May 17, 2002 - with the micro-cellular wireless network, and a handoff message when ...... cellular mobile networks", Vehicular Technology Conference, 1997, ...
Mobility Patterns in MicroCellular Wireless Networks
Suttipong
Thajchayapong 2002
Advisor: Prof. Peha
,~~ ~ectrical ~ Computer NGINEERING
Mobility Patterns in Micro-Cellular Wireless Networks
Suttipong
Department
Thajchayapong
of Electrical and Computer Engineering Carnegie Mellon University
Submitted in partial fulfillment of the requirements for the degree o]:Master of Science
May 17, 2002
Committee Professor
Jon M. Peha (Advisor)
Professor
Ozan K. Tonguz
Abstract
This study investigates
mobility patterns in micro-cellular
wireless networks, based on
measurements from the 802.11-based system that blankets the Carnegie Mellon University campus.Wecharacterize the distribution of dwell time, whichis the length of time that a mobile device remainsin a cell until the next handoff, and sign-on interarrival time, whichis the length of time between successive sign-ons from the same mobile device. Manyresearchers h~ave assumedthat these distributions are exponential, but our results based on empirical analysis show that dwell time and sign-on interarrival
time can be accurately described using heavy-tailed
arithmetic distributions that haveinfinite meanand variance. Oneimplication of this result is that mobility may cause or exacerbate long-range dependence of traffic
load in micro-cellular
networks. Anther implication is that the duration since the last handoff can be used to predict whenthe next handoff will occur, which could improve admission control and other algorithms. Wealso showthat the numberof handoffs per sign-on can be modeledaccurately with a heavytailed distribution.
Contents 1. Introduction
1
2. Description of Wireless Andrewand the Data
3
2.1 Overview of Wireless Andrew
3
2.2 Mobility in Wireless Andrew
4
2.3 Collection of Empirical Data and Processing of the Data
4
3. Analysis of Dwell Times
6
3.1 Distribution of Dwell Times
6
3.2 Estimation of the Distribution of DwellTimes
9
3.2.1 Estimationof the Distribution in Seconds
10
3.2.2 Estimationof the Distribution in Minutes
13
3.3 Predictions Using the Distribution of DwellTimes 4. Analysis of Sign-on Interarrival Times
19 23
4.1 Distribution of Sign-onInterarrival Times
23
4.2 Estimationof the Distribution of Sign-onInterarrival Times
27
4.2.1 Estimationof the Distribution in Seconds
28
4.2.2 Estimationof the Distribution in Minutes
3O
4.3 Predictions Usingthe Distribution of Sign-on Interarrival Times
36
4.4 Distribution of the Numberof Handoffper Sign-on
39
5. Conclusions
43
6. Future ResearchDirections
45
6.1 Possible Applicationsof this study
45
6.2 Possible Directions on this research
46
References
48
1. Introduction
A suitable mobility model is necessary for the study of micro-cellular wireless networks. Researchers whodo analysis or simulation of micro-cellular wireless networks, as is necessary whendetermining howmuchcapacity is needed or whether a given protocol is effective, makeassumptions about what realistic
must
mobility patterns are. As this thesis will show, these
assumptions have generally been incorrect.
Realistic models can only be found by analyzing
measureddata. In this study, we perform an empirical analysis on the data collected from the Wireless Andrewnetwork, the enterprise-wide broadband micro-cellular wireless network that blankets the Carnegie MellonUniversity campus[1]. Carnegie Mellon University was a pioneer of 802.1 l-based networks, which have recently becomeextremely popular, so it is likely that manyfuture enterprise networks will look like Wireless Andrew. Mobiledevices in the system weobserved generate a sign-on messageto establish a connection with the micro-cellular wireless network, and a handoff message whenthey movefrom cell to cell. Sign-on interarrival
time, knownin conventional cellular
wireless networks as call
interarrival time, is the length of time betweensuccessive sign-ons from the samemobile device. Dwell time, sometimesknownas cell residence time, is the period that a mobile device maintains a connectionwith a cell until the next handoff. Exponentialdistributions are often used to model dwell time and sign-oninterarrival time distributions [2,3,4]. Since few institutions
have long experience with vast micro-cellular systems like Wireless
Andrew,there have been few attempts to accurately characterize mobility patterns in these networks. In contrast,
there have been attempts in cellular
telephony systems [5,6,7].
An
empirical study of mobility in the commercialcellular of BCTel Mobility Cellular [7] found that dwell time could be well represented with a lognormal distribution.
Other studies have made
assumptions about device movementand then derived dwell time from these assumptions [5,6].
This thesis is organized as follows. In Section 2, wegive a brief description of the Wireless Andrewnetwork and our methodsof collecting and processing the empirical data. Sections 3 and 4 contain our analysis of dwell time and sign-on interarrival time, respectively. At the end of Section 4, wealso include our analysis on the numberof handoffs per sign-on, whichis defined as the numberof times that a mobile performs handoffs before signing on to the network again. Finally, Section 5 gives a conclusionand Section 6 showdirections for future research.
2. Description of Wireless Andrewand the Data
2.1 Overview of Wireless Andrew
Wireless Andrewis an enterprise-wide broadband micro-cellular
wireless network built to
provide a high-speed wireless service to Carnegie Mellon University campus. The Wireless Andrewproject began in 1993 and it was the world’s first large micro-cellular wireless network. The structure of Wireless Andrewconsists of a numberof Access Points (AP). Each access point supports a wireless link to mobile devices within range. It also serves as a bridge to the university’s wired network [8]. The-access points themselves are connected to one another and the rest of the campus’wired networkthrough an Ethernet [8]. The data examined in this study was measured from the Wireless Andrewnetwork in 1997. There were approximately 90 access points covering 6 buildings and approximately 100 user’s at that time [19,10]. Wireless Andrewin 1997 was operated at 915MHz using a precursor to the IEEE 802.11 standard [ 10]. Mostof the mobile devices used were laptop computers. Wireless Andrewhas grownsince 1997. Currently [8], with 650 access points, it covers the campusarea of 3 million square feet, whichincludes 65 administrative, academic,and residential buildings. It is available to faculty, staff and students across the campus.There are approximately 3000-4000users. Still, most of the mobile devices they use are laptop computers. Using the IEEE 802.11b standard with 2.4 GHzdirect-sequence
spread spectrum (DSSS) [8,11],
Wireless
Andrewtoday provides its users with a high-speed wireless data connection of up to 11 Mbps.
3
2.2 Mobility in Wireless Andrew
To establish a connection with Wireless Andrew,a mobile device sends a sign-on request packet to the access point that offers the strongest signal. A connection is initiated whena mohile successfully receives a sign-on response from the access point. A connection is maintained with the current access point until the received signal level becomessufficiently weakand there is another access point that offers a stronger signal [12,13]. Whenthe mobile device decides to switch the connectionto another access point, a handoffprocess is initiated. The handoff process consists of a request and a response [13]. The mobile device sends a handoff request packet to the target access point whoforwards the request packet, through the wired network,to the access point currently associated with the mobiledevice. Thecurrent acc, ess point sends a handoff response packet to the mobile directly over the wireless link and the mobile can switch to the target access point.
2.3 Collection of Empirical Data and Processing of the Data
FromJune 1997 to December1997, the packets associated with each sign-on or handoff that ’was observed at any access points were captured along with a timestamp that was accurate to the second. In this thesis, we consider a handoff or sign-on to occur at the instant whenan access point informs the mobile device that the handoff or sign-on was successful (even if the acknowledgement does not successfully reach the mobile for somereason). Sign-on interarrival time is the time betweena sign-on and the previous sign-on while dwell time is the time between a handoff and the previous handoff or sign-on. Each mobile device can be uniquely identified
by its MACaddress. These MACaddresses
were used to construct unique identifiers for the devices. To protect the users’ privacy, the
4
mapping between the identifiers
and other information about the device and the user was
deliberately hiddenfrom researchers. Devices that did not perform handoffs were excluded from consideration.
Fromtheir MAC
addresses, we were able to determine that 56 mobile devices performed handoffs during the collection period. Therefore, at least 56%of the users were mobile. Onemobile device performed approximately 70%of all the handoffs during the collection period. Althoughwe cannot knowfor certain, weassume that this mobile device was used for testing purposes, so we also discarded this mobile’sdata.
5
3. Analysis of Dwell Times
3.1 Distribution of Dwell Times
Wedefine a probability massfunction p(i) from our data wherep(i) is the fraction of dwell times whoseduration, rounded to the second, was i seconds. Figures 3.1a and 3.1b showthe plots of p(i) and Figure 3.2 showsthe correspondingcumulativedistribution function.
Table 3.1: Descriptive Statistics of Dwelldata
Mean Median Variance Standard Deviation Max Min Total number of samples
3004.79seconds (_= 50 minutes) 3 seconds 5,006,026,780.6 seconds2 2) (_= 386 hours 70,753.28 seconds(_= 20 hours) 5,115,456 seconds (_= 1421 hours) 0 seconds 9105 samples
0.18 0.16
0.14 012 0.1 0.08
0.06 0.04 0.02 0 15
20
25
(seconds)
Figure 3. la: Distribution of DwellTimeon a linear-linear scale
6
3O
10
~
1O0
001
0.001
0.0001 .......................................................................................................................................................... "----- --*--.----~’~ ......................................... ~ i (seconds)
Figure3.1b:Distributionof DwellTimesona log-logscale
0.6 r", 0.5 0 0.4 0.3 0.2
0 60
120
180
240
t (seconds) Figure 3.2: The Cumulative Distribution
7
Function of Dwell ’£imes
300
The graphs showthat most dwell times are small: approximately 54.62%are three seconds or less. Webelieve that many of these small dwell times occur when the mobile devices are receiving signals of comparablestrength from multiple access points. This causes the mobile devices to frequently switch connection from one access point to another. Despite the fact that 90%of all dwell times are less than two minutes, the meandwell time we observedwasover fifty minutes. This showsthat the tail of the dwell time distribution dominates the mean. in order to represent the tail of the distribution
more clearly, we define a ntew
probability massfunction q(j), whereq(j) is the probability that the dwell titne roundedup to the 60)
nearest minute equals j, i.e. q(j) = 2 p(i) Figures 3. 3a and 3. 3b show the pl ots of q(j ). i=60j-59
0.8
0.7
06
0.5
0.2
0.1
0 2
4
6
8
10
12
14
16
-01
j (minutes)
Figure 3.3a: Distribution of Dwell Time on a linear-linear
scale
18
20
10
100
1000
0.01
0001
00001
i (minutes)
Figure 3.3b: Distribution of Dwell Time on a log-log scale
3.2 Estimation of the Distribution of Dwell Times
In this section, wedetermineclosed-formexpressionsof the distribution for dwell time, whichis denotedby f(t). In our analysis, fit) is derivedusing the measured PMFs p(i), whichdescribesthe probability that dwell time roundedto the second equals I, and q(j), which describes the probability that dwell time is betweenj-1 andj minutes.Weassumethat q(j) correspondsto f(j0.5 minutes). Weconsidereda numberof possible shapes for the PDF.Weshowresults with the exponentialdistribution ae-°t, whichmanyresearchershaveassumedto be representativeof dwell time, andthe Paretodistributionat -~. In eachcase, the constantsa andb are set to minimizeleast meansquared(LMS)error. In order to determinehowwell our equationsfit with the distribution, in our plots, wealso showhigh andlow error bars aroundp(i) andq(j). Eachpoint on an error bar curveis twice the samplestandarddeviation aboveor belowprobabilities p(i) or q(j). Thesample
9
standard deviation has a value of ~/p(i)/N or ~/-q(j)/N where N -- 9105 is the total number of samples.
3.2.1 Estimation of the Distribution of Dwell Times in Seconds
Wefirst estimate the PDFf(t)of dwell time for small values of t. For the first three seconds, f(t) increases with respect to t. The exponential and Pareto distribution with constants selected to minimize LIMSerrors from p(i) between 4 seconds and 300 seconds are shownin Table 3.2.1 and in Figure 3.4. The Pareto distribution yields a muchlower error than the other distributions we tried, including the exponential. In addition, it is clear fromthe figures that the distribution fits the data well.
Table3.2. l: the Equationsthat fit the Distribution of DwellTimes Types of Estimated Equations
Optimal Equations
LMSErrors
Pareto Exponential
144 0.374t -°°3t 0.009e
-9 3.247x10 -8 5.13lxl 0
10
10
OOl
100
1 ooo
~~~.~.~...
0 001 ~0.374t^(-1 44) -- O.O09"ex#(-O.O3t) 0.0001
0000001 t (seconds)
Figure3.4a: TheEstimationof Dwells Distribution using Paretoand Exponential
---
004005
0.03
[~ p(t), data ~ 0.374t^(-1.4-4)
-4~
[
0.01002
0
t (seconds)
Figure3.4b: TheEstimationof Dwells Distribution using Pareto
11
uppererror bars
0.008
’--0.374t^(-1 44) lower error bars uppererror bars
0.002
80
100
120
140
160
180
t (seconds)
Figure3.4c: TheEstimation of DwellsDistributionusingPareto
0.004
~p(t), data ~ 0.374t^(-1 44) lowererror bars upper error bars
0.002
t (seconds) Figure 3.4d: The Estimation of Dwells Distribution using Pareto
12
3.2.2 Estimation of the Distribution of Dwell Times in Minutes
While the distribution in Section 3.2.1 accurately describes dwell time within the first five minutes, which account for 93.34%of the dwells we observed, the meanis dominated by the larger dwell times. To better describe the tail of the distribution, wedeterminethe distributions that minimize LMSerror from q(j) for periods of two minutes to 1000 minutes. The results are shownin Table 3.2.2 and in Figure 3.5. Once again, the Pareto distribution closely fits the empirical data while the exponential and other distributions wehave tried do not.
Table 3.2.2 the Equationsthat fit the Distribution of DwellTimes Types of Estimated Equations
Optimal Equations
LMSErrors
Pareto Exponential
0.065t ~.445
1.749× 109 .7 1.468×10
-0’062t 0.004e
1o
~oo
loo0
0.01
0.001
: q(t), data --0.065t^(-1.445) I-- O.O04_exC)(-O.O62t) o.oo01
0.00001t
o 000001 t (minutes)
Figure 3.5a: The Estimation of Dwells Distribution using Pareto and Exponential
13
0.045
0.04 0.035
0.03
0025 q(t), data --0.065t^(-I 445) lower error bars upperern:~r bars
0.02
0.015
001
0.005
0 5
10
15
20
25
-0.005 t (minutes)
Figure3.5b: TheEstimationof Dwells Distribution using Pareto
0.005
0.004
0003
q(t), data 0.065t^(-1 445) lower error bars upper error bars
0.002
0.001
60
90
-0001 t (minutes)
Figure3.5c: TheEstimationof DwellsDistribution using Pareto
14
1 iO
0.0015
0.001 q(t), data i-0 065t^(-1.445) i Lowererror b~arsi upper error bars I 0.0005
’
150
180
240
210
270
300
_0.00051201 t (minutes)
Figure 3.5d: The Estimation of DwellsDistribution using Pareto
As j increases, the probability that dwell time will be betweenj-1 andj minutesdecreases, and so does the accuracy of q(j). In order to reduce the noise, weaverage empirical data over periods even greater than one minute. Morespecifically, we determine the fraction r of dwell durations that were betweenX and X+Iwhere the averaging period 1 is greater than one minute, and we: set the probability that dwell time will be within a half minuteof X+I/2to be r/l. Figures 3.6a, 3.6b, 3.6c, and 3.6d showthe plot of the measureddistribution with averaging and the Pareto equation 0.065t 1445.
The averaging periods that are used with the dwell times in the Figures are shownin
Table 3.2.3 below. Wehave minimizedLMSerror over manyother intervals and always find that a similar Paretodistribution fits.
Table 3.2.3: The AveragingIntervals used in Figures 3.6a-d AveragingPeriods (I) 1 minute 10 minutes 100 minutes
Dwell Times (X) Less than l0 minutes BetweenI0 minutes and 100 minutes Between 100 minutes and 1000 minutes
15
10
100
1000
0.1
0.01
0.001 ~0 065t^(’1 445) lowererror bars uppererror bars ! 0.0001
0.00001
0.000001 t (minutes)
Figure3.6a: TheEstimationof DwellsDistributionusing Paretoandlarger averaginginterval for large dwells.
2
~
3
4
5
6
7
8
9
data i--0.065t^(-1,445 I Io /..... o~ bars upper error b~
0.01
0.001 t (minutes)
Figure3.6b: TheEstimationof DwellsDistribution using Paretoandlarger averaginginterval for large dwells.
t6
0 01 20
30
40
50
60
70
80
90
100
0.001
data --0 065t^(-1.445) lowererrorbars uppererror bars 0.0001
0,00001 t (minutes)
Figure3.6c: TheEstimationof DwellsDistribution using Paretoandlarger averaginginterval for large dwells.
0.0001 200
300
400
500
600
700
800
900
i
--00651^(-1"445) lowererrorbars uppererror bars
0.00001
’,, \
/
0000001¯ t (minutes)
Figure3.6d: TheEstimationof DwellsDistribution using Paretoandlarger averaginginterval for large dwells.
17
Comparisonbetween the Estimations in seconds and in minutes
As the analysis below shows, there is a little
difference between the Pareto results found in
Sections 3.2.1 and 3.2.2. Let px(t) be the probability that dwell is in the period of duration x centered at Let tmin be the dwell in unit of minute Let tsec be the dwellin unit of second
Therefore wehave tsec = 60train and fromour previous analysis, wehave (3.1)
P~ec(t~ec) 0.c) 374(t~e -144 p,~,, (t,~,,) = 0.065(tm~ . )-1.445
(3.21)
Assumingthat the distribution curve is roughly linear over one-minuteperiod and using Equation (3.2) Psec(tsec) = (l/60)Pmin(/’min)
(~0)Pmin(tSe~/~0)/
= 0 399(t Thus, Equation (3.1), which was derived by minimizing LMSerror from 4 seconds to 5 minutes and, Equation (3.2), which was derived by minimizingLMSerror from 2 to 1000 minutes, are almost identical.
Analysis of the Estimation of the Probability Density Function (PDF)of DwellTime
A distribution that has an arithmetic tail with exponents between-1 and -2 is heavy tailed, and has an infinite variance and infinite mean. Theinfinite meanseemsto contradict with our finite sample meanof 3004.789 seconds or approximately 50 minutes. However, in our analysis, we measuredwell data in a limited period and therefore exclude the users that did not handoff in
18
such period. It is likely that measuringover a inuch larger period wouldallow even larger dwell times to be collected, therefore yielding an even larger sample mean. As measurementperiod goes to infinity, samplemeanmayalso go to infinity.
3.3 Predictions Using the Distribution of Dwell Times
Giventhat it has beenIx since last handoff, wecan find the probability P(k, ~t) that a handoffwill occur within the next period of duration k from our empirical data as shownin Equation (3.3). Wealso calculate the same probability F(k, Ix) from the estimation of the distribution using Equation (3.4). The graphs of P(k, g) and F(k, g) for k = 1 minute, 5 minutes and I hour shownin Figures 3.7a, 3.7b, 3.7c, and 3.7d. The estimated close-form expressions seem to work well for this prediction.
k
P(k,ct)
=
y~ p(i + ,u i=o
(3.3)
£ p (i) i=~u k
F(k,/z)
[, f ( x + /~ ) 0
(3.4)
if (t)dt
19
0.9
I,e ".~
0.5tl
~
0.3
....... P(k,.u), k = 1minute i -- F(k,.ul, k = 1 minute P(k,.u), k = 5minutes I" F k,,u), k = 5minutes I ....... P(k,,u), k = I hour ~ F(k,.u), k = 1 hour
.-. k=l hour
~’-
k=l mnute
0.2
10
100
I~0
~0.~
~ (m~nute~) Figure 3.7a:
Predictions of the Probabilities that Dwell ends within 1 minute, 5 minutes and 1 hour using Empirical Data and Estimations of Dwell Time Distribution
P(k,~’.).k = 1minute~ i~ F(k,.u), k = 1minute/
Figure 3.7b: Predictions of the Probabilities that Dwell ends within 1 minute using Empirical Data and Estimations of Dwell Time Distribution
2O
~
. P(k,,u), k = 5minutes F(k,p), k = 5minu
(minutes)
Figure 3.7c: Predictions of the Probabilities that Dwellends within 5 minutes using Empirical Data and Estimations of Dwell TimeDistribution
IL-- F(k~,)r
*v~
o.2
........
k--
1hour
~,~
0
/~ (minutes)
Figure 3.7d: Predictions of the Probabilities that Dwellends within 1 hour using Empirical Data and Estimations of Dwell TimeDistribution
21
¯
Let E(K) and Mdenote the expected value and median of time until the next handoff respectively. Giventhat it has been g since the last handoff, E(K)andMcan be found fromthe empirical data using Equations(3.5) and (3.6). The graphs of E(K) andMare shownin 3.8. As ~t increases, both E(K)andMincrease, whichshowsthat, the longer it has beensince the last handoff,the longerit is likely to be until the nexthandoff. ~ i p(i+ ~u) i=0
=
E(K)
(3.5)
~ p(i) i=la M
Z p(i+,u)
=
i=0
0.5
(3.6)
........................................................................................................................................................
10000
’
9000
8000 7000
6000
5000 4000
3000 2000
!
1000
0 0
100
200
300
400
500
600
700
800
900
1000
p (minutes)
Figure3.8: Predictions of the ExpectedValues(E(K)) andMedians(M)of period that dwell end
22
4. Analysis of Sign-on Interarrival Times 4.1 Distribution of Sign-on Interarrival Times
As wedid with dwell time, we define a probability mass function w(i) from our data wherew(i) is the fraction of sign-on interarrival
times whoseduration was i seconds whenrounded to the
second. Figures 4.1a and 4.1b showthe plots of w(i) and Figure 4.2 shows the corresponding cumulativedistribution function.
Table 4.1: DescriptiveStatistics of Sign-onInterarrival data
Mean Median Variance Standard Deviation Max Min Total numberof samples
31,162.57seconds(_-- 8.65 hours) 23 seconds 66,408,552,869 seconds2 2) (~ 5,124 hours 257,698.57seconds (z~ 71.58 hours) 8,013,965 seconds (--2,226 hours) 0 seconds 13785 samples
0.06
0.05
0.04
0.03
0.02
0.01
10
:20
30
40
50
I (seconds)
Figure4. la: Distribution of Sign-onInterarrival Timeon a linear-linear scale
23
0.1
0.01
0.0001
QO0001 i (seconds)
Figure4. tb: Distributionof Sign-onInterarrivalTimeson a log-log scale
1
0.9 08
0.7
’~0.5
0.4
0.1
0 0
i 000
2000
3000
4000
5000
6000
7000
8000
9000
t (seconds)
Figure4.2: TheCumulativeDistribution Functionof Sign-onInterarrival Times
24
10000
The table and the graphs showthat most sign-on interarrival
times are small: approximately
50.42%are 23 seconds or less. Basedon the data, we believe that manyof these small sign-on interarrival
times occur when mobile devices were near the edge of the network, so they
frequently made and lost connection to the closet access points. It may also occur when computers went up and down The sample meanof sign-on interarrival
time we observed was over 520 minutes, while 80%of
all sign-on interarrival times are less than 8 minutes. This showsthat the tail of the sign-on interarrivals distribution dominatesthe meanas it did with dwell time. In order to represent the tail of the distribution moreclearly, wedefine a newprobability massfunction v(j), wherev(j) is the probability that the sign-on interarrival time roundedup to the nearest minute equals j, i.e. 60 j
= ~ w(i) Figures 4. 3a and 4. 3b show the pl ots of i=60j-59
0.6
0.5
0.4
0.2
o.1
10
12
14
1~
18
(minutes)
Figure 4.3a: Distribution of Sign-onInterarrival Timein linear-linear scale
25
20
10
100
1000
10~00
O.1
o 001
0.001
0.0001
0.00001 j (minutes)
Figure4.3b: Distributionof Sign-onInterarrival Timein log-log scale
In order to reducenoise andpresent the distribution of large sign-on interarrival times even moreclearly, we define another probability massfunction y(d) for integer d correspondingto 360d
successive six-hour periods, i.e. y(d) = ~_~v(j). Figure 4.4 showsthe plot ofy(d). Wefind j=360d-.359
that the distribution has a wave-likecurve. Thepeaksof this curvecorrespondto multiples of 24 hours. This showsa daily cycle pattern of users’ behaviors, whichhappenswhenusers sign-on at approximately the sametimes of the day. Forexample,a user arrives at his workplace andsignson once and then comes back and signs-on again at approximately the same time the next morning.In addition, the 3rd peak is higher than the 2 nd peak, whichshowsthat sign-ons occur moreoften in a period of 3 days than 2 days. Sign-ons over 3-day periods could happenduri[ng weekendswhenusers sign-on on Friday and then again on Monday.
26
0.01
0.001
0.0001
0.00001
Figure 4.4: Distribution of Sign-on Interarrival Timesaveraged over 360-minuteperiods, shown in linear-log scale
4.2 Estimation of the Distribution of Sign-on Interarrival Times
In this section, wefind closed-formexpressionsfor the distribution for sign-on interarrival time, denotedby g(t). In our analysis, g(t) is derived using the measuredPMFsw(i) whichdescribes the probability that sign-on interarrival time roundedto the secondequals i, v(j) whichdescribes the probability that sign-on interarrival time is betweenj-1 and j minutes, and y(d) whichdescribes the probability that sign-on interarrival time is between6(d-l) and 6d hours. Morespecifically, we assumethat w(i) corresponds to g(i seconds), v(j) corresponds to g(j- 0.5 minutes), and y(d) corresponds to g(6d-3hours). Weconsidered a numberof possible shapes for the PDF. In this paper, we show results with the exponential distribution
aebt, which manyresearchers have
assumedto be representative of sign-on interarrival time, and the Pareto distribution at +. In each case, the constants a and b are set to minimize least meansquared (LMS)error. In order
27
determine howwell our equations fit with the distribution, in our plots, wealso showhigh and low error bars around w(i) and v(j). Eachpoint on an error bar curve is twice the samplestandard deviation aboveor belowprobabilities w(i) or v(j). This samplestandard deviation has a value of ~
U
or
where N=13785is the total numberof samples.
4.2.1 Estimation of the Distribution of Sign-on Interarrival Times in Seconds
Wefirst estimate the PDFg(t) of sign-on interarrival time for small values of t. For the first two seconds, g(t) increases with respect to t. The exponential and Pareto distribution with minimum LMSerrors from w(i) between 3 seconds and 300 seconds are shownbelow in Table 4.2.1 and in Figure 4.5. The Pareto distribution yields a muchlower error then the exponential and other distributions wetried. In addition, it is clear fromthe Figures that the distribution fits the data well.
Table 4.2.1: TheEquationsthat fit the Distribution of Sign-onInterarrival Times Types of Estimated Equations
Optimal Equations
LMSErrors
Pareto Exponential
-~3a5 0.205t -°°~lt 0.004e
.9 2.356×10 .7 1.069×10
28
0.1 10
looo
100
0.01
!
w(t),
i--
da~
O.O04exp^(-O.011 t)J
0.001
0.0001 t (seconds)
Figure 4.5a: The Estimation of Sign-on Interarrivals Distribution using Pareto and Exponential
0.06
0.05
0.04 I~w~t),data
0.03
i~0’205t^(’1"165) lower bounds upper bounds
0.02
0.01
10
20
30
40
-001 ~ ..............................................................................................................
50
60 i
t(seconds)
Figure 4.5b: The Estimation of Sign-on Interarrivals Distribution using Pareto
29
0008
0 OO6
0.2o5t^(-1.165) lower bounds up_perbounds
0004
0.002
0
~"
,’ 120
180
240
3~0
-0.002 ....................................................................................................................................................................................................................... ’ t (seconds)
Figure4.5c: TheEstimationof Sign-onInterarrivals iDistribution using Pareto
4.2.2 Estimation of the Distribution of Sign.on Interarrival Times in Minutes
Whilethe distribution in Section 4.2.1 accuratelydescribes sign-on interarrival time within the first five minutes,whichaccountfor 76.73%of the sign-on interarrivals weobserved,the meanis dominated by the larger sign-oninterarrival times. Tobetter describethe tail of the distribution, we determinethe distribution that minimizes LMSerror from v(j) for periods ranging from2 minutes to 10000 minutes. The results are shownbelow in "Fable 4.2.2 and in Figure 4.6 for periods from2 minutesto 720 minutes (=12 hours). Wealso do another estimation for periods from 1 minute to 10000 minutes to characterize the daily cycle. Once again, the Pareto distribution closely fits the empiricaldata while the exponentialandother distributions wetried do not.
3O
Table4.2.2: TheEquationsthat fit the Distributionof Sign-onInterarrival Times
Typesof EstimatedEquations
OptimalEquations
LMSErrors
Pareto Exponential
0.131t-1,3~5 -°°4t 0.006e
-8 1.052×10 .7 5.945× l 0
0.1
~
10
100 1~0
0.01
0.001
~0.131t^(-1.345) I-- O’O06exp(’O’O4t) 0.0001
0.00001
0.000001 t (minutes)
Figure4.6a: TheEstimationof Sign-onInterarrivals Distribution using ParetoandExponential
31
0.07
006
0.05
0.04 --0 1311^(-1.345) upper bounds J
003
0.02
0.0~
0 5
10
15
20
25
-0.01 t (minutes)
Figure4.6b: TheEstimationof Sign-onInterarrivals Distributionusing Pareto
0.005
0 004
0.003
v(t}, data ~0 1311^(-1 345) lower bounds upper bounds
0.002
0.001
60
90
120
150
180
-0001 t (minutes)
Figure4.6c: TheEstimationof Sign-onInterarrivals Distributionusing Pareto
32
0002
00O15
ooo~
00005
v(t), data 0.131t^(-1,’]45) lower bounds upper bou£~ s
.....
~o
___
] ~ -00005, t (minutes)
Figure 4.6d: The Estimation of Sign-on Interarrivais Distribution using Pareto
0.0008
0.0006
v{t], data O, 131t^(-1.345) lower bounds upper bounds
00004
0.0002
I 0
I
.....
/
t (minutes)
Figure 4.6e: The Estimation of Sign-on Interarrivals Distribution using Pareto
33
0.00~
00008
0.0006
~0"131t^(-1"345) lower bounds upperb2:,unds
00004
00002
-0.0002 t (minutes)
Figure4.6f: TheEstimationof Sign-on]ntcrarrivals Distribution using Pareto
Estimation of Daily Cycle For periods extending well beyond 12 hours, we must consider the daily cycle. Wetherefore employa distribution of the formh(t) = 0.131t~345 ¯ s(t) where0.131 t -1345is the distribution that workedwell for a 12-hour period and s(t) is a sine wavewith a 24-hour (1440 minutes) period. Wefurther assumethat the sine wavehas a peak at time 0 so that s(t) = msin(2~t/~1440 + %) + n. The constant m and n are selected to minimize LMSerror with y(d), the fraction of sign-on interarrivals occurring in the dt~ 6-hour period. Weassumetlhat y(d) corresponds with h(d ¯ 6 hours - 3 hours). Weimplicitly ignore the fact that sign-on interarrivals of roughly three days are more common than sign-on interarrivals of two days. The plot of the estimation h(t) is shownin Figure 4.7.
34
t (hours)
0.1
0.01
0.0001
0.00001
Figure 4.7: The Estimation of Daily Cycle Pattern of Sign-on Interarrival TimesDistribution using a sine function and a Pareto as its envelope
Comparisonbetween the estimations in seconds and in minutes Theanalysis belowshowsthat the results found in Section 4.2.1 and 4.2.2 are different. Let px(t) be the probability that sign-on interarrival is in the period of duration x centeredat t. Let train be the sign-on interarrival in unit of minutesand tsec be the sign-on interarrival in unit of seconds. Therefore wehave tsec = 60t,~n. Our analysis yielded the following estimates
Psec(Gc) -I = ~65 0"205(t~)
(4.1)
p,,~,(t,,
(4.2)
m) = 0.131(t~,)-1345
Assumingthat the distribution curve is roughly linear over a one-minute period and using Equation (4.2)
35
Psec (tsec)
= (~/~60)Pmin (train)
= (~//~)Prnin
0)/
_= 0.538(t~e~)-l.345
The result is different from Equation (4.1). Apparently, the PDFdecays slower whenLMS minimizedfor sign-on interarrival
times under 5 minutes, than whenlonger sign-on interarrival
times are considered. However,it can be seen that the distribution is still heavy-tailed and has infinite meanand variance.
4.3 Predictions Using the Distribution of Sign-on Interarrival Times
Given that it has been 13 since the last sign-on, we can use our empirical data to find the probability W(s, 13) the device will sign-on within s. This is calculated using Equation(4.3). also find the sameprobability G(s, 13) fromour estimation of the distribution using Equation(4-.4). The graphs of W(s, 13) and G(s, 13) are shownin Figure 4.8a for s -- 1 and 6 hours, and in Figure 4.8b for s = 12 hours, 1 day and 3 days. The estimated closed-form expressions seem to work well for this purpose. ~ w(i + fl)
’=°
w (s,/~ )
(4.3)
~ w(i)
G(s,13)
=
i g(t + fl)dt 0
(4.4)
i g (t)dt
36
....... W(s,l~), s = 1hour
--G(s,I]),
W(s,l~), s = 6hours........... 1000
2000
3000
4000
5000
8000
7000
1hour
G(s,l~,), 6hou rs 8000
9000
~
00
s = 6hours
o.1
0.001 ~ (minutes)
Figure4.8a: Predictionsof the Probabilitiesthat sign-on wi]] occuragainwilhin 1 hourand6 hours fromEmpiricalData and Estimationsof Sign-onIntera~ivals Dis~ibution
......
~G(s,I]), = 1day
W(S,13), s = 1day W(s,~),s = 3days..................
I~
1000
2000
3000
4000
5000
6000
7000
8000
G(s,~),s = 3days 9000
10000
0.01 (minutes)
Figure4.8b: Predictionsof the Probabilitiesthat sign-on will occuragainwithin12 hours1 day and3 days fromEmpiricalData andEstimationsof Sign-onInterarrivals Distribution
37
Let E(S) and H denote the expected value and median of time until the next sign-on, respectively. Giventhat it has been 13 since the last sign-on, E(S) and H can be found from the empirical data using Equations(4.5) and (4.6). The graphs of E(S) and H are shownin Figure As in dwell time, as 13 increases, both E(S) and H increase, i.e. the longer it has beensince the last sign-on, the longer it is likely to be until the next sign-on. In Figure 4.10, wealso showthe predictions of E(S) for 13 up to 10000minutes or approximatelya week. It can be seen that E(S) still increases even higher. In contrast, the meanand medianswouldremain constant with respect to 13 if sign-oninterarrivals distribution wereexponential.
E(S)
~ iw(i+fl) ,=0
=
(4.5)
~ w(i) i=~ H
Z w(i+fl) i=0
0.5
(4.6)
~ w(i)
15000
12000
9000 E(S) .......
6000
3000
0
360
720
1080
1440
1800
2160
2520
2880
3240
3600
3960
4320
~ (minutes)
Figure 4.9: Predictions of the ExpectedValues (E(S)) and Medians(H) of period until the sign-on
38
25000
20000
~15000
g t~ ]0000
5OO0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
~3 (minutes)
Figure 4.10: Predictions of the ExpectedValuesof period until the next sign-on
4.4 Distribution of the Numberof Handoffs per Sign-on
Wedefine the probability mass function of the numberof handoffs per sign-on, gt(z), wherez is the numberof handoffs per sign-on. Weincluded all sign-ons that were followed by one or rnore handoffs. Moreover, as described in Section 3.1, in the system we employed, the majority of dwell times were short, presumably because a device that is receiving signals of comparable strength from multiple access points sometimes oscillates
between them. Since these rapid
handoffs are probably not indicative of actual user mobility, we show in Figure 4.11 the distribution of the numberof handoffs wherewe exclude those handoffs that follow dwell times less than four seconds.
39
Table 4.4.1 : Descriptive Statistics of Number of Handoffper sign-on data
Mean Median Variance Standard Deviation Max Min Total numberof samples
5
2.44 calls 1 call 210.62calls 3.26 calls 44 calls 1 call 1583 samples
10
15
20
25
30
35
40
0.1
0.01
0.001
z (number of handoff
per sign-on)
Figure 4.11: The Distribution of the Numberof Handoffsper Sign-on
40
45
,~0
Weestimate the distribution using the geometricabz, and Zeta az-b distributions, wherea and b are found from minimizing least meansquared (LMS)error from z = 1 to z = 10. These curves are also shownin Figure 4.12. Table 4.4.2 showsthe resulting equations, and their corresponding LMSerrors. It can be seen from Table 4.4.2 and from the figure that the Zeta distribution is a close fit, while the geometric is not. Indeed, even though LMSwas only minimizedfor z in the range of 1 to 10, Figure4.12 showsthat the Zeta is still a goodfit for larger values ofz. The Zeta distribution
with exponent -2.165 is heavy-tailed with finite meanand infinite
-2 variance [14]. Wenote that 7’ = Z 0 "165 805~ " = 1.218, so this expression cannot exactly represent z=l
the distribution for all z. This maybe causedby the significant uncertainty in our data for large values of z due to the fact that we have only 1583samples, or it maybe because the tail of the PMFdecays faster whenz > 10 than it does whenz < 10.
Table 4.4.2 the Equations that fit the Distribution of Numberof Handoffsper Sign-on Types of Estimated Equations
Optimal Equations
LMSErrors
Zeta Geometric
-1165 0.805z z1.131(0.478)
0.007656 0.026636
41
2
3
4
5
6
7
8
9
1~0
0.1
~- ~u(z), data --=-- 0.805z"(-2.165) ~ 0.131 (0.478)"-(z) ~ lower error bars
0.01
upper error
bars
0.001
0.0001 z (number of handoff
per sign-on)
Figure 4.12a: Estimationof the Distribution of Number of Handoffsper Sign-onusing Geometric and Zeta
2
4
6
8
10
12
14
16
18
2~0
O.1
0.01
...... qJ(z), data I + 0.805ZA(-2.165) ---X-- 0.131 (0.478)A(Z)
0.001
lower error upper error
0.0001
bars bars
0.00001
0.000001
0.0000001 z (number
of
handoff
per
sign-on)
Figure 4.12b: Estimationof the Distribution of Number of Handoffsper Sign-on using Geometric and Zeta
42
5. Conclusions In this stttdy,
we use data measured from Carnegie Mellon University’s Wireless Andrew
network [1,8], which is representative of the enterprise-wide 802.11-based broadband microcellular wireless networksthat are rapidly growingin popularity. Our purposeis to characterize users’ mobility patterns, by determining from empirical analysis the distributions of dwell time, sign-on interarrival time, and numberof handoffs per sign-on. Mostdwell times and sign-on interarrival times are small as shownby the mediansof 3 seconds and 23 seconds, respectively. However,the meandwell time was 50 minutes and the meansignon interarrival time was almost 9 hours, whichshowsthat the tails of these distributions dominate the means. Indeed, both distributions can be well characterized with an arithmetic tail,
and
parameters that imply infinite variance and even infinite mean. This surprising result contrasts greatly with the commonassumptions that dwell time and sign-on interarrival modeledwith a light-tailed
time can be
exponential distribution, or even a lognormal distribution [7]. We
have demonstratedthat the exponential workspoorly in both cases. There has been a great deal of research [15-21] on the use of heavy-tailed distributions with long-range dependenceand self-similarity
to describe traffic flows in wired networks. The tact
that the length of time each mobile device is present in a particular 802.11 LANmicro-cell is heavy-tailed mayimply that the period whena device is sending traffic
to that LANis also
heavy-tailed. Even if individual streams do not produce traffic with long-range dependence,the aggregation of traffic from multiple streams, someof whosetime in the cell is heavy-tailed, may create long-range dependence. As a result, networks mayneed greater communicationscapacity and buffer space than one might predict based on models that assu~ne dwell times are exponentially distributed. Moreover,it has been shownthat reliable transport protocols such as TCPcan create or further exacerbate the appearance of long-range dependence[17, 18, 21], and
43
other protocols mayhave similar effects, so this distribution
of dwell time mayalso have
implications for design of protocols that operate over micro-cellular networks. The distribution of dwell time increases from 0 to 3 seconds, whichis the median.Basedon our data, webelieve that the medianis very low because there are circumstances where a device is receiving signals of comparablestrength from multiple base stations, and it oscillates amongthem wheneversignal strengths or interference levels change. Fromfour seconds to roughly 17 hours, the Pareto distribution with an exponent of-1.445 fits well with the observed distribution of dwell time. For sign-on interarrival time, the distribution similarly increases as time goes from 0 lto 2 seconds, and then it follows a Pareto distribution for durations of a few seconds, minutes, and even hours. In addition, wefind a daily cycle of users’ behaviors. The long-term distribution can be described well by multiplying the same Pareto function with a sine wavethat has a 24-hour cycle. Knowingthat these distributions are arithmetic and heavy-tailed allows us to makepredictions about dwell time and sign-on interarrival time. The longer it has been since the last handoff or sign-on, the longer it is likely to be until the next handoffor sign-on. This wouldnot be the case if both dwells and sign-on interarfivals distributions were exponentials. Finally, we characterize the numberof handoffs per sign-on. Wefind that a Zeta distribution is a reasonably accurate estimate of the distribution. Withthe parameterswefound, this distribution is also heavy-tailed with infinite
variance, but unlike the case of dwell times and sign-on
interarrival times, it has a finite mean.
44
6. Future Research Directions In this section, we showfuture work that would build on this research. Section 6.1 discusses possible applications of mobility modelsin the design of micro-cellular networks. In section .6.2, we showpossible future directions for empirical study of Wireless Andrewto learn more about mobility.
6.1 Possible Applications of this study
Using the distributions of dwell time, sign-on interarrival time and numberof handoffs per signon that we have found, we could develop simulation software for micro-cellular networks that includes morerealistic modelsof mobility. Our discovery that dwell times are heavy tailed can be used to improve the admission control algorithm. Currently, 802.11-based enterprise-wide micro-cellular
networks like Wireless
Andrewhave no mechanismto protect quality of service (QoS). Base stations always accept both newsign-on and handoff requests, even though a larger numberof mobiles per cell would result in performance degradation for the mobiles already in these cells.
Manyadmission control
algorithms have been considered to prevent such degradation [22]. A simple examplewould be to block newcalls in a cell wherethe numberof active mobiles exceeds c for someconstant c < N, where N is the maximum number of simultaneously active mobiles in each cell with adequate QoS (handoffs would be accepted even whenthe number of mobiles exceeds c). Our work shows that moreeffective algorithms are possible by considering howlong each of these mobile devices has been in a cell. For example,if manyof the devices recently entered the cell, the probability that they will remain long enoughto exceed available resources is muchlower than if all devices have already been stationary for several minutes.
45
Another possible use of these results is the design of forwarding mechanisms.Whena mobile device does a handoff to a newcell, the base station in the old cell mayforward any packet that arrives to the base station in the newcell. If the mobile does several handoffs, then packets arriving at the initial cell maybe forwardedalong a multi-hoppath before reaching the mobile. In networksthat operate this way, it is occasionally necessaryto simplify these multi-hop paths. Our results showthat the length of time that a mobilewill remain in a cell dependson howlong it has already been in that cell. Therefore, the simplifications of multi-hop paths should only be made after the mobile has remainedin the samecell for a sufficiently long period of time. Otherwise, the probability is too high that the mobilewill handoffagain soon.
6.2 Possible Directions on this research
A goal of this research is to enable simulation software, or a networkalgorithm, to predict for a given mobile at a given time whenthe next handoff will occur and whenthe next sign-on will occur. Wehave estimated the appropriate distributions in Sections 3 and 4. It maybe possible to produce even moreaccurate distributions by using additional information about the mobile device and its recent history, or about the networkat that particular time. It maybe possible to makemoreaccurate predictions about the duration of the next dwell time. Onewayto investigate this is by finding whetherthere are correlations betweenthe i t~ dwell time and the jt~ dwell time that followthe samesign-on for i ~ j. There are also other possibilities that wouldhelp predict the next dwell time, such as finding whether the next dwell time is correlated with previous sign-on interarrival times, or the period since the last sign-on, or the numberof handoff since the last sign-on. Anotherwayto use the numberof handoffs since the last sign-on is to separately estimate the distribution of the i t~ dwell time after a sign-on for all i. Thei th dwell time mayhavea different distribution fromthe jt~ dwell time for i ~ j.
46
Similarly, it wouldbe useful if moreaccurate predictions of the next sign-on interarrival time can be determined. Onepossible direction is to determine whether there are correlations among sign-on interarrival times, i.e. whether the next sign-on interarrival time is correlated with the previous sign-on interarrival times or previous dwell times. Anotherpossibility is to determine whetherthe next sign-on interarrival time is correlated with the numberof handoffs since the last sign-oninterarrival time. Finding how dwell time or sign-on interarrival interesting.
For example, the distribution
time varies with time of a day would be
of dwell times during the working period 8am-5pm
mightbe different from the distribution during the nighttime. Finding whether the distribution of dwell time or sign-on interarrival time heavily dependson whichaccess point the device is associated with wouldalso be useful. It is also possible that dwell time will be different depending on the direction that the mobile was moving whenit entered the micro-cell. Direction is a newarea that could be explored in its ownright. Further analysis could determine whetherit is possible to predict the direction of the next handoff, perhaps using the direction of previous handoffs, the duration of previous dwell times, the numberof handoffs since the last sign-on, the duration of previoussign-on interarrival times, and other factors.
Acknowledgement This work was supported in part by the National Science Foundation under Grant NCR9706491.
47
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50
