fa decreases gradually as the SNR increases.(ii) Given SNR, as 1/ decreases,   decreases slightly while  fa increases significantly.For example, for SNR = 5 dB, when 1/ decreases from 1/2 to 1/16,  fa grows from 0.0098 to 0.0301 significantly, while   just drops from 0.9997 to 0.9754 slightly.This will lead to a serious adverse impact on packet detection.(iii) The detection performance is almost perfect (i.e.,   = 1 and  fa = 0) when SNR = 10 dB, which is easy to achieve in real environments [15].Figures 4(a) and 4(b), respectively, plot   and  fa as  1 varies, when 1/ = 1/2, 1/4, 1/8, 1/16.From Figure 4, we have the following observations.(i) Given 1/, as  1 increases,   always decreases, but  fa first decreases to 0 and then remains unchanged.The reason is as follows: increasing  1 will decrease the successful detection probability from (9) as well as the false-alarm probability from (14).(ii) Give  1 , as 1/ decreases,   decreases significantly, while  fa decreases gradually.

Wireless Communications and Mobile Computing
Finally, from these figures, the close match between the theoretical and simulation curves manifests that our performance model is very accurate.

Conclusion
In mobile edge computing, various battery-powered mobile nodes desire to acquire information technology services at the network edge under power saving.WiFi downclocking is such a promising technique.In this paper, we investigate a novel WiFi downclocking technique called SRID and first theoretically study the impact of SRID attributes (namely, tolerance threshold, correlation threshold, and energy ratio threshold) on the detection performance of packet arrival.This study is helpful in designing better WiFi downclocking protocols.

Figure 3 :
Figure 3: (a)   and (b)  fa vary when the SNR varies.
3.3.Calculation of  and  fa via Monte Carlo Method.In the previous two subsections, we give expressions of   and  fa .However, they involve 2 1 random variables and therefore are hard to solve.Below we adopt the Monte Carlo method [14] to calculate them.Algorithm 1 lists the computation process in

Table 1 :
Parameter settings in simulation.