April 2006
IEEE P802.19-06/0021r0
IEEE P802.19
Wireless Coexistence
Project
IEEE P802.19 Coexistence TAG
Title
Probability of Bluetooth/WLAN Packet Collision
Date
Submitted
[April 14, 2006]
Source
[Stephen J. Shellhammer]
[Qualcomm, Inc.]
[5775 Morehouse Drive]
[San Diego, CA 92121]
Re:
[IEEE 802.11-05/0330r3]
Abstract
[Derivation of formula for the probability of a Bluetooth/WLAN packet collision]
Purpose
[]
Notice
This document has been prepared to assist the IEEE P802.19. It is offered as a
basis for discussion and is not binding on the contributing individual(s) or
organization(s). The material in this document is subject to change in form and
content after further study. The contributor(s) reserve(s) the right to add, amend or
withdraw material contained herein.
Release
The contributor acknowledges and accepts that this contribution becomes the
property of IEEE and may be made publicly available by P802.19.
Submission
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E-mail:
Page 1
[(858) 658-1874]
[(858) 651-3004]
[shellhammer@ieee.org]
Steve Shellhammer, Qualcomm, Inc.
April 2006
IEEE P802.19-06/0021r0
1 Introduction
In the 802.11n coexistence assurance (CA) document [1] there is a formula for the
probability that a Bluetooth packet collides with a WLAN packet. In reviewing the 802.11n CA
document we have derived the formula for a packet collision which differs from the formula
given in [1]. The purpose of this document is to give the derivation of the probability of packet
collision.
Section 2 gives the derivation for the probability of a temporal collision. Section 3 gives
the formula for the probability of a joint temporal and frequency collision.
2 Probability of Temporal Collision
The probability of a temporal collision is the probability that during a WLAN
transmission that the Bluetooth Piconet also transmits, at least for a portion of the WLAN packet.
Figure 1 illustrated the timing between the WLAN and Bluetooth packets. In this figure the
WLAN packet considers with the beginning of a Bluetooth packet. This is only for illustration
purposes. In practice there is an offset between the beginning of the WLAN packet and the
beginning of the next Bluetooth packet.
N T2
R
T1
WLAN
BT
BT
BT
BT
T3
T2
Figure 1: Timing of WLAN and Bluetooth Transmissions
The timing parameters are as follows,
T1 is the duration of the WLAN packet.
T2 is the period of the Bluetooth transmissions.
T3 is the duration of the Bluetooth transmission.
Submission
Page 2
Steve Shellhammer, Qualcomm, Inc.
April 2006
IEEE P802.19-06/0021r0
Typical values for these times are: T1 is several ms, T2 is 625 s and T3 is 359 s.
Let us define the full Bluetooth periods that divide into the WLAN packet duration,
T 
N  1
 T2 
Where x  is the largest integer less than or equal to x. We define R as the remainder of
the WLAN packet that is more than N times the Bluetooth period.
R  T1  N T2
The number of temporal packet collisions is a random variable, call it M. We need to
find the probability mass function for this random variable. In order to do this we need to
consider that the time between the beginning of the WLAN packet and the beginning of the next
Bluetooth packet is a random variable. Figure 2 illustrates this random alignment.
WLAN
d
BT
BT
BT
BT
T2
Figure 2: Alignment of WLAN and Bluetooth Packets
The offset, d, is a random variable which is uniformly distributed between 0 and T2.
f (d )  U (0, T2 )
We now determine the number of packet collisions as a function of d, and then average
over the distribution of d.
The Bluetooth packet that begins before the WLAN packet will collide with the WLAN
packet if d  (T2  T3 ) . The next N Bluetooth packets always collide with the WLAN packet.
The next packet after those collides with the WLAN packet if d  R .
Submission
Page 3
Steve Shellhammer, Qualcomm, Inc.
April 2006
IEEE P802.19-06/0021r0
Since there are two values of d at which the number of colliding packets changes we end
up with different answers depending on which value of these values are larger. So let us consider
these two conditions separately.
Case 1: R  (T2  T3 )
In this case for values of d  R there are N+1 collisions. For R  d  (T2  T3 ) there are
N collisions. And for d  (T2  T3 ) there are N+1 collisions. Hence in this case the probability
mass function for the number of Bluetooth packet collisions is given by,
P( M  N ) 
T2  T3  R
T2
P( M  N  1) 
T3  R
T2
Case 2: R  (T2  T3 )
In this case for values of d  (T2  T3 ) there are N+1 collisions. For (T2  T3 )  d  R
there are N+2 collisions. And for d  R there are N+1 collisions. Hence in this case the
probability mass function for the number of Bluetooth packet collisions is given by,
P( M  N  1) 
2T2  T3  R
T2
P ( M  N  2) 
R  (T2  T3 )
T2
3 Probability of a Joint Time/Frequency Collision
As was done in [1], since Bluetooth hops we need to consider not only the Bluetooth and
the WLAN transmitting at the same time but also transmitting on common frequency. We can
also include in this analysis the effect of Bluetooth not actually transmitting during each
opportunity.
Let us define several probabilities. Let us define the probability that a given Bluetooth
packet hops into the WLAN channel as p F . As was pointed out in [1] the value of this
Submission
Page 4
Steve Shellhammer, Qualcomm, Inc.
April 2006
IEEE P802.19-06/0021r0
probability is given by the channel bandwidth divided by the number of Bluetooth hopping
frequencies.
20 / 79 for 20 MHz
pF  
 40/79 for 40 MHz
Let us define the probability that Bluetooth transmits during a giving time slot (duty
cycle) as p DC .
The for each Bluetooth packet the probability that Bluetooth transmits within the WLAN
channel is just the product of these two probabilities, since they are independent.
p  p F p DC
We would now like to find the probability that there is a joint time/frequency packet
collision. Letting the collision event be referred to as event C, the probability of a packet
collision in both time and frequency is given by,
P(C )   P(M  m)[1  (1  p) m ]
m
Using the probability of a temporal collision derived in Section 2, we can write the
probability of collision as,
Case 1: R  (T2  T3 )
 T T  R 
T  R 
[1  (1  p) N ]   3
[1  (1  p) N 1 ]
P(C )   2 3
T2


 T2 
Case 2: R  (T2  T3 )
 2T  T  R 
 R  (T2  T3 ) 
[1  (1  p) N 1 ]  
[1  (1  p) N 2 ]
P(C )   2 3
T
T
2
2




Submission
Page 5
Steve Shellhammer, Qualcomm, Inc.
April 2006
IEEE P802.19-06/0021r0
4 Conclusions
The formula for probability that a Bluetooth packet collides with a WLAN packet was
derived.
5 References
[1] Eldad Perahia and Sheung Li, P802.11n Coexistence Assurance Document, IEEE
802.11-06/0330r3, March 2006
Submission
Page 6
Steve Shellhammer, Qualcomm, Inc.