The basic concept of Bluetooth Low Energy (BLE) is short packet transmission and transient connection. It can quickly establish a connection, send data, and quickly disconnect, so that neighbor discovery is frequent and becomes an important issue. In the neighbor discovery which includes advertising and scanning, the BLE specification defines several important parameters. The parameters on the advertiser side include advertising interval, advertising duration, etc. On the scanner side, there are scan interval, scan window, etc. How to configure these parameters for quick neighbor discovery has been troublesome for BLE implementers. Prior analyses on BLE discovery process also showed some disagreements or made some incorrect assumptions. In this paper, we use rigorous probability-theory based derivations to obtain different kinds of successful discovery probabilities. We clarify disagreements in prior works and also provide insights on how to configure parameters for maximizing discovery probability. In particular, we prove that the discovery probabilities on each of the three channels are correlated. We also find that, when the advertising duration is set close to some multiples of the scan interval, an ill-fated synchronization problem will occur. To have a high discovery probability, both scan window and scan interval should be set at a large value, though it might not be good for energy saving.
It has been nearly 30 years since the Bluetooth short range wireless communication technology was proposed in the early 1990s. Over the years, Bluetooth technology has progressed from solving the cabling issue between the host computer and its peripheral devices to being widely used in various communication applications for personal hand-held devices. Because Bluetooth has been used mostly in battery-powered devices, the Bluetooth SIG (Special Interest Group) issued the Bluetooth 4.0 specification in 2010 for low power operation of Bluetooth [
Many BLE applications are run between battery-powered portable devices. Before any kind of connections can be set up, a pair of BLE devices need to go through the neighbor discover process to know the existence of each other. BLE specifications allocate three (37, 38, 39) out of the 40 total channels for discovery. One device does the advertising on the three channels alternately and the other device does the scanning also on the three channels alternately. Since the two devices are not synchronous, they need to be in the matching roles (one advertising and the other scanning) and at the right time and the right place (on the same channel) for the discovery to be successful. If one round of advertising does not result in successful discovery, a random delay is introduced before the next round so that a different advertising/scanning matching is tried. This is different from the infrastructure-mode WiFi (Wireless Fidelity), where an always-on access point (AP) exists and serves as a reference point for other WiFi stations to scan and connect to. Several advertising/scanning parameters (e.g., advertising interval, advertising duration, scan interval, scan window) are involved in the BLE discovery process. BLE specifications do not mandate each parameter’s value but rather provide a range of possible values for each parameter so that different applications can choose their appropriate ones. However, this also causes headaches for BLE implementers as the dimension of parameter values is too large to handle. Many rely on trial and error. It is desirable to have some parameter-setting guidance so that an efficient BLE neighbor discovery process can be realized and the power consumption is minimized.
There have been quite a few studies on analyzing the performance of the BLE discovery process. The performance metrics include successful discovery probability, discovery latency, and power consumption. In [
From the above literature survey, we identified several issues worthy of more careful investigation. First, there is disagreement on the successful discovery probabilities on individual channels within an advertising event (to be more thoroughly explained in the next section). Second, individual channel discoveries are assumed to occur independently. Third, there are some abrupt changes in discovery probability and discovery latency at certain parameter settings, which none of prior works can explain. There is still not much insight shed on proper setting of discovery parameters. In this paper, we address all the above and claim the following contributions:
1) We give a rigorous proof that the successful discovery probabilities on three individual channels within an advertising event are the same.
2) We show that successful discovery on an individual channel is dependent on the discovery result on the prior channel within the same advertising event.
3) We find the successful discovery probability for an advertising event (which includes discoveries on three individual channels) and show what discovery related parameter settings are to avoid. The rest of this paper is organized as follows: In Section 2, we give a short introduction on the BLE discovery operation. Section 3 first reviews certain prior analyses to get a feel about how they might go wrong, and then we use rigorous probability-theory based derivations to obtain different kinds of successful discovery probabilities. Section 4 shows verification of the analysis by simulations. Section 5 is the conclusion.
BLE minimizes the device power consumption by allowing devices to go into the sleep state. When devices attempt to communicate with each other, they wake up to discover each other. Three channels (channels 37, 38, and 39) are allocated for the discovery purpose. Consider the scenario of two BLE devices. One device enters the advertising state and periodically sends advertising PDUs (protocol data units; or simply called packets) on the three advertising channels. The other device enters the scanning state or the initiating state. It also takes turn scanning on the three advertising channels. There are 4 types of packets that the advertiser may send to start the discovery: ADV_SCAN_IND, ADV_DIRECT_IND, ADV_NONCONN_IND, andADV_IND. ADV_SCAN_IND is for scanning only. A device entering the scanning state (a scanner) can respond to it with a SCAN_REQ packet to request further information. The advertiser will then reply with a SCAN_RSP packet. ADV_DIRECT_IND is for fast connection setup. A device entering the initiating state (an initiator) can respond with CONNECT_REQ to quickly set up a connection with the advertiser. ADV_NONCONN_IND is for one-way advertising only. Subsequent scan or connect request is not allowed. ADV_IND is the general-purpose advertising packet. Scanner/initiator can reply with a scan/connect request. The discovery process between an advertiser and a scanner via ADV_SCAN_IND/ADV_IND - SCAN_REQ-SCAN_RSP exchange is the most lengthy one and is usually used as the example scenario for BLE discovery analysis. Our analysis in later sections also assumes this type of packet exchange. It can be easily modified to adapt to other types of advertising packet exchanges.
A scanner listens on channels ch37, ch38, ch39 alternately for advertising packets. It switches scanning channels at every scan interval (scanInterval). In each scanInterval, it listens only for a duration of the scan window (scanWindow). The idle time in the scanInterval can save energy and reduce reply competitions among multiple scanners, if any. When an advertiser advertises and a scanner scans on the same channel, and the advertising packet is successfully received by the scanner, the scanner will reply with a SCAN_REQ. The advertiser then replies with a SCAN_RSP to complete the discovery process.
For later analysis, we need a reference to the duration of a complete discovery process. We use the ADV_IND - SCAN_REQ - SCAN_RSP exchange as an example. The maximum lengths of ADV_IND, SCAN_REQ,SCAN_RSP packets are 47, 22, 47 bytes, respectively. At the 1Mbps channel rate, their transmission times are 376, 176, 376 us, respectively. There is also a 150 us IFS (inter-frame space) between packet transmissions. Therefore, a complete discovery process for this example takes 0.376 + 0.150 + 0.176 + 0.150 + 0.376 = 1.228 ms.
To facilitate comparisons with [
| Notation | Meaning | Recommended Values |
|---|---|---|
| τ W A | Length of advertising duration | ≤10 ms |
| τ A I | Length of advertising interval | [20, 10, 240] ms |
| τ S I | Length of scanning interval | [2.5, 10, 240] ms |
| τ S W | Length of scanning window | [2.5, 10, 240] ms |
| T S | Time for discovery packets exchange | ≤1.228 ms |
| τ S W ′ | Length of effective scan window | τ S W ′ = τ S W − T S |
| t 0 | Arrival time of ch37 ADV_IND packet | |
| α 1 , α 2 , α 3 | Successful discovery probability on channel 37, 38, 39 |
This is the scenario considered in Section 4.1.1 of [
E 1 : S 1 is synchronous with A 1 , meaning a channel x (x = 37, 38, or 39) ADV_IND packet arrives when the scanner is also scanning channel x.
E 4 : S 1 has enough time to reply to ADV_IND before the current scan window ends.
In a 1:1 Network, the necessary and sufficient condition of a successful discovery is E 1 ∩ E 4 . In continuous scanning, Pr [ E 1 ] = τ S W / 3 τ S I = 1 / 3 since we assume τ S W = τ S I . For E 4 to occur, the remaining time in the scan window when the ADV_IND first arrives at the scanner must be greater than T S , the time required for the ADV_IND packet to be successfully received by S 1 and the subsequent SCAN_REQ and SCAN_RSP exchanges between A 1 and S 1 to be completed. Let t 0 be the arrival time of channel 37 ADV_IND packet. Hence, Pr [ E 4 ] = Pr [ τ S W − t 0 > T S ] = Pr [ t 0 < τ S W − T S ] = ( τ S W − T S ) / τ S I . Therefore, the probability that a channel 37 discovery is successful is
α 1 = Pr [ E 1 ∩ E 4 ] = Pr [ E 1 ] Pr [ E 4 ] = 1 / 3 ( τ S W − T S ) / τ S I (1)
(because E 1 and E 4 are independent events), which is identical to Equation (1) in [
Using the same argument that E 1 ∩ E 4 is what makes a successful discovery, the probability that the second (on channel 38) ADV_IND results in a successfully discovery is α 2 = Pr [ E 1 ∩ E 4 ] = 1 / 3 Pr [ τ S W − t 0 − τ W A > T S ] = 1 / 3 Pr [ t 0 < τ S W − τ W A − T S ] , or
α 2 = ( τ S W − τ W A − T S ) / 3 τ S I , (2)
which is Equation (2) in [
α 3 = Pr [ E 1 ∩ E 4 ] = 1 / 3 Pr [ τ S W − t 0 − 2 τ W A > T S ] = 1 / 3 Pr [ t 0 < τ S W − 2 τ W A − T S ] = ( τ S W − 2 τ W A − T S ) / 3 τ S I .
In the above discussion, we assume τ S I > τ W A . On the other hand, if we consider τ S I < τ W A ,
α 2 = ( τ S I + τ S W − τ W A − T S ) / 3 τ S I , (3)
which matches Equation (2) in [
It is necessary to have some explanations as to why the single-channel successful discovery probability is dependent on the relation between τ S I and τ W A , and also why there is disagreement between [
We first use a simple argument to explain why successful single-channel discovery probabilities α 1 , α 2 , α 3 are equal. The conditions that the advertiser A 1 ’s Ch37 ADV_IND results in a successful discovery by the scanner S 1 are: 1) A 1 ’s Ch37 ADV_IND is successfully received by S 1 and 2) the subsequent exchange of SCAN_REQ and SCAN_RSP are also successful. We call the two conditions an event E c h 37 occurs. E c h 37 occurs when the arrival time ( t 0 ) of Ch37 ADV_IND falls within the Ch37 scan window but at least T S time before the scan window ends. Since the scan interval is periodically repeating at three times of the scan interval ( τ S I ), it suffices to consider a scan cycle τ S C ( = 3 τ S I ) , which t 0 might fall within. After a successful reception of the ADV_IND, the subsequent exchange of SCAN_REQ and SCAN_RSP will always be successful because there are no other devices to interfere with their transmission/reception. Therefore the probability that A 1 ’s Ch37 ADV_IND results in a successful discovery by S 1 , α 1 = ( τ S W − T S ) / τ S C = ( τ S W − T S ) / 3 τ S I . The argument and the result are no different from [
What is wrong in the derivation of α 2 and α 3 in [
In this section, we first show that there is a dependency between adjacent successful discovery events. We then prove with rigorous probability derivations that successful single-channel discovery probabilities are equal. Previously we define E c h 37 as the event that a Ch37 discovery succeeds. Similarly, let E c h 38 , E c h 39 be the event that a Ch38, Ch39 discovery succeeds, respectively. Here E c h 37 , E c h 38 , E c h 39 are restricted to be the three successful discovery events (in the context of probability) in the same advertising event (advEvent). We now discuss the dependency of E c h 38 on E c h 37 . We do not apply any restriction to the relationship among τ W A , τ S W , τ S I , except that τ S W ≤ τ S I , by system design. Without loss of generality, let time 0 be the start time of S 1 ’s Ch37 scanning window which event E c h 37 occurs. t 0 falls within (0, τ S W − T S ). That is,
0 < t 0 < τ S W − T S (4)
Then event E c h 38 occurs if t 0 + τ W A falls within any one of the ( i τ S C + τ S I , i τ S C + τ S I + τ S W − T S ), i = 0 , 1 , 2 , 3 , ⋯ , or
i τ S C + τ S I − τ W A < t 0 < i τ S C + τ S I + τ S W − T S − τ W A (5)
The term i τ S C accounts for the multiple candidate Ch38 scanning windows after the first Ch37 scanning window. For example, consider i = 0.
Given Equation (4) (event E c h 37 occurs), Equation (5) may hold (i.e., event E c h 38 occurs) only when the two intervals specified by (4) and (5) overlap, or in other words, if (4)’s upper bound is greater (5)’s lower bound (expressed in Equation (6)) AND (5)’s upper bound is greater than (4)’s lower bound (expressed in Equation (7)).
τ S W − T S > i τ S C + τ S I − τ W A ,or τ W A > i τ S C + τ S I − ( τ S W − T S ) (6)
i τ S C + τ S I + τ S W − T S − τ W A > 0 , or τ W A < i τ S C + τ S I + ( τ S W − T S ) (7)
In the BLE specification, τ W A ≤ 10 ms , and the maximum possible T S = 1.228 ms . For the complete discovery procedure (ADV_IND, SCAN_REQ, and SCAN_RSP successfully finish their transmission and reception) to have a chance to succeed, both τ S W and τ W A must be greater than T S . That is, τ S W − T S > 0 and τ W A − T S > 0 . Define the effective scan window τ S W ′ = τ S W − T S . Then (6) and (7) can be combined into
i τ S C + τ S I − τ S W ′ < τ W A < i τ S C + τ S I + τ S W ′ (8)
Equation (8) specifies the possible intervals for τ W A such that event E c h 38 may
occur. We first consider i = 0. τ S I − τ S W ′ < τ W A < τ S I + τ S W ′ . There are two distinct cases.
1) τ S I − τ S W ′ ≤ τ W A ≤ τ S I : The intersection of the intervals specified by (4) and (5) is ( τ S I − τ W A , τ S W − T S ) .
Pr [ E c h 38 | E c h 37 ] = ( τ S W − T S − τ S I + τ W A ) / ( τ S W − T S ) .
2) τ S I < τ W A ≤ τ S I + τ S W ′ : The intersection of the intervals specified by (4) and (5) is ( 0 , τ S I + τ S W − T S − τ W A ) .
Pr [ E c h 38 | E c h 37 ] = ( τ S I + τ S W − T S − τ W A ) / ( τ S W − T S ) .
We next consider i = 1. That is, E c h 38 may occur in the next scan cycle. τ S C + τ S I − τ S W ′ < τ W A < τ S C + τ S I + τ S W ′ .
3) τ S C + τ S I − τ S W ′ < τ W A < τ S C + τ S I : The intersection of the intervals specified by (4) and (5) is ( τ S C + τ S I − τ W A , τ S W − T S ) .
Pr [ E c h 38 | E c h 37 ] = ( τ S W − T S − τ S C − τ S I + τ W A ) / ( τ S W − T S ) .
1) τ S C + τ S I < τ W A < τ S C + τ S I + ( τ S W − T S ) : The intersection of the intervals specified by (4) and (5) is ( 0 , τ S C + τ S I + τ S W − T S − τ W A ) .
Pr [ E c h 38 | E c h 37 ] = ( τ S C + τ S I + τ S W − T S − τ W A ) / ( τ S W − T S ) .
Continue on. We consider i = 2 , 3 , 4 , ⋯ . That is, E c h 38 may occur in the third, fourth, fifth, … scan cycle. Following the same approach as above, it should not be too difficult to verify that there is a general formula for the conditional probability Pr [ E c h 38 | E c h 37 ] .
2) i τ S C + τ S I − τ S W ′ < τ W A < i τ S C + τ S I :
Pr [ E c h 38 | E c h 37 ] = ( τ S W − T S − i τ S C − τ S I + τ W A ) / ( τ S W − T S ) .
3) i τ S C + τ S I < τ W A < i τ S C + τ S I + τ S W ′ :
Pr [ E c h 38 | E c h 37 ] = ( i τ S C + τ S I + τ S W − T S − τ W A ) / ( τ S W − T S ) .
Next, we consider the case that E c h 37 does not occur, specified by E c h 37 ¯ . t 0 falls within ( τ S W − T S , 3 τ S I ). Then E 38 occurs if t 0 + τ W A falls within any one of the intervals ( i τ S C + τ S I , i τ S C + τ S I + τ S W − T S ), i = 0 , 1 , 2 , 3 , ⋯ , or
τ S W − T S < t 0 < 3 τ S I (9)
i τ S C + τ S I − τ W A < t 0 < i τ S C + τ S I + τ S W − T S − τ W A (10)
The condition that the two intervals specified by (9) and (10) overlap is
( i − 1 ) τ S C + τ S I < τ W A < i τ S C + τ S I , i = 0 , 1 , 2 , 3 , ⋯ (11)
The condition that the two intervals specified by (9) and (10) overlap is
( i − 1 ) τ S C + τ S I < τ W A < i τ S C + τ S I , i = 0 , 1 , 2 , 3 , ⋯ (11)
We first consideri = 0. Equation (11) becomes − 2 τ S I < τ W A < τ S I . But τ W A is always positive. We then consider two distinct cases.
1) 0 ≤ τ W A ≤ τ S I − τ S W ′ : The intersection of the intervals specified by (9) and (10) is ( τ S I − τ W A , τ S I − τ W A + τ S W − T S ) .
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ S W − T S ) / ( 3 τ S I − τ S W + T S ) .
2) τ S I − τ S W ′ < τ W A ≤ τ S I : The intersection of the intervals specified by (9) and (10) is ( τ S W − T S , τ S I + τ S W − T S − τ W A ) .
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ S I − τ W A ) / ( 3 τ S I − τ S W + T S ) .
We next consider i = 1. Equation (11) becomes τ S I < τ W A < τ S C + τ S I . We consider three cases.
1) τ S I < τ W A < τ S I + τ S W ′ : The intersection of the intervals specified by (9) and (10) is ( τ S C + τ S I − τ W A , τ S C ) .
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ W A − τ S I ) / ( 3 τ S I − τ S W + T S ) .
2) τ S I + τ S W ′ < τ W A < τ S C + τ S I − τ S W ′ : The intersection of the intervals specified by (9) and (10) is ( τ S C + τ S I − τ W A , τ S C + τ S I − τ W A + τ S W − T S ) .
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ S W − T S ) / ( 3 τ S I − τ S W + T S ) .
3) τ S C + τ S I + τ S W ′ < τ W A < τ S C + τ S I : The intersection of the intervals specified by (9) and (10) is ( τ S W − T S , τ S C + τ S I + τ S W − T S − τ W A ) .
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ S C + τ S I − τ W A ) / ( 3 τ S I − τ S W + T S ) .
Continue on. We consider i = 2 , 3 , 4 , ⋯ . Following the same approach as above, the general formula for the conditional probability Pr [ E c h 38 | E c h 37 ¯ ] is:
1) ( i − 1 ) τ S C + τ S I < τ W A < ( i − 1 ) τ S C + τ S I + τ S W ′
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ W A − τ S I − ( i − 1 ) τ S C ) / ( 3 τ S I − τ S W + T S ) .
2) ( i − 1 ) τ S C + τ S I + τ S W ′ < τ W A < i τ S C + τ S I − τ S W ′ :
Pr [ E c h 38 | E c h 37 ¯ ] = ( τ S W − T S ) / ( 3 τ S I − τ S W + T S ) .
3) i τ S C + τ S I − τ S W ′ < τ W A < i τ S C + τ S I :
Pr [ E c h 38 | E c h 37 ¯ ] = ( i τ S C + τ S I − τ W A ) / ( 3 τ S I − τ S W + T S ) .
We now verify that the unconditional ch38 discovery probability Pr [ E c h 38 ] is the same as the unconditional ch37 discovery probability Pr [ E c h 37 ] , regardless of the value of τ W A . Due to the periodicity of Pr [ E c h 38 | E c h 37 ] and Pr [ E c h 38 | E c h 37 ¯ ] , shown in
1) τ S I − τ S W ′ ≤ τ W A ≤ τ S I :
Pr [ E c h 38 ] = Pr [ E c h 38 | E c h 37 ] ⋅ Pr [ E c h 37 ] + Pr [ E c h 38 | E c h 37 ¯ ] ⋅ Pr [ E c h 37 ¯ ] = ( τ S W − T S − τ S I + τ W A ) / ( τ S W − T S ) ⋅ ( τ S W − T S ) / 3 τ S I + ( τ S I − τ W A ) / ( 3 τ S I − τ S W + T S ) ⋅ ( 3 τ S I − τ S W + T S ) / 3 τ S I = ( τ S W − T S ) / 3 τ S I
2) τ S I < τ W A ≤ τ S I + τ S W ′ :
Pr [ E c h 38 ] = Pr [ E c h 38 | E c h 37 ] ⋅ Pr [ E c h 37 ] + Pr [ E c h 38 | E c h 37 ¯ ] ⋅ Pr [ E c h 37 ¯ ] = ( τ S I + τ S W − T S − τ W A ) / ( τ S W − T S ) ⋅ ( τ S W − T S ) / 3 τ S I + ( τ W A − τ S I ) / ( 3 τ S I − τ S W + T S ) ⋅ ( 3 τ S I − τ S W + T S ) / 3 τ S I = ( τ S W − T S ) / 3 τ S I
3) τ S I + τ S W ′ < τ W A < τ S C + τ S I − τ S W ′ :
Pr [ E c h 38 ] = Pr [ E c h 38 | E c h 37 ] ⋅ Pr [ E c h 37 ] + Pr [ E c h 38 | E c h 37 ¯ ] ⋅ Pr [ E c h 37 ¯ ] = 0 ⋅ ( τ S W − T S ) / 3 τ S I + ( τ S W − T S ) / ( 3 τ S I − τ S W + T S ) ⋅ ( 3 τ S I − τ S W + T S ) / 3 τ S I = ( τ S W − T S ) / 3 τ S I
In
As to the joint behavior of ch37 and ch38 discoveries, we expect both to complement each other for maximum discovery probability. It is the probability of E c h 37 ∪ E c h 38 .
Pr [ E c h 37 ∪ E c h 38 ] = Pr [ E c h 37 ] + Pr [ E c h 38 ∩ E c h 37 ¯ ] = Pr [ E c h 37 ] + Pr [ E c h 38 | E c h 37 ¯ ] ⋅ Pr [ E c h 37 ¯ ]
We now discuss the dependency of E c h 39 on E c h 37 . Event E c h 37 occurs when t 0 falls within (0, τ S W − T S ), previously specified in Equation (4). Then event E c h 39 occurs if t 0 + 2 τ W A falls within any one of the ( i τ S C + 2 τ S I , i τ S C + 2 τ S I + τ S W − T S ), i = 0 , 1 , 2 , 3 , ⋯ , or
i τ S C + 2 τ S I − 2 τ W A < t 0 < i τ S C + 2 τ S I + τ S W − T S − 2 τ W A , i = 0 , 1 , 2 , 3 , ⋯ (12)
Similar to the derivation in Section 3.3, given Equation (4) (event E c h 37 occurs), Equation (12) may hold (i.e., event E c h 39 occurs) only when the two intervals specified by (4) and (12) overlap, or in other words, if (4)’s upper bound is greater (12)’s lower bound (expressed in Equation (13)) and (12)’s upper bound is greater than (4)’s lower bound (expressed in Equation (14)).
τ S W ′ > i τ S C + 2 τ S I − 2 τ W A or τ W A > i τ S C / 2 + τ S I − τ S W ′ / 2 (13)
i τ S C + 2 τ S I + τ S W ′ − 2 τ W A > 0 or τ W A < i τ S C / 2 + τ S I + τ S W ′ / 2 (14)
which can be combined into
i τ S C / 2 + τ S I − τ S W ′ / 2 < τ W A < i τ S C / 2 + τ S I + τ S W ′ / 2 , i = 0 , 1 , 2 , 3 , ⋯ (15)
Due to the space limitation we directly give the general formulas for the conditional probability Pr [ E c h 39 | E c h 37 ] . The derivations follow the same approach as in Section 3.3,
1) i τ S C / 2 + τ S I − τ S W ′ / 2 < τ W A < i τ S C / 2 + τ S I :
Pr [ E c h 39 | E c h 37 ] = ( τ S W − T S − i τ S C − 2 τ S I + 2 τ W A ) / ( τ S W − T S ) .
2) i τ S C / 2 + τ S I < τ W A < i τ S C / 2 + τ S I + τ S W ′ / 2 :
Pr [ E c h 39 | E c h 37 ] = ( i τ S C + 2 τ S I + τ S W − T S − 2 τ W A ) / ( τ S W − T S ) .
The general formulas for the conditional probability Pr [ E c h 39 | E c h 37 ¯ ] are:
1) ( i τ S C − τ S I ) / 2 < τ W A < ( i τ S C − τ S I ) / 2 + τ S W ′ / 2 , i = 1 , 2 , 3 , ⋯
Pr [ E c h 39 | E c h 37 ¯ ] = ( 2 τ W A + τ S I − i τ S C ) / ( 3 τ S I − τ S W + T S ) .
2) ( i τ S C − τ S I ) / 2 + τ S W ′ / 2 < τ W A < i τ S C / 2 + τ S I − τ S W ′ / 2 , i = 0 , 1 , 2 , 3 , ⋯
Pr [ E c h 39 | E c h 37 ¯ ] = ( τ S W − T S ) / ( 3 τ S I − τ S W + T S ) .
3) i τ S C / 2 + τ S I − τ S W ′ / 2 < τ W A < i τ S C / 2 + τ S I , i = 0 , 1 , 2 , 3 , ⋯
Pr [ E c h 39 | E c h 37 ¯ ] = ( i τ S C + 2 τ S I − 2 τ W A ) / ( 3 τ S I − τ S W + T S ) .
Pr [ E c h 39 | E c h 37 ¯ ] as a function of τ W A . They show repeating pattern at the cycle of 1.5 τ S I . Readers can also verify that the unconditional ch39 discovery probability Pr [ E c h 39 ] is the same as the unconditional ch37 discovery probability Pr [ E c h 37 ] , regardless of the value of τ W A .
Pr [ E c h 39 ] = Pr [ E c h 39 | E c h 37 ] ⋅ Pr [ E c h 37 ] + Pr [ E c h 39 | E c h 37 ¯ ] ⋅ Pr [ E c h 37 ¯ ] = ( τ S W − T S ) / 3 τ S I
That the three consecutive ADV_IND packets in an advEvent are transmitted on different channels is to maximize the probability of discovery and therefore speed up the discovery process. This is equivalent to maximizing the probability of a union event E c h 37 ∪ E c h 38 ∪ E c h 39 , representing that at least one single-channel discovery succeeds in an advEvent. We call it a successful advEvent discovery.
Pr [ E c h 37 ∪ E c h 38 ∪ E c h 39 ] = Pr [ E c h 37 ] + Pr [ E c h 38 ] + Pr [ E c h 39 ] − Pr [ E c h 37 ∩ E c h 38 ] − Pr [ E c h 37 ∩ E c h 39 ] − Pr [ E c h 38 ∩ E c h 39 ] + Pr [ E c h 37 ∩ E c h 38 ∩ E c h 39 ]
We have known that Pr [ E c h 37 ] = Pr [ E c h 38 ] = Pr [ E c h 39 ] = ( τ S W − T S ) / 3 τ S I . Pr [ E c h 37 ∩ E c h 38 ] = Pr [ E c h 38 | E c h 37 ] ⋅ Pr [ E c h 37 ] , and is plotted in
E c h 37 ∩ E c h 38 ∩ E c h 39 occurs when Equation (4), Equation (5), and Equation (12) hold. From earlier discussion in Section 3.3, Pr [ E c h 37 ∩ E c h 38 ] is non-zero when
i τ S C + τ S I − τ S W ′ < τ W A < i τ S C + τ S I + τ S W ′ , i = 0 , 1 , 2 , 3 , ⋯
Taking i = 0 for example, we discuss two situations: τ S I − τ S W ′ ≤ τ W A ≤ τ S I and τ S I < τ W A ≤ τ S I + τ S W ′ .
1) τ S I − τ S W ′ ≤ τ W A ≤ τ S I : The intersection of the intervals specified by (4) and (5) is ( τ S I − τ W A , τ S W − T S ) . For Equation (12) to also hold ( E c h 39 occurs), t 0 must be in ( 2 τ S I − 2 τ W A , 2 τ S I − 2 τ W A + τ S W ′ ) . The intersection of these two intervals is ( 2 τ S I − 2 τ W A , τ S W − T S ) and τ W A is further restricted to be τ S I − τ S W ′ / 2 ≤ τ W A ≤ τ S I . Therefore, we have Pr [ E c h 39 | E c h 37 ∩ E c h 38 ] = ( τ S W − T S − 2 τ S I + 2 τ W A ) / ( τ S W − T S − τ S I + τ W A ) , for τ S I − τ S W ′ / 2 ≤ τ W A ≤ τ S I . Previously, we have found that
Pr [ E c h 38 ∩ E c h 37 ] = ( τ S W − T S − τ S I + τ W A ) / 3 τ S I .
Therefore Pr [ E c h 37 ∩ E c h 38 ∩ E c h 39 ] = ( τ S W − T S − 2 τ S I + 2 τ W A ) / 3 τ S I , for τ S I − τ S W ′ / 2 ≤ τ W A ≤ τ S I .
2) τ S I < τ W A ≤ τ S I + τ S W ′ : The intersection of the intervals specified by (4) and (5) is ( 0 , τ S I + τ S W − T S − τ W A ) . For Equation (12) to also hold, t 0 must be in ( 2 τ S I − 2 τ W A , 2 τ S I − 2 τ W A + τ S W − T S ) . The intersection of these two intervals is ( 0 , 2 τ S I − 2 τ W A + τ S W − T S ) and τ W A is further restricted to be τ S I ≤ τ W A ≤ τ S I + τ S W ′ / 2 . Therefore, we have Pr [ E c h 39 | E c h 37 ∩ E c h 38 ] = ( τ S W − T S − 2 τ S I + 2 τ W A ) / ( τ S W − T S − τ S I + τ W A ) , for τ S I ≤ τ W A ≤ τ S I + τ S W ′ / 2 . Previously, we have found that
Pr [ E c h 38 ∩ E c h 37 ] = ( τ S I + τ S W − T S − τ W A ) / 3 τ S I .
Therefore Pr [ E c h 37 ∩ E c h 38 ∩ E c h 39 ] = ( τ S W − T S + 2 τ S I − 2 τ W A ) / 3 τ S I , for τ S I ≤ τ W A ≤ τ S I + τ S W ′ / 2 .
Using similar technique we can find other Pr [ E c h 37 ∩ E c h 38 ∩ E c h 39 ] with i = 1 , 2 , 3 , ⋯ . With them calculated, we can finally obtain Pr [ E c h 37 ∪ E c h 38 ∪ E c h 39 ] as
1) | τ W A − τ S I − i τ S C | ≤ τ S W − T S , i = 0 , 1 , 2 , 3 , ⋯
Pr [ E c h 37 ∪ E c h 38 ∪ E c h 39 ] = ( τ S W − T S + 2 | τ S I + i τ S C − τ W A | ) / 3 τ S I .
2) | τ W A − τ S I − i τ S C / 2 | ≤ ( τ S W − T S ) / 2 , i = 0 , 1 , 2 , 3 , ⋯
Pr [ E c h 37 ∪ E c h 38 ∪ E c h 39 ] = ( 2 τ S W − 2 T S + 2 τ S I + i τ S C − 2 τ W A ) / 3 τ S I .
3) otherwise,
Pr [ E c h 37 ∪ E c h 38 ∪ E c h 39 ] = ( τ S W − T S ) / τ S I
This is the main result of this paper.
successful advEvent discovery is maximized at ( τ S W − T S ) / τ S I , which is three times of the minimum value. In this case, the three single-channel discoveries are disjoint events. They complement one another to make the discovery most fruitful.
We wrote a python simulation program to verify the accuracy of the discovery probability analysis. In each simulation run with certain specified τ W A , τ S I , τ S W values, 100,000 advEvents were generated. Successful discoveries on channels 37, 38, 39 were individually recorded. This allowed us to obtain successful single-channel discovery probabilities Pr [ E c h 37 ] , Pr [ E c h 38 ] , and Pr [ E c h 39 ] . They were found to be nearly identical and very close to ( τ S W − T S ) / 3 τ S I (within the simulation accuracy tolerance) for all parameter values chosen. Successful discoveries from the advEvent point of view were also recorded. This allowed us to obtain successful advEvent discovery probabilities. They are compared with analytical results and one example is shown in
scan window resolution 1.25 ms) and analytical results (with scan window resolution 0.625 ms) match well both visually and actually (the difference has a mean value of 0.3 percent and standard deviation 0.48 percent). Again, the saddle points in
In Section 4, we show that the maximum successful advEvent discovery probability is ( τ S W − T S ) / τ S I . Since T S is a fixed value and scan window τ S W is bounded by scan interval τ S I , ( τ S W − T S ) / τ S I is maximized when τ S W is equal to τ S I and both as large as possible. This agrees with what we observed in
BLE has been in use for many years. However, how to configure its parameters for quick neighbor discovery has been troublesome for many implementers. We use rigorous probability derivations to clarify disagreements in prior works and also provide insights on how to configure parameters for maximizing discovery probability. To be specific, we show that the probabilities of successful discoveries on each of the three discovery channels are equal. But those three discovery events are dependent events. In addition, we show that there are several parameter settings that will cause low discovery probability. In particular, if the advertising duration τ W A is set to equal to some multiples of scan interval ( τ S I , 4 τ S I , 7 τ S I , ⋯ ), which we call ill-fated advertising/scanning synchronization, the discovery process will be the least efficient. For maximum discovery probability, the length of the scan window should be large, which leads to large scan interval too. However, this is not necessarily good for energy saving. Currently, our analysis applies to the 1:1 (one advertiser and one scanner) scenario. Future works are to extend it to multiple advertisers or multiple scanners. This study can also serve as a base for discovery latency and energy consumption analysis.
This work was supported by grants from MOST (Most 108-2221-E-194-009-MY2, MOST 108-2218-E-194-007), Taiwan.
The authors declare no conflicts of interest regarding the publication of this paper.
Hou, T.-C. and Huang, K.-C. (2020) Accurate Analysis on Bluetooth Low Energy Neighbor Discovery. Journal of Computer and Communications, 8, 231-250. https://doi.org/10.4236/jcc.2020.812020