Abstract-We provide a closed-form expression for the packet loss probability in Bluetooth networks accounting for capture ef- fects due to propagation losses.
Evaluation
of Packet Loss Probability
in Bluetooth Networks
Franc0 Mazzenga, Dajana Cassioli, Pierpaolo Loreti and Francesco Vatalaro University of Rome “Tor Vergata” and RADIOLABS, Consorzio Universith Industria - Laboratori di Radiocomunicazioni Via de1 Politecnico 1, 00133 Rome - Italy e-mail: {mazzenga, cassioli, loreti, vatalaro)@ing.uniroma2.it
Abstract-We provide a closed-form expression for the packet loss probability in Bluetooth networks accounting for capture effects due to propagation losses. The effectiveness of the proposed approach is assessed comparing the analytical results with those obtained from Monte Carlo simulations. By considering different scenarios, we show the dependence of results on the geometry and on the characteristics of the environment. It is observed that packet loss probability can significantly change with the position of the reference receiver in the area, as well as with the extension of the served area as compared with the coverage area of the receiver.
1. INTRODUCTION
A
D-HOC wireless networks allow terminals to flexibly and autonomously organize themselves to communicate without a pre-existing infrastructure. To this aim, in an ad-hoc network connectivity is based on a peer-to-peer paradigm, and there is no difference between terminals. Bluetooth is a transmission standard designed to support ad-hoc connectivity in a local area [ 11. When Bluetooth terminals get close enough, they can cluster into a piconet and temporarily designate one master unit to coordinate transmissions with up to seven slave units. Bluetooth is based on packet transmission and frequency hopping (FH) technologies to provide channelization among different piconets within the same area. Terminals belonging to the same piconet communicate over the channel identified by a frequency hopping code. According to the Bluetooth standard, terminals are allowed to hop within up to 79 bands in the unlicensed 2.4 GHz Industrial-Scientific-Medical (ISM) band [2]. Based on different FH code patterns, several piconets can coexist in the same area, possibly realizing a network of linked piconets, known as scattevnet. Typical scenarios for scatternets are conference halls or airports, that gather a large number of people willing to connect their portable terminal to other terminals, or to the fixed network(s). In these cases, the nodes aggregate randomly to form a large number of piconets with a different number of slaves per piconet. Within a scatternet, packet collisions can occur with significant probability, and this kind of interference degrades link performance. As a consequence, packet loss probability (PLP) increases and the overall throughput of each piconet is reduced. In addition, capture effects due to the dependency of the interference level on the spatial dis-
tribution of terminals and on the characteristics of the environment make the throughput inhomogeneous over the area. Some contributions have been already provided in the literature to evaluate interference effects on Bluetooth performance [3], [4]. An analytical approach for the PLP calculation was presented in [4], where results have been obtained by restricting the number of overlapping piconets to three and assuming a simple propagation model. Then, a worst case analysis was provided which led to an upper bound for the PLP without considering the mitigation effect of propagation losses [5]. In this paper we provide a semi-analytical approach to evaluate interference effects on PLP. The proposed procedure accounts for the geometry of the environment and its propagation characteristics, and for the position of the reference receiver (RR). The paper is organized as follows. In Section 11 first we derive a closed-form expression for the PLP, and then we illustrate the numerical procedures to evaluate the parameters in the PLP equation. In Section 111we provide some simulation results to validate the proposed approach and evaluate PLP for different scenarios. Finally, in Section IV we draw our conclusions. II. SEMI-ANALYTICALPROCEDURE In a Bluetooth system PLP is commonly defined as the probability that the signal to interference plus noise ratio at the output of the RR falls below a threshold, po, which accounts for the fast fading characteristics of the environment:
where C is the received power, I and N are the interference power and the thermal noise power, respectively. In the following we assume that C, I and N are three statistically independent random variables (r.v.‘s). In (1) .fz((:r)is the probability density function (p.d.f.) of 2 = C ~ p0(1+ N). In this Section we present a semi-analytical procedure to evaluate (1). We have [6]:
0-7803-7400-2/02/$17.00 © 2002 IEEE
313
1
.x
.fz(.x.)= .fc(.x.)@-.fw ~PO ( PO1 where @ denotes convolution, .fc((:r)is the p.d.f. of C, and .fi+T(x)is the p.d.f. of W = 1 + N. (x) = .fi (x)@.fN(:I;),where Due to statistical independence .fi+T .fI(x) and .fN(x) are the p.d.f.‘s of 1 and N, respectively.
A. Intevfevence
due to packet collisions
Y I
We consider M active piconets in the area. Packet collisions take place when two or more piconets simultaneously transmit over the same frequency slot. Depending on the dimensions of the area where piconets are located, propagation distance mitigates interference effects due to packet collisions. FH patterns of different piconets can be represented through statistically independent time-discrete random processes, in each time slot as.fNfpl}. The NJ frequensuming values in the set {.fo. fr cies .fi are the carrier frequencies used for hopping. We assume that each Bluetooth unit transmits the same power level (i.e., absence of power control), IVY,, and that each interfering unit can be arbitrarily located in the area. The overall interference power, I, suffered by the RR due to the M active piconets, is:
Fig. 1.
Example
of a geometrical
arrangement
in the local area
where xm, ~1 = 1. M, are independent, identically distributed, binary r.v.‘s accounting for the occurrence of the frequency-collision events, and Y, is the power received at the RR coming from a transmitter belonging to the m-th piconet. We can model xm as: 1. x m- ~ i 0.
with probability p,,,, with probability qrrr
(4)
where p, is the probability that the m-th piconet in the area transmits on the same frequency slot of the RR and qrn = 1 ~ p,. Consider only one-slot packet transmission and assume that each slot always contains a packet. For the case in which packet length is equal to the time slot, we have:
pm=\+L)2
urisyrichroriizctl
picoricts
(5)
From (5) we have that p, = p, and consequently that xIrr = x (independent of m). Other expressions of p for different values of the packet length compared to the time slot duration are reported in [5]. The Y,,,, depends on the propagation losses due to the transmitter-receiver distance and on the geometry of the obstacles. Therefore:
I
Fig. 2.
Procedure
to obtain
the probabilities
of interference
and power
levels.
into two components, i.e. L = A.R, where we evidence a deterministic component usually referred to as path loss, A, related to the transmitter-receiver distance, and a random component, R, accounting for shadowing. We restrict our analysis to a 2-D environment model and we assume that A only depends on the distance rJ of the interferer from the RR. B. Packet loss pmhahility
Assuming .fy(.r) and .fc(.x.)are known, the PLP due to iU active piconets in the area is: Pp
(Al)
= Prob{
ZA[ < O}
(7)
where Y,,,, = rvy . L,.
(6)
The power loss L,,,,, depends on the position of the RR and on the position of the m-th interferer. Since the power transmitted by each interfering user goes through the same propagation environment, the statistics of the interfering power measured at the RR are independent of m. Therefore, in the following we simply consider Y = IVY, . L. When the map of the area is available, L could be evaluated using ray-tracing techniques for each position of the RR and of the interferer (see Fig. 1). When a stochastic propagation model is assumed, L can be factored
314
(8) m=l
The following recurrence equation in the number of piconets holds: ZAI = zhf-1 ~ EM (9) where EAT = poxYh1 and Zt, = C ~ poN. Without loss of generality, in the following derivations we omit N. The p.d.f. of Zh, is:
with .f~,~(.x.)= .fc((.x.). After some calculations the packet loss probability can be expressed in a compact form as:
where p is given in (5) and q = 1 ~ p. The coefficients &
are:
where grn (.x.) = p;“‘.f~~ (-x/p()) cs . . . cs .fy,,, (-x/p()) for rn = 1. 2. Al , and !~(~(.r)= 6(.r). In the Appendix we illustrate numerical approximations for .fy (.x.) and for .fc (.x.). For .fu (.x.)we have:
A. Validation ofthepmposed appmach In this subsection, we describe the simulation procedure adopted to compare the theoretical and simulation results, in terms of packet loss probability for a Bluetooth scatternet in a typical high interference scenarios. We performed Monte Carlo simulations to evaluate the effects of piconet interference only. In each snapshot we generate nf masters uniformly located in the area. Each master forms a piconet with N,s active slaves, where N,s is a random number, uniformly distributed between 1 to 7. Following the recommendations in [2], the transmitted power IVY, is set to 0 dBm. We assumed the following dual slope model for path loss, [7]:
qzi. TiTI,. = ;- NF’ y i=o I=0
Ii/[1P E Ok.. (Xi. y) E s,pxay
$.fR
(;)
‘?‘j
where J is a suitably chosen integer and rj, ,j = 1. probabilities evaluated as: N,-1
d < 8.h d ; 8.h
K, are
(14) where N, and Ng are the number of points along the x and :I/ axes, respectively, located in the grid superimposed on the local area (see Fig.2), and =(:I;. :I/IP g RI;. (x. :I/) g 5’~) is an indicator function identifying the points in the area (5’~) such that the power P = Tl$A, received by the RR, lies in an interval 02k: (see Appendix). To characterize the received power p.d.f .fc(x) we follow the same approach as before (see Appendix). In this case we have:
.fC(:I;)= $
xz 40 + 20 log(d) xz 25.3 + SCilog(d)
(17) Finally we neglected the presence of noise and shadowing. As-
(13) where K is a suitably chosen integer and ok, k: = 1. probabilities evaluated as:
20 . log (fy) 36 . log (fy)
A(d) =
z;‘o08 % ; y6 1:: $004 002 01
2
3
4 5 5 7 B Number d plconets inthearea (M)
9
10
Fig. 3. Packet Loss Probability vs the number of piconets in the area - Area dimensions: 20 x 20 m - synchronous piconets
(15) J, are
NT,-1
observing that the indicator function E now accounts for both the served area SA and the coverage area CA of the RR. The proposed numerical approach to evaluate .fy (x) and .fc(x) is quite general and can be used for any type of environment and for any configuration of obstacles in the area. The presence of obstacles in the area restrict the possible positions of the interferers and of the RR in the area. III.
Fig 4 Packet LOS Probablhty vs the number of pIcone& III the are5 - Area dmenslons: 20 x 10 m - synchronous pIcone&
RESULTS
In the following we provide computer calculation results to illustrate the effectiveness of the proposed approach.
315
suming a receiver sensitivity of P70dBm using (17), we obtain
a circular coverage area CA with radius 10 m [7]. The N,> slaves participating in a piconet are located randomly, according to a uniform distribution, in a circular area of 20 meters diameter, centered in the position of the piconet master. In generating the slaves coordinates we ensured that they were confined within the served area ,!?A. To evaluate .fc(.x.) in (15) we considered a circular coverage area CA centered in the RR position with a radius of 10 m. Each piconet transmission begins in a randomly selected time slot. In each piconet the master begins the transmission by sending an ACL packet to one of the N,> slaves belonging to its piconet. We assumed the conventional round-robin scheduling policy. For each master we randomly generated its own channel hopping sequence assuming a uniform distribution over the 79 frequency-carriers. The length of the frequency hopping sequence for each master was taken equal to the duration of the simulation snapshot. We averaged the performance metrics over a large number of simulation snapshots for each scenario. In each snapshot we regenerated the users’ positions and the piconet loads (the number of slaves in each piconet). In each time slot we compute the signal-to-interference ratio (C/1) of the RR. When the interference 1 is zero, i.e. piconets transmit over different frequencies, we assume that no packet loss occurs, since the transmitter was placed in the coverage area of the RR. In Figs. 3-4 the PLP obtained by Monte Carlo simulations are compared with the results of the proposed semi-analytical procedure. We considered different positions of the RR as illustrated in Fig. 1. The semi-analytical results shown Fig.3-4 were obtained assuming for both C and Y a discrete p.d.f. (see Appendix) based on the following values (in dBm): IL = IIk = II0 + ka where k is an integer and a (dBm) is the power step increment. We assumed IIt, = -8OdBm and the maximum values for C and Y were set to IVY, = OdBm. The histogram bins were assumed to be 0~ = [IIk. IIk + a). The grid steps a.~ and a!/ were assumed to be 2cm in every case. A very good agreement between the simulated and the semianalytical results is shown in Fig. 3-4. As expected, improving the resolution of the discrete approximation of the continuous pdf of Y and C, i.e. reducing the step width a, the numerical and semi-analytical results become closer. B. Semi-analytical
evaluation
ofthe PLP in the scattevnet
In this subsection we analyze the behavior of the PLP by varying the position of the RR and the characteristics of the area where the RR and the interferers are located. We consider the path loss model in (17) and the dimensions of the area are varied from 10 x 10m2 up to 20 x 20m’. To account for packet collision effects, three different transmission conditions have been considered: synchronous and asynchronous piconet transmissions with packet duration equal to the time slot T, and asynchronous transmission with packet duration lower than T,. The ratio between the packet duration and T, was y = Xici/ciSO (see [51X
316
In Fig.5-6 we plotted the PLP as a function of the number of piconets for the different packet collision events. From Fig.5 it
Fig. 5. Packet - synchronous, events
Loss Probability vs number of piconets asynchronous and packet length lower
in the area -Area 10x10 than time slot, collision
Fig 6 Packet - synchronous, event5
Los Probablhty vs number of pIcone& asynchronous and packet length lower
III the area - Area 20x20 than tune qlot, colh~~on
is evident that when the dimensions of the area are comparable with the coverage area of the terminals, the PLP is practically independent on the position of the RR. As expected, the asynchronous case gives the worst results in terms of PLP. Indeed, the results shown in Fig.5 are identical to the upper bounds in [5] that were obtained by neglecting every possible effects due to propagation distance. Increasing the dimensions of the area, effects due to distance between the interferers and the RR become important and this leads to large variations of the PLP with the RR position. The plots in Fig. 6 represent the maximum and the minimum PLP in the area. In fact, it is straightforward to observe that the maximum and the minimum value for the PLP are obtained when RR is in the center and in the corner, respectively (see Fig. 1).
Finally, it can be observed that the differences between the PLP in the synchronous and asynchronous cases with packet duration lower than time slot, can be considered always negligible except when the area dimensions are larger than the coverage area (Fig.6). IV. CONCLUSIONS We presented a semi-analytical procedure to evaluate the packet loss probability in Bluetooth networks accounting for the capture effects due to the environment geometry and to its propagation characteristics. Results are in good agreement with the packet loss probability obtained through Monte Carlo simulations. We showed that the capture effects strongly depend on the geometry and on the propagation characteristics of the environment and we investigated on the sensitivity of the PLP with the RR position. It was observed that when the dimensions of the area are comparable with the coverage area of the terminal, capture effects are practically negligible so that synchronous and asynchronous transmissions with packet duration lower than time slot yield almost the same PLP. Instead, if the area dimensions are larger than the terminal coverage area, the synchronous transmission performs better and PLP significantly changes with the RR position. V. ACKNOWLEDGEMENTS The authors wish to thank Prof. A. Mecozzi for collaborating in devising eq. (11).
where =(:I;. :I/IPI E RI;. (x. :I/) E 5’~) is an indicator function identifying the points in the surface (SA) such that the power PI, received by the RR, lies in Ok. The function f (x. y) is the location probability density of the interferer. The integral in (19) can be approximated as follows. We grid the area as illustrated in Fig. 2 and we fix the position of the RR. For each point in the grid we evaluate the power PI received at the RR using the selected path loss model A(d). Then we arrange the power values PI in a histogram whose bins are identical to {Ok}. The probabilities 7~ are obtained from the fraction of samples collected within each bin of the histogram. If we vary the position of the RR in the area, the statistical distribution of distance d changes leading to a different histogram for PI. Therefore, the probabilities fink.depend on the position of the RR and they need to be recalculated when the position of the RR changes. Using a rectangular uniformly-spaced grid with steps a:~;, a:~ along the x axis and the :I/ axis, respectively, and assuming a uniform distribution of the users in the local service area, i.e. f (:I;. :I/) = l/S, where S is the surface area, then the approximation (14) for equation (19) holds. The presence of obstacles in the area restricts the possible positions of the interferers and of the RR in the area and this can be accounted for in f (x. y). The statistical characterization of the received power C follows the same approach as above. Similarly, the received average power PC can be expressed as PC = TVy,A(d), where now d is the distance between the RR and the transmitter in the same piconet. Again we discretize the power PC on J levels In {Ii. 12. r,~}, each one having probability yl. y2. this case .fc(x) is provided in eq. (15). To evaluate the probabilities rj we use the same approach as in (14) observing that the indicator function E in (14) now accounts both for the local served area 5’~ and for the coverage area CA of the RR. 7.J.
VI. APPENDIX We introduce PI = TVy,A(rJ), the average power at distance rJ from the interferer neglecting shadowing losses, and we assume that the p.d.f. of the shadowing .fR((.x.) is known. We consider PI as a discrete r.v. assuming a finite set of values {II,. IIa. IIK} with probabilities ~1. TK, respectively. In this case eq. (13) holds. The probabilities 7rk depend on the position of the RR and on the physical characteristics of the area described by the path loss model A(d). In this Appendix we evaluate 7rk using a procedure suitable for computer calculation, and assuming a fixed position for the RR. The probabilities ok are defined as: ~~ = Prob(PI
E Ok}.
k = 1.
K
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Y. Lim, .I. Kim, S. L. Min and .I. SooMa, “Performanceevaluationof the Bluetooth-based publicinternetaccesspoint,” in Proceeding.~ ofthe 15th Internutionul Cmference on Networking (ICOIN-15) ,Beppu,Japan, 200 1.
(18)
A. KarnikandA. Kumar,“Performance analysisof theBluetoothphysical layer,”Proceeding.~ of the IEEE ICY’ WC, 2000. 151 A. El-Hoiydi,“InterferencebetweenBluetoothNetworks- upperbound on thepacketerrorrate,”IEEE Comm. Mt., vol. 5, No. 6, June2001. 161 A. Papoulis,Probability, Random Variables and Stochastic Proce.we.s, McGrawHill, 1965. and S. Mattisson,“Bluetooth- A new low-powerradio in171 J.Haartsen terfaceproviding short-rangeconnectivity,”Proceeding.~ of the IEEE, vol.XX,no. 10,October2000. 141
where 0~ is the power interval [IIk ~ .p”7”‘. IIk + at”‘) and p”““, a(v) define the lower and the upper limit of Ok, resp%ctively.‘The afo7r”‘Lp) can be selected to have a superset of disjoint sets 0~ covering the considered power interval. The equation (18) can be expressed as:
317
