Deriving the Vehicle Speeds from a Mobile
Telecommunications Network
Ren-Huang Liou, Yi-Bing Lin, Fellow, IEEE, Yu-Long Chang, Hui-Nien Hung, Nan-Fu Peng, and Ming-Feng Chang
Abstract—Vehicle speeds are often measured by intelligent transportation systems (ITSs) by utilizing sensors or soft-ware solutions. Our previous work proposed the Lin–Chang– Huangfu (LCH) scheme to compute the cell residence times by the standard counter values in mobile telecommunications switches. In this paper, we use mathematical and statistical developments to investigate the accuracy of the LCH scheme by deriving the bias of the cell residence times computed in this scheme. Then, we extend the LCH scheme with some filtering and compensation techniques for vehicle speed estimation and validate our approach with vehicle detector (VD) measurements at National Highway 3, Longtan Township, Taoyuan County, Taiwan. Our study indicates that the LCH scheme is an effective approach to vehicle speed estimation.
Index Terms—Lin–Chang–Huangfu (LCH) scheme, mobile switching center (MSC), telecommunication, vehicle speed.
I. INTRODUCTION
M
OST intelligent transportation systems (ITSs) measure vehicle speeds to assist drivers in estimating travel times and to avoid traffic jams. To provide this service, an ITS server is responsible for collecting and computing vehicle speeds. This traffic information can be accessed by users through networks such as the Internet.The ITS server can obtain traffic information from a mo-bile telecommunications network. The intuition behind this
Manuscript received October 28, 2012; revised February 4, 2013 and March 15, 2013; accepted March 20, 2013. Date of publication April 12, 2013; date of current version August 28, 2013. The work of R.-H. Liou was supported by the MediaTek Fellowship. The work of Y.-B. Lin was supported in part by the National Science Council under Grant NSC 101-2221-E-009-032, by the Academia Sinica under Grant AS-102-TP-A06, by Chunghwa Telecom, by the National Chiao Tung University/Industrial Tech-nology Research Institute Joint Research Center Research Project, by the Information and Communications Research Laboratory/Industrial Technology Research Institute (ICL/ITRI) Project under Grant C352BW1100 and Grant C301EA3600, by the Department of Industrial Technology through the Aca-demic Technology Development Program under Grant 101-EC-17-A-03-S1-193, and by the Minstry of Education Aim for the Top University and Elite Research Center Development Plan. The work of M.-F. Chang was supported in part by Chunghwa Telecom and in part by the ICL/ITRI Project under Grant TL-101C019. The Associate Editor for this paper was F. Zhu.
R.-H. Liou, Y.-L. Chang, and M.-F. Chang are with the Department of Com-puter Science, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: rhliou@cs.nctu.edu.tw; ylchang@cs.nctu.edu.tw; mfchang@cs.nctu.edu.tw).
Y.-B. Lin is with the Department of Computer Science, National Chiao Tung University, Hsinchu 300, Taiwan, and also with the Institute of Information Science and the Research Center for Information Technology Innovation, Academia Sinica, Taipei 115, Taiwan (e-mail: liny@cs.nctu.edu.tw).
H.-N. Hung and N.-F. Peng are with Institute of Statistics, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: hhung@stat.nctu.edu.tw; nanfu@stat.nctu.edu.tw).
Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TITS.2013.2255039
approach is described as follows. When a person runs faster, the person passes telephone poles on the side of the road more frequently. Similarly, when a car travels down a road, it will pass cell phone towers [base stations (BSs)] more often. The length of time a cell phone is connected to a particular BS will vary inversely with speed. In fact, the speed of the vehicle can be estimated just by dividing the length of the road covered by a particular BS by the difference between the time when it left the cell (radio coverage of the BS) and when it entered that cell. Information about when “handing over” from one BS to another is available in the mobile telecommunications network so that the speed can be estimated if the intersections of cells relative to the road on which is being travelled is also known. Of course, the vehicle itself already has more accurate means of estimating its velocity, but speedometer readings are not typically accessible externally. In addition, the Global
Position-ing System (GPS) provides a much more accurate method of
estimating speed, but running GPS continuously takes a lot of power, and some cell phones do not even have a GPS receiver. The alternative method based on frequency of handovers could provide speed estimation for map-based information on traffic to other travelers. Such a system would require installation of a central server (i.e., the ITS server) to gather all the information and to make the resulting summary traffic information available over the Internet. These statistics include 1) the number of handovers into each cell, 2) the number of handovers out of each cell, and 3) the total voice traffic. Here, voice traffic is measured as the sum of the call holding times for all calls.
Our previous work [1]–[3] proposed the Lin–Chang–
Huangfu (LCH) scheme to estimate the cell residence time (the
time periods that user equipment (UE) stays in the cells) by the ratio of the voice traffic and the number of handovers into the cell. The cell residence time in turn can be used to estimate the speed. This paper is a major extension of our previous conference paper [4]. We compute the cell residence times of the LCH scheme to estimate the vehicle speeds and validate the LCH scheme with the vehicle detector (VD) measurements at National Highway 3, Longtan Township, Taoyuan County, Taiwan. A major contribution of this paper is the investigation on the bias for the cell residence times computed by the LCH scheme. This kind of bias derivation has not been found in the literature, which shows that the LCH scheme is an appropriate approach to vehicle speed estimation.
This paper is organized as follows. Section II describes the related work. Section III describes how the LCH scheme computes the cell residence times. Section IV derives the bias of the cell residence times computed by the LCH scheme. Section V proposes several techniques for improving the 1524-9050 © 2013 IEEE
Fig. 1. Simplified mobile telecommunications network architecture. accuracy of the vehicle speed estimation. Section VI inves-tigates the performance of the LCH scheme by numerical examples, and the conclusions are given in Section VII.
II. RELATEDWORK
The vehicle speed measurement approaches can be classified into three categories.
• VDs [5]: The VDs [see Fig. 1(a)] are installed in roads to measure the speeds of the vehicles [see Fig. 1(b)], and the speeds are reported to the ITS server [see Fig. 1(c)] through a wireline or a wireless network.
• GPS-based vehicle probes (GVPs) [6], [7]: The user
equipment (UE; i.e., mobile devices) in the vehicles are
equipped with GPS receivers, which send the GPS coordi-nates and time information through the mobile telecom-munications network [i.e., the BSs in Fig. 1(d) and the
mobile switching center (MSC) in Fig. 1(e)] to the ITS
server. The ITS server computes the speeds according to the received GPS coordinates.
• Cellular floating vehicle data (CFVD) [8]–[10], [16], [17]: The network probes [see Fig. 1(f)] are installed to monitor the signals between the BSs and the MSC and then to send them to the ITS server. The network probe also replaces the user identities and the phone numbers in the signals by their hash values from the one-way hash function to protect the user privacy [18]. Based on the call activities of the UE devices, the ITS server tracks the locations of the UE devices at the cell level and estimates the cell residence times of the UE devices to derive their moving speeds. The VD approach is typically deployed by the transporta-tion department of the government. On the other hand, the GVP and CFVD approaches are typically developed by the telecommunication operators. The VD approach suffers from high construction and maintenance costs for detectors [8], [10]. The GVP approach requires that the UE devices are equipped with the GPS receivers, and it consumes extra radio resources to transmit the GPS data. Furthermore, our experience (with Chunghwa Telecom) indicates that not many vehicles with GPS receivers travel in some suburb/country roads, and the traffic information for these roads is difficult to obtain from the GPS
data. Therefore, the GVP approach is typically used in urban areas. For suburb/country areas, the vehicle speeds are obtained by the CFVD approach, which requires extra signaling links and network probes in the mobile telecommunications network to monitor the signaling messages delivered between the UE devices and the mobile core network.
Our previous work [1]–[3] proposed the LCH scheme. The LCH scheme derives the cell residence times from the standard counter values that are automatically and periodically collected by the MSCs. The details of the LCH scheme will be described in Section III.
III. LIN-CHANG-HUANGFUSCHEME
The LCH scheme was described in [1], and the details are reiterated here for the reader’s benefit. In Fig. 1, the MSC [see Fig. 1(e)] is responsible for the call processing and mobility management [11]. The MSC is connected to a group of BSs. The radio coverage of a BS is a cell [see the dashed circles in Fig. 1(g)]. During a phone conversation, the UE in a cell connects to the MSC through the BS. If the UE in conversation moves from one cell to another, then the call path is switched from the old cell to the new cell. This process is referred to as
handover.
In a standard commercial mobile telecommunications oper-ation, the MSC records the call activities of the UE devices (e.g., when the UE makes/receives a call or when the UE in a conversation hands over from one cell to another). The MSC collects the statistics of the activities in each cell for every Δt interval that are typically ranging from 15 min to several hours. Two of the statistics are the number of handovers in and out of the cells and the voice traffic (in Erlang) of the cells. Consider a time point τ . We define Δτ as the time slot ((τ/Δt)Δt, (τ/Δt) + 1Δt). For purposes of description, we define the road segment for speed estimation as the target
road segment [see Fig. 1(h)]. The average cell residence time
of cell i [see Fig. 1(g)] is derived as follows. Let ρ(τ ) be the carried traffic of cell i in Δτ . In other words, ρ(τ ) is the number of calls arriving at cell i in Δτ multiplied by the expected carried call holding times (in minutes). The carried call holding time is different from the offered call holding time. The offered call holding time is the duration (in minutes) of a call if there were unlimited radio channels in all BSs and if the new calls and the handovers are always successful. In reality, the capacity of a BS is limited; therefore, a call attempt may be blocked at the beginning or may be forced to terminate in a handover. For a call that is connected, the call minutes are actually measured at the MSC in the ρ(τ ) statistics and are defined as the carried call holding time tc. In the duration tcof a carried call measured
in the MSC, the call is not blocked at the beginning and is not forced to terminate at handovers occurring in tc (although it
may be forced to terminate at the end of tc). Let γ(τ ) denote
the number of handovers into cell i in Δτ . Let tmbe the cell
residence time. In [1], we estimated the cell residence time
t∗m(τ ) of the UE arriving at cell i in time slot Δτ as
t∗m(τ ) = ρ(τ )
Based on (1), the average vehicle speed of the one-way target road segment is derived as follows. Suppose that the vehicle with the UE [see Fig. 1(b)] in cell i− 1 moves to cell i + 1 through cell i. Let x be the length of the target road segment covered by cell i. From (1), the average vehicle speed v(τ ) of the target road segment in Δτ can be computed as
v(τ ) = x t∗m(τ ) =
xγ(τ )
ρ(τ ) . (2)
Now consider a two-way road. To compute the average speed of each direction in the two-way road, we first determine the moving directions of the UE devices by their handover sequences of cells. Then, based on the moving directions, we compute the γ(τ ) and ρ(τ ) of each direction. The average speed of each direction is computed by the γ(τ ) and ρ(τ ) of each direction by using (2).
In the following, we derive the bias of the cell residence times computed by the LCH scheme. The bias will indicate that the LCH scheme is a good measure for estimating the vehicle speeds.
IV. BIAS OF THECELLRESIDENCETIMEESTIMATION Here, the accuracy of the LCH scheme is evaluated by estimating the bias of this scheme. We will show that the LCH scheme has a smaller bias for the vehicle speed estimation than the general cell residence time estimation; therefore, the LCH scheme is an appropriate approach to estimate the vehicle speeds.
Let tm be the real cell residence time and t∗m(τ ) be the
estimator of tm. Then, the bias of the cell residence time
estimator is defined as
btm(t
∗
m(τ )) = E [t∗m(τ )]− E[tm]. (3)
Note that, in statistics, the bias and the error are not the same. The bias of an estimator is the difference between the estimator’s value and the real value [12]. On the other hand, the error is the discrepancy between an exact value and its approximation. In the following, we first derive γ(τ ) and ρ(τ ) constrained by Δτ . Then, we express the cell residence time estimator t∗m(τ ) by the derived γ(τ ) and ρ(τ ). Finally, (3) is used to compute the bias.
Fig. 2 shows the timing diagram of the activities of six UE devices in an observation time period [t0, t19]. For example,
consider the behavior of UE 3. UE 3 moves to cell i at t8
(marked by), and a call is connected to UE 3 at t11(marked
by◦). UE 3 leaves cell i at t13(marked by), and the call for
UE 3 is completed at t15(marked by•). The cell residence time
for UE 3 is tm= t13− t8. The carried call holding time for
UE 3 is tc= t15− t11.
Consider time slot Δτ = t19− t5 (see the gray area in
Fig. 2). A call observed in cell i in Δτ can be one of the following three types:
• a new call that is originated in Δτ , e.g., the call arrivals at time t10for UE 6 and time t11for UE 3 in Fig. 2;
• an existing call that is already connected in cell i at the beginning of Δτ , e.g., the call for UE 1 originated at t4
Fig. 2. Timing diagram for UE movements and call arrivals (: UE moves to cell i;: UE leaves cell i; ◦: a call arrives; •: a call completes; : Δτ).
before time slot Δτ when UE 1 was in cell i; in other words, UE 1 arrived at cell i before t5, and the call started
in the cell at t4< t5is still in progress;
• a handover call that is switched from another cell to this cell, e.g., the handovers at time t6for UE 2, t9for UE 4,
and t12for UE 5.
Let α(τ ), β(τ ), and γ(τ ) be the numbers of new calls, existing calls, and handover calls in Δτ in cell i, respectively. In Fig. 2, α(τ ) = 2 (i.e., the calls that originated at t10and t11), β(τ ) = 1 (i.e., the call that originated at t4), and γ(τ ) = 3 (i.e.,
the calls that were handed over at t6, t9, and t12). Let λ be the
call arrival rate. It is clear that
E [α(τ )] = λΔτ. (4)
Let tmbe a random variable with mean 1/η and tcbe a random
variable with mean 1/μ. In the Appendix, we prove five facts that lead to the following important theorem.
Theorem 1: Assume that tc and tmhave arbitrary
distribu-tions with the means 1/μ and 1/η, respectively. If time slot Δτ = 1/δ is fixed and if we observe the call activities for a long period t, then
btm(t
∗ m(τ )) =
μδ
λη2. (5)
Equation (5) indicates that the LCH scheme has better ac-curacy when the call holding time is long (i.e., μ is small), Δτ is long (i.e., δ is small), the call arrival rate λ is large, or the cell residence time is short (i.e., η is large). Because the vehicles typically have much shorter cell residence times (i.e., higher speeds) than the pedestrians, (5) indicates that the LCH scheme has higher accuracy in estimating the vehicle speeds than the general cell residence time estimation that also includes pedestrians.
V. TECHNIQUES FORIMPROVING THEACCURACY OF THE VEHICLESPEEDESTIMATION
Several techniques are described here to further improve the accuracy of the vehicle speed estimation expressed in (2). We first introduce two filtering techniques to exclude the UE
Fig. 3. Two scenarios that affect the accuracy of the vehicle speed estimation. (a) Cell may cover several roads and pedestrians. (b) Vehicle may move to another road without leaving the la of a cell.
devices that are not in the target road segment. Then, we describe two compensation techniques to increase the number of observed calls based on the leaky-bucket integration strategy [15]. Note that these techniques may not cover all realistic road configurations. Here, we demonstrate that the accuracy of speed estimation can be improved by the concepts of “filtering” and “compensation.” Based on the filtering/compensation concepts, we can further develop other potential techniques to improve the accuracy and hope that these concepts can be used to develop appropriate techniques in user’s scenarios.
If cell i [1 in Fig. 3(a)] also covers the area other than the target road segment [2 and3 in Fig. 3(a)], γ(τ ) and ρ(τ ) also
include the call activities of the UE devices that are not in the target road segment. These activities may reduce the accuracy of (2). To resolve this issue, we first introduce the standard loca-tion update procedure in a mobile telecommunicaloca-tions network and then show how to identify the UE devices in the target road segment from the UE devices outside the target road segment but in cell i. In the mobile telecommunications network, the cells are grouped into location areas [LAs; e.g., LA j contains cells i− 1 and i. See 4 in Fig. 3(a)]. When a UE [5 in Fig. 3(a)] moves from one LA to another, the UE executes the location update procedure to inform the MSC of its new LA [11], [14]. The location update messages are delivered from the BS to a mobility database (specifically, a visitor location
register) through the MSC. Based on the location update, we
propose the following technique to identify the UE devices in the target road segment.
Filtering Technique 1. For every UE that has call activities in
cell i, let cell A be the cell where the UE performs the location update when entering the LA of cell i, and let cell B be the cell where the UE performs another location update when leaving the LA of cell i. If both cells A and
B cover the road, then the UE is identified as in the target
road segment.
For example, when the UE moves from LA j− 1 to LA
j + 1 through LA j [see 5 in Fig. 3(a)], one location update is performed in cell i− 1 of LA j (i.e., cell A), and another location update is performed in cell i + 1 of LA j + 1 (i.e., cell
B). Because both cells i− 1 and i + 1 cover the road, the UE
is identified as a vehicle moving in the target road segment. Depending on their destinations, some vehicles may be stopped within the LA of cell i or may have moved from the target road to another road [see 1 in Fig. 3(b)]. Those UE devices may not be identified by filtering technique 1 but can be detected by the following technique.
Filtering Technique 2. For every UE that has the call activities
in cell i, if the UE’s handover sequence of cells contains at least three cells which cover the road, the UE is identified in the target road segment.
In Fig. 3(b), if the UE’s handover sequence of cells contains cells {i − 2, i − 1, i}, cells {i − 1, i, i + 1}, or cells {i, i + 1, i + 2}, the UE is identified in the target road segment.
For the UE devices identified by filtering techniques 1 or 2, we use their handover information and call holding times in cell
i to compute γ(τ ) and ρ(τ ) of the target road segment. Then,
the average speed of the target road segment is computed by using (2).
Although these two filtering techniques effectively exclude the UE devices that are not in the target road segment, they also reduce the number of the observed calls. It is clear that, if few calls are observed in Δτ , the samples cannot reflect the actual vehicle speeds of the target road segment. To resolve this is-sue, we consider the far-history and near-history compensation techniques based on the leaky-bucket integration strategy. The far-history compensation technique uses the “same Δτ ” on the same days of the past weeks to compensate for the number of the observed calls. On the same days of the past weeks, the traffic patterns are typically similar, e.g., the traffic patterns of this Monday are similar to those of last Monday. Similarly, for national holidays (e.g., the New Year’s day), we may use far-history compensation of the last national holiday (e.g., the last New Year’s day). For time slot Δτ , let Δτk be the same
time slot on the same day of the most recent kth week, and let Δτ0= Δτ . For example, if Δτ is the time slot on this Monday,
then Δτ1is the same time slot on last Monday. The far-history
compensation technique guarantees that v(τ ) is computed by at least Kh samples of handovers. The details are given in the
following.
Far-History Compensation Technique. The average speed
of the target road segment with the threshold Kh is
computed as v(τ ) = xKk=0γ(τk) K k=0ρ(τk) (6)
where K = min N : N n=0 γ(τn)≥ Kh . (7)
Note that, in (7), if γ(τ ) is no less than Kh, (6) is the same
as (2) (i.e., no historical datum is used).
Because the MSC collects ρ(τ ) and γ(τ ) for every Δt interval, some calls may cross the time slot (i.e., the existing calls). These “crossing time slot” calls may cause a nonsmooth effect on the consecutive time slots. The leaky-bucket strategy can smooth the results if the traffic patterns of the consecutive time slots do not significantly change. However, if the speeds of the consecutive time slots dramatically change (e.g., the difference between these two speeds is larger than a threshold
Vs), then this technique should not be used. The details are
described in the following.
Near-History Compensation Technique. The average speed
of the target road segment is computed as
v(τ )← ⎧ ⎨ ⎩ W v(τ ) + (1− W ) ×v(τ − Δt), for|v(τ) − v(τ − Δt)| < Vs v(τ ), for|v(τ) − v(τ − Δt)| ≥ Vs (8) where 0≤ W ≤ 1 is a weighting factor, v(τ − Δt) is the speed of the preceding time slot of Δτ , and Vs is the
threshold to detect the significant traffic change.
In (8), a larger W value means that the past speed v(τ− Δt) has less of an effect on the current speed v(τ ).
VI. NUMERICALEXAMPLES
Here, the vehicle speeds derived from the LCH scheme and those measured by the VD scheme are compared. We have ob-tained the vehicle speed data of National Highway 3 at Longtan Township, Taoyuan County, Taiwan [see Fig. 4(a)]. The speed data were published by the Ministry of Transportation and Communications of Taiwan, which were measured by the VD at the 66 km of the highway [see Fig. 4(b)]. The wideband
code-division multiple-access BSs are deployed along the highway.
From a sector (cell) of a BS at Longtan Township [about 66.8 km of the highway, which is about 800 m away from the VD; see Fig.4 (c)], we utilize the LCH scheme with filtering techniques 1 and 2, and far-history and near-history compensation techniques to estimate the vehicle speeds from a cell of length x = 1 km and Δt = 1 h. The x length is obtained through measurement [16], [17]. We use the average of the numbers of handovers into the cell and out of the cell to estimate the speeds. Our experiments indicate that, using the average of the numbers of handover into and out of the cell has better accuracy for the LCH scheme. To eliminate the ping-pong effect, we adopt the algorithm in [9] to filter out the interhandover intervals, which are less than 10 s. In this paper, we consider Kh= 10, Vs= 40 km/h, and W = 0.5.
Let vsbe the vehicle speeds measured by the VD (s = VD)
and computed by the LCH scheme (s = LCH), respec-tively. Fig. 5(a) and (b) plot vs from 8:00 to 20:00 on
Fig. 4. Experimental environment in National Highway 3 at Longtan Town-ship, Taoyuan County, Taiwan. (The VD is marked by×, and the BS is marked by◦.)
Fig. 5. Vehicle speeds of the VD and LCH schemes on September 16, 2011 (Δt = 1 h and x = 1 km). (a) Northbound. (b) Southbound.
September 16, 2011, in the northbound and southbound direc-tions, respectively. In these figures, both the VD and the LCH approaches indicated that the traffic did not significantly vary from 8:00 to 20:00 (i.e., vs> 75 km/h). Fig. 6(a) and (b) plots
vsfrom 8:00 to 20:00 on September 25, 2011, in the northbound
and southbound directions, respectively. In Fig. 6(a), both ap-proaches captured a traffic jam occurred from 18:00 to 20:00 (i.e., vs< 60 km/h). In Fig. 6(b), both approaches indicated that
the traffic did not significantly vary during 8:00 to 20:00 (i.e.,
vs> 75 km/h). Figs. 5 and 6 show that the trends of speeds of
the LCH scheme are consistent with those of the VD scheme in both directions.
We further define the discrepancy as
= vLCH− vVD vVD
. (9)
Based on Figs. 5 and 6, Fig. 7 plots the curves in the obser-vation time periods. The figure indicates that are reasonally small in most cases. When the traffic jam occurs, [in Fig. 7(b), the◦ curve from 19:00 to 20:00], is slightly larger. The reason is that, when the traffic jam occurs, the cell residence times of
Fig. 6. Vehicle speeds of the VD and the LCH scheme on September 25, 2011 (Δt = 1 h and x = 1 km). (a) Northbound. (b) Southbound.
Fig. 7. Discrepancy between the LCH and the VD schemes (Δt = 1 h and x = 1 km). (a) Experiment on September 16, 2011. (b) Experiment on September 25, 2011.
the vehicles are typically longer, and (5) already implies that the LCH scheme has lower accuracy for longer cell residence times.
We have also collected γ(τ ) and ρ(τ ) statistics during 8:00 to 20:00 over 49 days (from September 13, 2011 to October 31, 2011) from the mobile telecommunications network. Based on the observed data, this paper indicates that E[] is 14.46% if none of the techniques in Section V is applied, E[] is reduced to 12.7% if filtering techniques 1 and 2 are applied, and E[] is further reduced to 7.51% if both filtering and compensation techniques are all applied.
VII. CONCLUSION
This paper extended the previously proposed LCH scheme [1], [2] to derive the vehicle speeds. The bias analysis indicated that the LCH scheme is an appropriate approach to vehicle speed estimation (as compared with the pedestrian speed es-timation). We further utilized several techniques to improve the accuracy of the LCH scheme and then compared the vehicle speeds derived from the improved LCH scheme with those mea-sured by a VD installed by the government. The comparison study showed that the trends of speeds derived from the LCH and VD schemes are consistent, and the discrepancies between these two schemes are reasonably small in most cases, where
the average discrepancy E[] is 7.51%. Our study indicates that the LCH scheme can appropriately capture the vehicle speeds of the roads and can avoid expensive deployment costs of exist-ing approaches (e.g., sensor deployment of the VD scheme).
One potential issue of CFVD methods is that it only applies to cell phones that are actually connected. This will be typically only a fraction of the cell phones within the cell. This paper has indicated that the number of the connected phones is reasonably large to produce data for our measurements and compensation techniques. Another potential issue is the selection of the time interval Δt for statistical data collection. One may question that speed calculated in a period of 15 min or longer is not real time. Our experience indicates that the 15-min interval (or even longer) is appropriate. The traffic information broadcast from the government in Taiwan uses the 15-min interval or longer. Many GVP approaches also use similar intervals to report the traffic information.
In the future, we will investigate the speed estimation in the Long-Term Evolution environment and evaluate the per-formance of the LCH scheme based on packet-switching data connections. We will also develop more techniques to enhance the accuracy of the speed estimation for several scenarios (e.g., parallel road effect, few LA crossings, tourist effect, etc.).
APPENDIX
DETAILEDPROOF OFTHEOREM1
Here, we describe the proof for Theorem 1. We first prove the following facts.
Fact 1: Assume that tcis exponentially distributed and tm
has arbitrary density function fm(tm) with rate η. If time slot
Δτ has an arbitrary distribution with mean 1/δ, then E[β(τ )] =
λ/μ, and E[γ(τ )] = (λη)/(δμ).
Proof: Consider the start time of time slot Δτ (e.g., t5 in
Fig. 2). This time point observes the call activities of cell i. The expected number E[β(τ )] of existing calls is proportional to
E[α(τ )] and the expected call holding time 1/μ and is inversely
proportional to the expected length E[Δτ ] of the time slot. Therefore, we can express E[β(τ )] as
E [β(τ )] =E [α(τ )] (1/μ) E[Δτ ] =
λ
μ. (10)
Intuitively, if (1/μ) < E[Δτ ], then 1/(μE[Δτ ]) is the proba-bility that a call is an existing call in Δτ . Then, the total number of existing calls E[β(τ )] is E[α(τ )] multiplied by 1/(μE[Δτ ]), which results in (10).
Obviously, E[γ(τ )] can be expressed as E[α(τ )] multiplied by the average number of handovers of a call, Due to the memo-ryless property of the exponential distribution, the residual call holding time (e.g., τc= t15− t13in Fig. 2) of a handover call
is also exponential with the same mean 1/μ. Let the distribution function of tm be Fm(tm). Let τm (e.g., τm= t13− t11 in
Fig. 2) be the residual cell residence time. According to the
residual life theorem [13], the density function of τm is (1−
Fm(tm))/(1/η) = [1− Fm(tm)]η. Note that the measured tc
periods are carried call holding times. Therefore, if a handover occurs, a radio channel is available in the new cell, and the
handover is always successful. Define Pias the probability that
a new call completes after it has handed over i times. Define
q1= Pr[tc> τm] as the probability that a new call is handed
over, and define q2= Pr[τc> tm] as the probability that a call
is handed over under the condition that it has been handed over before. Let Pi= q1qi−12 (1− q2) where i≥ 1. Then, the
average number of handovers of a call can be expressed as
∞ i=1 iPi= ∞ i=1 iq1qi2−1(1− q2) = q1 1− q2 . (11) In (11), probability q1is derived as q1= Pr[tc> τm] = ∞ τm=0 ∞ tc=τm μe−μtc[1− F m(τm)] ηdtcdτm = ∞ τm=0 e−μτm[1− F m(τm)] ηdτm (12)
and 1− q2can be expressed as
1− q2= Pr[τc< tm] = ∞ τc=0 ∞ tm=τc μe−μτcf m(tm)dtmdτc = ∞ τc=0 μe−μτc[1− F m(τc)] dτc = μ η q1. (13)
From (12) and (13), the average number of handovers of a call is q1/(1− q2) = η/μ. In a typical vehicle environment, the
average number of handovers of a call is between 2 and 3. The expected number E[γ(τ )] can be derived as
E [γ(τ )] = E [α(τ )] η μ = λη δμ. (14)
From the given proof, the derivation of E[α(τ )], E[β(τ )], and
E[γ(τ )] are independent of the Δτ distribution.
Based on Fact 1, we have Facts 2 and 3.
Fact 2: Assume that both tc and Δτ are exponentially
distributed with means 1/μ and 1/δ, and tm has arbitrary
density function fm(tm) with rate η. Then, E[ρ(τ )] = λ/(δμ).
Proof: As defined in Fact 1, let the residual life of tc be
τc, and let the residual life of tmbe τm. For time slot Δτ of
cell i, ρ(τ ) is contributed by the conversation minutes of new, existing, and handover calls. Consider the new call for UE 3 during the time slot Δτ in Fig. 2. This call arrives at t11 with tc= t15− t11. At t11, the residual life of UE 3’s cell residence
time is τm= t13− t11. Since the call arrivals are a Poisson
process, t11is a random observer of UE 3’s cell residence time tm. The time point t11is also a random observer of Δτ . From
the residual life theorem and the memoryless property of the exponential distribution, the residual life Δτnof Δτ seen by t11
is also exponentially distributed with the mean 1/δ. Therefore, the expected call minutes E[ρn(τ )] of this new call in Δτ is
E [ρn(τ )] = E [min(tc, τm, Δτn)]
= E [min (min(tc, Δτn), τm)] . (15)
Clearly, random variable xn= min(tc, Δτn) is exponentially
distributed with the rate μ + δ. Let the Laplace transform of
fm(tm) be fm∗(tm). From the derivation in [14], (15) is
rewrit-ten as E [ρn(τ )] = E [min(xn, τm)] = 1 μ + δ − η (μ + δ)2 [1− fm∗(μ + δ)] . (16) Now consider the existing call for UE 1 during Δτ in Fig. 2. This call arrives at t4, At t5, the residual call holding time is τc= t16− t5, and the residual life of UE 1’s cell residence
time is τm= t18− t5. Since t5can be considered as a random
observer of UE 1’s call holding time tc, τc is exponentially
distributed with mean 1/μ. Then, the expected call minutes
E[ρe(τ )] of this existing call in Δτ is
E [ρe(τ )] = E [min(τc, τm, Δτ )] .
Since both τc and Δτ are exponentially distributed with the
rates μ and δ, E[ρe(τ )] can be expressed as (16); in other words,
E[ρe(τ )] = E[ρn(τ )].
Finally, consider the handover call for UE 4 during Δτ in Fig. 2. This user moves to cell i at t9 with the cell residence
time tm= t17− t9. At t9, the residual life of the call holding
time is τc= t14− t9. Since t9is a random observer of UE 4’s
call holding time tc, which is exponentially distributed with
mean 1/μ, τc has the same distribution as tc. The time point
t9is also a random observer of Δτ , and the residual life Δτhof
Δτ seen by t9is also exponentially distributed with mean 1/δ.
Therefore, the expected call minutes E[ρh(τ )] of this handover
call in Δτ is
E [ρh(τ )] = E [min(τc, tm, Δτh)]
= E [min (min(τc, Δτh), tm)] . (17)
Let xh= min(τc, Δτh), which is exponentially distributed
with the rate μ + δ. Then, similar to the derivation of (16), (17) is rewritten as E [ρh(τ )] = E [min(xh, tm)] = 1 μ + δ [1− fm∗(μ + δ)] . (18)
From (4), (16), (18), and Fact 1, E[ρ(τ )] is expressed as
E [ρ(τ )] = E [α(τ )] E [ρn(τ )] + E [β(τ )] E [ρe(τ )] + E [γ(τ )] E [ρh(τ )] = (E [α(τ )] + E [β(τ )]) × 1 μ + δ− η (μ + δ)2 [1− fm∗(μ + δ)] + E [γ(τ )] 1 μ + δ [1− fm∗(μ + δ)] = λ δμ. (19)
In (19), E[ρ(τ )] is independent of the tmdistribution.
Fact 3: Let Δτ be a fixed value (1/δ). If the call holding
time tc and the cell residence time tm are exponentially
dis-tributed, then E[ρ(τ )] = λ/(δμ).
Proof: Like the proof of Fact 2, ρ(τ ) is contributed by the
conversation minutes of new, existing, and handover calls. The expected call minutes E[ρn(τ )] of the new call in Δτ is
E [ρn(τ )] = E [min(yn, Δτn)]
= E [min (min(tc, τm), Δτn)] (20)
which is the same as (15), except that Δτn is uniformly
distributed over the interval [0, (1/δ)] with density function δ. Since both tc and τm are exponentially distributed with
rates μ and η, respectively, random variable yn= min(tc, τm)
is exponentially distributed with rate μ + η, and (20) is rewritten as E [ρn(τ )] = 1 δ Δτn=0 Δτn yn=0
δyn(μ + η)e−(μ+η)yndyndΔτn
+ 1 δ Δτn=0 ∞ yn=Δτn δΔτn(μ+η)e−(μ+η)yndyndΔτn = 1 μ + η 1− δ μ + η+ δe−(μ+ηδ ) μ + η . (21)
Now, consider the expected call minutes E[ρh(τ )] of the
han-dover call in Δτ , i.e.,
E [ρh(τ )] = E [min(τc, tm, Δτh)] . (22)
Both τc and tm are exponentially distributed with the rates
μ and η, and Δτh is uniformly distributed over the
inter-val [0, (1/δ)]. Therefore, (22) can be expressed as (20); in other words, E[ρh(τ )] = E[ρn(τ )]. The expected call minutes
E[ρe(τ )] of the existing call in Δτ is
E [ρe(τ )] = E min ye, 1 δ = E min min(τc, τm), 1 δ (23) where ye= min(τc, τm) is exponentially distributed with the
rate μ + η, and (23) is rewritten as
E [ρe(τ )] = 1 δ ye=0 ye(μ + η)e−(μ+η)yedye + ∞ ye=1δ 1 δ (μ + η)e−(μ+η)yedy e = 1 μ + η 1− e−(μ+ηδ ) . (24)
From (4), (21), (24) and Fact 1, E[ρ(τ )] is expressed as
E [ρ(τ )] = E [α(τ )] E [ρn(τ )] + E [β(τ )] E [ρe(τ )] + E [γ(τ )] E [ρh(τ )] = (E [α(τ )] + E [γ(τ )]) 1 μ + η × 1− δ μ + η+ δe−(μ+ηδ ) μ + η + E [β(τ )] 1 μ + η 1− e−(μ+ηδ ) = λ δμ.
Fact 4: Assume that Δτ , tc, and tm have arbitrary
dis-tributions with means 1/δ, 1/μ, and 1/η, respectively. Then,
E[γ(τ )]≈ (λη/δμ).
Proof: Consider a long observation time period t 1/δ,
1/μ. Let t∗ be the time that is occupied by the calls to a user during period t. Then, there are(t∗/(1/μ) = μt∗ call
arrivals within t. Note that the user is expected to move across cells t/(1/η) times in period t, and among these crossings,
t∗/(1/η) of them occur during conversations. Therefore, the
number of handovers is t∗/(1/η) = ηt∗, and the average number of handovers of a call is E[n1] = (ηt∗)/(μt∗).
Clearly, limt∗→∞E[n1] = η/μ; therefore, if t is long enough,
we have
E [γ(τ )] = E [α(τ )] E[n1]≈ λη
δμ. (25)
The result of Fact 4 is consistent with that of Fact 1.
Fact 5: Assume that Δτ = 1/δ is fixed, and tc and tm
have arbitrary distributions with means 1/μ and 1/η, respec-tively. Let t be the observation period. Then, limt→∞E[ρ(τ )] = λ/(δμ).
Proof: We assume that the UE devices are moving around n2cells, and their behaviors are observed in a long time period t n2/δ, n2/μ. Since the UE devices will not leave n2cells,
handovers of these calls (therefore, the tmdistribution) will not
affect the statistics of call minutes. There areλtn2 calls that
arrive in n2cells during period t. Some of the calls are initiated
before the beginning of t, which are not counted (we call it the initial effect). Some of the counted calls do not complete at the end of t (we call it the end effect). Let tc, ibe the call holding
time of the ith call, and let fc(tc, i) be the density function
of the call holding time of the ith call with the rate μ. Then, the measured call minutes θ of these n2 cells during t can be
approximated as θ = λtn2 i=1 ∞ tc,i=0 tc, ifc(tc, i)dtc, i=λtn2 μ .
There are a total of n2t/(1/δ) = n2tδ time slots. Since E[ρ(τ )] is the expected call minutes of traffic in Δτ of cell i,
it can be approximated as E[ρ(τ )]≈ θ/(n2tδ). If t becomes
large, the initial and end effects become insignificant, and lim
t→∞E [ρ(τ )] =
λ
δμ. (26)
Note that the result in Fact 5 is the same as those in Facts 2 and 3. These results are also consistent with the Erlang equa-tion, which defines the traffic per minute as λ/μ. Therefore,
λ/(δμ) is the traffic in time slot Δτ . Theorem 1: Assume that tc and tm have arbitrary
distri-butions with the means 1/μ and 1/η, respectively. If time slot Δτ = (1/δ) is fixed and if we observe the call activities for a long period t, then btm(t∗m(τ )) = (μδ)/(λη2).
Proof: Suppose that we observe the call activities for a long
period t→ ∞. Since the measurements of the LCH scheme are conducted in Δτ , from (25) and (26), both E[γ(τ )] and
E[ρ(τ )] go to infinity if 1/δ becomes infinity, and (1) cannot
be used to compute t∗m(τ ). To fix this problem, let c1=
(E[ρ(τ )])/(1/δ) = λ/μ and c2= (E[γ(τ )])/(1/δ) = (λη)/μ.
Then, we can rewrite (1) as function f (r, s), such that
t∗m(τ ) = f (r, s)|r=c1,s=c2 = r s r=c1,s=c2 . (27) To compute (27), we use the Taylor series for f (r, s) around the point (c1, c2), which is shown in the following:
f (r, s) = f (c1, c2) + ∂f (r, s) ∂r r=c1,s=c2 (r− c1) + ∂f (r, s) ∂s r=c1,s=c2 (s− c2) + 1 2 ∂2f (r, s) ∂r2 r=c1,s=c2 (r− c1)2 + ∂ ∂s ∂f (r, s) ∂r r=c1,s=c2 (r− c1)(s− c2) + 1 2 ∂2f (r, s) ∂s2 r=c1,s=c2 (s− c2)2+· · · . (28) Since we consider the f (r, s) function around point (c1, c2)
and the expected value E[r] = c1and E[s] = c2, we have E[r− c1] = 0, E[s− c2] = 0 (29) E [(r− c1)(s− c2)] = 0. (30)
In addition, from (27), we have
∂2f (r, s) ∂r2 r=c1,s=c2 = 0 (31) ∂2f (r, s) ∂s2 r=c1,s=c2 =2c1 c3 2 . (32)
From (29)–(32), the expectation of (28) can be written as
E [f (r, s)] = f (c1, c2) + 1 2 ∂2f (r, s) ∂s2 r=c1,s=c2 × E(s− c2)2 =c1 c2 + c1 c3 2 E(s− c2)2 =1 η + μ2 λ2η3 E(s− c2)2 . (33)
In (33), E[(s− c2)2] = Var[γ(τ )/(1/δ)] = δ2Var[γ(τ )]. From
(14), γ(τ ) is a Poisson random variable with the mean
E[γ(τ )] = (λη)/(δμ) and variance Var[γ(τ )] = (λη)/(δμ).
Therefore, (33) is rewritten as E [f (r, s)] =1 η + μ2 λ2η3 Var γ(τ ) 1 δ =1 η + μδ λη2. (34)
Since the mean of tmis 1/η, from (3) and (34), btm(t∗m(τ )) is
btm(t∗m(τ )) = E [t∗m(τ )]− E[tm] = 1 η + μδ λη2 −1 η = μδ λη2. (35) ACKNOWLEDGMENT
The authors would like to thank the four anonymous review-ers. Their comments have significantly improved the quality of this paper.
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Ren-Huang Liou received the B.S. and M.S.
de-grees in computer science from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2007 and 2009, respectively. He is currently working toward the Ph.D. degree with NCTU.
His current research interests include voice over Internet Protocol, mobile computing, and perfor-mance modeling.
Yi-Bing Lin (M’96–SM’96–F’03) received the
B.S.E.E. degree from National Cheng Kung Uni-versity, Tainan, Taiwan, in 1983, and the Ph.D. de-gree in computer science from the University of Washington, Seattle, WA, USA, in 1990.
He is currently a Senior Vice President and a Life Chair Professor with the College of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. He is also with the Institute of Information Science and the Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan. He is also a member of the Board of Directors of Chunghwa Telecom. He is the author of Wireless and Mobile Network Architecture (Wiley, 2001),
Wireless and Mobile All-IP Networks (Wiley, 2005), and Charging for Mobile All-IP Telecommunications (Wiley, 2008).
Mr. Lin is a Fellow of the Association for Computer Machinery, the American Association for the Advancement of Science, and the Institution of Engineering and Technology. He received numerous research awards, including the 2005 National Science Council Distinguished Researcher and the 2006 Academic Award of Ministry of Education.
Yu-Long Chang received the B.S. degree in
com-puter science from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 2011. He is currently working toward the M.S. degree with NCTU.
His current research interests include cellular floating vehicle data, design and analysis of personal communications service networks, mobile comput-ing, and performance modeling.
Hui-Nien Hung received the Ph.D. degree in
statis-tics from the University of Chicago, Chicago, IL, USA, in 1996.
He is currently a Professor with the Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan. His research interests include applied prob-ability, biostatistics, statistical inference, statistical computing, and industrial statistics.
Nan-Fu Peng received the B.S. degree in applied
mathematics from National Chiao Tung University (NCTU), Hsinchu, Taiwan, in 1981 and the Ph.D. degree in statistics from Ohio State University, Columbus, OH, USA, in 1989.
He is currently an Associate Professor with the Institute of Statistics, NCTU. His research interests include Markov chains, population dynamics, and queueing theory.
Ming-Feng Chang received the B.S. and M.S.
de-grees in electrical engineering from National Taiwan University, Taipei, Taiwan, in 1982 and 1984, respec-tively, and the Ph.D. degree in computer science from the University of Illinois at Urbana-Champaign, Urbana, IL, USA, in 1991.
From 2003 to 2005, he served as the Chair of the Department of Computer Science and Informa-tion Engineering, NaInforma-tional Chiao Tung University, Hsinchu, Taiwan, where he is currently a Professor with the Department of Computer Science. His cur-rent research interests include the design and analysis of Internet communica-tions, personal communications networks, and traffic information systems.
![Fig. 2 shows the timing diagram of the activities of six UE devices in an observation time period [t 0 , t 19 ]](https://thumb-ap.123doks.com/thumbv2/9libinfo/7569260.124964/3.891.462.831.102.351/fig-shows-timing-diagram-activities-devices-observation-period.webp)
