Hui-Nien Hung1, Yi-Bing Lin2*and Chao-Liang Luo2,3 1Institute of Statistics National Chiao Tung University, Hsinchu, Taiwan
2Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan 3Telecommunication Laboratories Chunghwa TelcomCo., Ltd., Taiwan
ABSTRACT
In the broadband era, narrowband short message service (SMS) is still the most popular wireless data service. Many studies have been conducted to investigate the performance of SMS based on the arrival rates of short messages. From Chunghwa Telecom’s commercial SMS call data records, we observed that even if the SMS arrival rates are the same, the distribu-tions for the number of SMS arrivals per half hour are quite different for various observed days. We further identify that for the SMS traffic in a specific day, there are non-burst and burst periods. This paper investigates the SMS behaviors on weekdays, weekends, and holidays (specifically, new years’ days and eves). With the assistance of kernel-based fitting method, we derive the SMS arrival number distributions of various traffic types and observed days. Our approach fits each SMS arrival number distribution by three cubic polynomial functions that can accurately capture the SMS behaviors. On the basis of the SMS arrival number distributions derived from our model, the mobile operators have better understanding about the volumes of short messages in different times and days, which can be used to design more flexible short message charging rates. Copyright © 2012 John Wiley & Sons, Ltd.
KEYWORDS
arrival distribution; kernel-based fitting; mobile telecommunications network; short message service (SMS) *Correspondence
Yi-Bing Lin, Department of Computer Science and Information Engineering, National Chiao Tung University. E-mail: [email protected]
1. INTRODUCTION
Short message service (SMS) contributes about 60% of the mobile data service revenue today [1]. This statistic indi-cates that even in the broadband area, narrowband SMS is still the most popular wireless data service. Many business applications, such as stock service, personal identification verification service, weather casting service or daily news service, can be delivered to the customers through SMS [2–5]. Figure 1 shows the SMS architecture for univer-sal mobile telecommunications system (UMTS) [6–9]. In this architecture, a short message sent to a user equip-ment (UE, Figure 1(c)) can be originated from another UE (Figure 1(a)), where the short message is first sent to the short message-service center (SM-SC, Figure 1(g)) through the originating UMTS terrestrial radio access net-work (Figure 1(d)), the mobile-originating mobile switch-ing center (Figure 1(e)) and the inter-workswitch-ing mobile switching center (Figure 1(f)). Upon receipt of a short mes-sage, the SM-SC sends the message to the gateway MSC
(GMSC, Figure 1(h)). The GMSC interrogates the home subscriber server (Figure 1(i)) to identify the mobile termi-nating MSC (Figure 1 (j)) of the recipient and forwards the message to the mobile terminating MSC. Finally, the short message is delivered to the terminating UE (Figure 1(c)) via the terminating UMTS terrestrial radio access network (Figure 1(k)). In Chunghwa Telecom, (CHT, the largest telecom operator in Taiwan), the SM-SC and SMS-GMSC are collocated.
The short message can also be sent from an exter-nal application (typically on the Internet; see Figure 1(l)) to the SM-SC through the external short message entity (Figure 1(b)) by using short message peer-to-peer protocol [6]. In Chunghwa Telecom, 30.48% of the short messages are originated from the external short message entity.
When the SM-SC receives a short message, a call data record (CDR) is created for billing and other adminis-tration purposes. The CDR records the SMS arrival time information, which is important for SMS traffic engineer-ing. For example, in the work of Petros et al., the SMS
Figure 1. UMTS-based SMS architecture.
arrival rate was used for modeling SMS reliability to investigate a significant risk for emergency alerts in New Year during 2005 [10]. Sou et al. investigated the SMS re-sending traffic through the SMS arrival rate [11]. Wu et al. proposed a new mechanism for real-time content monitor-ing and filtermonitor-ing through SMS arrival [12]. Although these studies provide very useful insights into the SMS beha-viors, conclusions of these studies could be strengthened if the SMS arrivals were described with more accurate distributions for the numbers of SMS arrivals instead of their means (i.e., arrival rates). For the description purpose, we use the short term ‘arrival distribution’ to represent ‘distribution for the number of SMS arrivals’.
In this paper, the collection data from Chunghwa Telecom’s SMSC are more than 10 million short message generated from 100,000 users between 2007 and 2010. We show that the SMS arrival behaviors are significantly dif-ferent for various traffic types and observed days. Then, with the assistance of the kernel-based fitting method [13], we derive eight types of SMS arrival distributions. The paper is organized as follows. Section 2 describes his-tograms for the number of SMS arrivals from a macro view. Section 3 presents the fitting model to derive the SMS arrival distributions. Finally, Section 4 concludes this study and outlines the future work.
2. SHORT MESSAGE SERVICE
ARRIVAL BEHAVIOR: A
PRELIMINARY VIEW
This section presents the SMS arrival histograms based on more than 10 million SMS CDRs created in different time periods. Let NF ;d.T ) be the average number of SMS arrivals during the 30-min period [T , T C 30 min) on the d th day of February 2010. Let S1be the set of the week-days in February 2010, and S1be the set of the weekends in February 2010. Denote NF.T / and NF.T / as
NF.T / D P d 2S1NF ;d.T / jS1j and NF.T / D P d 2S 1NF ;d.T / ˇ ˇS 1 ˇ ˇ (1)
That is, NF.T / is the average number of SMS arrivals during [T , T C 30 min) for a weekday in February 2010, and NF.T / is that for a weekend.
Figure 2 plots the histograms of Log(NF.T // and Log(NF.T //. Several trends are observed in the figure.
ing the 30-min period [T , T C 30 min) on the New Year’s Day for year y, and NN;y.T / as that on the
New Year’s Eve for year y. Similarly, denote NL;y.T / and NL;y.T / as those in Lunar New Year’s Day and
Lunar New Year’s Eve for year y, respectively. Let S2D f2007; 2008; 2009; 2010g, and let NN.T / D P y2S2NN ;y.T / jS2j ; NN.T / D P y2S2NN;y.T / jS2j NL.T / D P y2S2NL;y.T / jS2j ; and NL.T / D P y2S2NL;y.T / jS2j (2) That is, NN.T / is the average number of SMS arrivals dur-ing the 30-min period [T , T C 30 min) in the New Year’s Days for 2007, 2008, 2009, and 2010. Similarly, NN.T /
is the average number of New Year’s Eves, NL.T / is that of Lunar New Years’ Days, and NL.T / is that of Lunar
New Years’ Eves, respectively.
Figure 3(a) plots the Log(NN.T // and the Log(NL.T //
curves for (Lunar) New Year’s Eve. Figure 3(b) plots the Log(NN.T // and the Log(NL.T // curves for (Lunar) New Year’s Day. These curves show that the trends of SMS traffic for the New Year’s Day (Eve) and the Lunar New Year’s Day (Eve) are similar. Both NN.T / and NL.T /
tend to decrease from 0:00 to 6:00, and then increase from 6:00 to midnight. The peak traffic occurs at midnight of the (Lunar) New Year’s Eve. In general, the NL.T / and
the NN.T / values are larger than the NL.T / and the
NN.T / values. Also, the trends of the (Lunar) New Year curves are similar to that of the (Lunar) New Year’s Eve curves except that NN.T / (NL.T // decreases from 10:00 to midnight.
To closely investigate the histograms of Log(NF.T // and Log(NF.T // curves in Figure 2, we replace the
30-min period T by 1-min period t between 8:00 and 9:59 in Figure 4.
In Figure 4, NF.t / and NF.t / represent the
aver-age number of SMS arrivals during [t , t C 1 min) in a weekday and a weekend, respectively. We found that a large volume of short messages are issued periodically in Figure 4. In every 30-min period, a large volume of short messages occur in a 3-min period called the burst period (see areas (a), (b), (c), and (d) in Figure 4). The traffic in the remaining 27 min is much smaller than that in the 3-min
burst traffic during (T , T C 30 min) in the Lunar New Year’s Day for year y 2 S2, and NL;N ;y.T / be the average number of non-burst traffic. That is
NL;y.T / D NL;B;y.T / C NL;N ;y.T /: Similarly, we define NL;B;y.T / and NL;N ;y.T / for
Lunar New Years’ Eves. Denote NF ;B.T / D P d 2S1NF ;B;d.T / jS1j ; NF ;N.T / D P d 2S1NF ;N ;d.T / jS1j ; NF;B.T / D P d 2S 1ˇNF ;B;d.T / ˇS 1 ˇ ˇ ; NF;N.T / D P d 2S1NF ;N ;d.T / ˇ ˇS 1 ˇ ˇ ; NL;B.T / D P y2S2NL;B;y.T / jS2j ; NL;N.T / D P y2S2NL;N ;y.T / jS2j ; NL;B.T / D P y2S2NL;B;y.T / jS2j ; and NL;N.T / D P y2S2NL;N ;y.T / jS2j :
Figure 5 shows the curves of NF ;B.T / (weekday burst) and NF ;N.T /(weekday non-burst). The figure indicates that most burst periods occur in (6:00, 18:00). Compared with the period (6:00, 18:00), there are fewer business activities in (22:00, 6:00), and therefore much less burst periods are observed in this time interval. On the aver-age, the SMS volumes in the burst periods are 39.29% (weekday) and 45.58% (weekend) larger than that in the non-burst periods. To address such variance of SMS traffic in different time intervals, it is desirable to derive various arrival distribution functions based on different observed days and burst types. We will focus on eight traffic types (see Table I) in the remainder of this paper.
3. FITTING OF ARRIVAL
DISTRIBUTIONS
The volume of SMS arrivals to a mobile telecom network is typically very large, and these arrivals can be viewed as
452 Wirel. Commun. Mob. Comput. 2014; 14:450–459 © 2012 John Wiley & Sons, Ltd.
(a)
New Year’s Eve and Lunar New Year’s Eve(b)
New Year’s Day and Lunar New Year’s DayFigure 3. Average numbers of SMS arrivals per half hour for a (Lunar) New Year’s Day and a (Lunar) New Year’s Eve.
Figure 5. The numbers of SMS arrivals in the burst and the non-burst periods in a weekday.
Table I. Eight types of SMS arrival distributions.
Type Description
F , B Burst traffic in a weekday F , N Non-burst traffic in a weekday F,B Burst traffic in a weekend F,N Non-burst traffic in a weekend L, B Burst traffic in a Lunar New Year’s Day L, N Non-burst traffic in a Lunar New Year’s Day L,B Burst traffic in a Lunar New Year’s Eve L,N Non-burst traffic in a Lunar New Year’s Eve statistically independent. Therefore, for each type-x traffic
(where x 2 f.F ; B/, (F ; N ), (F; B), (F; N ), (L; B), (L; N ), (L; B), (L; N /g), the traffic can be modeled by a non-homogeneous Poisson process with the arrival rate function rx.T /. Let S be the set of the days considered in (1) and (2), then Nx.T / is the average number of type-x SMS arrivals in (T ; T C 30 min) over the days in S . Statis-tically, Nx.T / can be seen as a random sampled data, and rx.T / D limjsj!1Nx.T /, which is the expected number of SMS arrivals that occur per unit time. Because the mea-sured data from CHT’s commercial operation are drawn from a limited size of S D S1,S1, or S2 in this study, Nx.T / is an approximation of rx.T /.
The Nx.T / function can be fit by a non-parametric estimation Orx.T / through a kernel-based approximation method [13,14]. For every T , this method considers the points T2 T .; T / D ŒT .=2/; T C .=2/ and then scales Twith the Nadaraya–Waston kernel weighted fac-tor K.T; T / [10,11] to derive the estimation Orx.T /, where K T; TD 8 ˆ ˆ ˆ ˆ ˆ < ˆ ˆ ˆ ˆ ˆ : 3 4 " 1 jT T j 2# ; if ˇ ˇ ˇ ˇ jT T j ˇˇˇ ˇ < 1 0; otherwise (3)
On the basis of (3), we express Orx.T / as
Orx.T / D P T2T .;T /K.T; T / Nx.T/ P T2T .;T /K.T; T / (4)
In (4), K.T; T / is a weight assigned to Tbased on its distance from T , and parameter specifies the width of the neighborhood used to estimate rx.T /. The value of parameter can be determined by using cross validation method [13] or observed directly from the measured data. When is larger, the Orx.T / curve becomes smoother. As a non-bust traffic example, Figure 6(a) shows the curves for NF ;N.T / and OrF ;N.T / with D 2, 4, and 6. When is larger, the OrF ;N.T / curve becomes farther away from NF ;N.T /. The errors between OrF ;N.T / and NF ;N.T / are 3.03%, 10.61% and 22.92% for D 2, 4, and 6, respec-tively. Clearly, when D 2, the OrF ;N.T / curve is close to NF ;N.T / and still smooth enough. Therefore, we choose D 2 for nonparametric estimation.
As a burst traffic example, Figure 6(b) shows the curves for NL;B.T / and OrL;B.T / with D 2, 4, and 6. The
figure indicates that D 2 is not smooth, and D 4 is a better choice for OrL;B.T /.
The nonparametric Orx.T / effectively smoothens the arrival rate function and provides useful insight to describe the SMS arrival distribution. However, for computational
454 Wirel. Commun. Mob. Comput. 2014; 14:450–459 © 2012 John Wiley & Sons, Ltd.
(a)
x = (F ,N)(b)
x = (L* ,N)Figure 6.Nx.T /, Orx.T / and segments for Orx.T /.
purposes, it is desirable to approximate NX.T / by a parametric function Orx.T / that satisfies two criterions: Criterion 1. Orx.T / is a continuous and piecewise
poly-nomial function with some known knots and degree less than or equal to three.
Criterion 2. Orx.T / is close to NX.T / in the sense
that the objective function D PTOrx.T / NX.T /2
is minimized.
Criterion 1 provides the guideline for generating the parametric form of Orx.T / such that the degree of the poly-nomial function is limited to 3. The objective function in Criterion 2 measures the distance between Orx.T / and NX.T /. To find the appropriate knots for Orx.T / that sat-isfy Criterion 1, the non-parametric Orx.T / plays an assis-tance role. For example, the OrF ;N.T / curve ( D 2) in Figure 6(a) suggests that NF ;N.T / curve can be appro-priately partitioned into 3 segments with two knots at T D 6 W 00 and T D13 W 00, where each segment can be
fit by a cubic polynomial function that satisfies Criterion 1. For the purpose of deriving Orx.T /, we first replace the time period T by an index j . Define set S3D f.T =30 min/ C 1g D f1; 2; 3; : : : ; 48g. Then, both NX.T / and Orx.T /can be transformed to NX.j / and Orx.j / where j 2 S3 For the NN ;F curve in Figure 6, we select two knots at j D 13 (i.e., T D 6 W 00) and j D 27 (i.e T D 13 W 00), which are around the breakfast and the lunch times. For 0 j 13, OrN ;F .j / is a decreasing curve. For 13 j 27, OrN ;F .j / increases and then decreases. For 27 j 48, OrN ;F .j / also increases and then decreases. Other traffic types show similar trends, and can be seg-mented by the same knots. Therefore, Orx.j / is partitioned into three segments fit by
Orx.j / D 8 < : Orx;1 .j /; 1 j 13 Orx;2 .j /; 13 j 27 Orx;3 .j /; 27 j 48 (5) Where
(8) and
Orx;3 .j / D a3;1j3C a3;2j2C a3;3j C a3;4for 27j 48 (9) From (6), we have
a2;4D 2197a1;1C 169a1;2C 13a1;3C a1;4
2197a2;1 169a2;2 13a2;3 (10) a3;4D 2197a1;1C 169a1;2C 13a1;3C a1;4
17486a2;1C 560a2;2C 14a2;3 19683a3;1
729a3;2 27a3;3 (11)
Substitute (10) and (11) into (7)–(9) to yield
Orx;1 .j / D a1;1j3C a1;2j2C a1;3j C a1;4 (12)
Orx;2 .j / D 2197a1;1C 169a1;2C 13a1;3C a1;4 Cj3 2197 a2;1C j2 169 a2;2 C .j 13/ a2;3; (13) and
Orx;3 .j / D 2197a1;1C 169a1;2C 13a1;3C a1;4 C 17486a2;1C C560a2;2C 14a2;3 Cj3 19683 a3;1C
j2 729 a3;2 C .j 27/ a3;3:
(14) Equations (12)–(14) can be represented in a matrix format
OrxD Ja (15)
where OrxDhOrx;1 .1/ ; Orx;1 .2/ ; : : : ; Orx;1 .12/ ; Orx;2 .13/ ; : : : ; Orx;2 .26/ ; Orx;3 .27/ ; : : : ; Orx;3 .48/iTis a 48-component column vector, a Da1;1; a1;2; a1;3; a1;4; a2;1; a2;2; a2;3; a3;1; a3;2; a3;3T is a 10-component parameter vector, and J is a 4810 matrix that consists of three sub-matrixes J1J2, and J3. That is JD 2 4 J1 J2 J3 3 5 (16)
For n D 2 and i D 1; : : : ; 14, we have
J2;iD
h
2197 169 13 1 .i C 12/3 2197 .i C 12/2 169 .i 1/ 0 0 0i
1x10
For n D 3 and i D 1; : : : ; 22, we have
J3;iD
h
2197 169 13 1 17486 560 14.i C 26/319683 .i C 26/2 729 .i 1/i
1x10
On the basis of (15), we derive the parameter vector a by using Criterion 2. Specifically, we plug (15) into the objective function DX T Orx.T / Nx.T /2 DX j 2S3 Orx.j / Nx.j /2 ; where S3D T 30 minC 1 DOrx Nx T Orx Nx
where Nxis a 48-component vector representing Nx.j /
NxD ŒNx.1/ ; Nx.2/ ; : : : ; Nx.48/T (17)
From (15) and (17), we have
D .JaNx/T.JaNx/ (18)
where .Ja Nx/ is a 48-component vector. To minimize this objective function, we take partial derivatives of with respect to parameters a and set them equal to 0. That is, from (18), we solve
456 Wirel. Commun. Mob. Comput. 2014; 14:450–459 © 2012 John Wiley & Sons, Ltd.
0D @h.Ja Nx/T.Ja Nx/ i @a D @haTJT NTx .Ja Nx/ i @a D 8 < : 2 4@ aTJT NTx @a 3 5 .Ja Nx/ 9 = ; T C .Ja Nx/T @ .Ja N x/ @a DhJT.Ja Nx/ iT C .Ja Nx/TJ D 2 .Ja Nx/TJ (19)
If we set 0D 0, then from (19) we have ŒJa NxTJD aTJT J NTxJD, which leads to
aTJTJD NTxJ (20)
By multiplyingJTJ 1 from right in both sides of (20), we have
aT D NTxJ
JTJ 1 (21)
By transposing the matrices of both sides of (21), we have aD JTJ 1 T JTNx DJTJ 1JTNx (22)
Equation (22) guarantees that Criterion 2 is satisfied. On the basis of this equation, the computed a, a2;4, and a3;4 values for eight types of traffic are showing in Table II. Define the error between Orx.j / and Nx.j / as
errorD Or
x.j / Nx.j / Nx.j /
(23)
Figures 7(a) and (b) show the Nx.j / and the Orx.j / curves, where x D .F ; N ) and (L; B). In Figure 7(a), the error (i.e., Equation (23)) between the two curves is 1.09%, and in Figure 7(b), the error is 8.11%.
Table III shows the errors between Orx.T / and Nx.T / for different x. The table indicates that Orx.j / accurately fits Nx.j / for non-burst traffic types with errors between 0.25% and 6.79%. On the other hand, the errors for burst traffic types are between 8.11% and 15.43%. The larger error incurred by the burst Orx.j / than the non-burst one on the same day is due to the fact that the variance of burst traffic is larger than that of non-burst traffic. Furthermore, the burst traffic happens once every half an hour and the Table II. Thean;lvalues for differentx.n D 1, 2, 3, and l D 1, 2, 3, 4).
a1;1 a1;2 a1;3 a1;4
F , B 0.42 6:30 17.37 38.22 F , N 0:02 0.60 7:21 31.53 F,B 0.03 1.76 36:82 167.58 F,N 0.01 0.48 11:45 64.59 L, B 6:00 228.00 2596:00 9182.00 L, N 9:00 260.10 2370:10 7130.10 L,B 4:38 115.99 923:53 2173.90 L,N 0:61 16.40 135:56 353.15
a2;1 a2;2 a2;3 a2;4
F , B 0:61 33.77 588:86 3415.30 F , N 0:24 13.35 229:73 1257.90 F,B 0:14 4.79 10:41 309:42 F,N 0:15 8.66 151:35 845.96 L, B 0:10 40:10 2180.10 1983:70 L, N 0.01 44:00 1500.10 12; 875:10 L,B 5:77 320.50 5559:90 30,945.00 L,N 0:75 44.66 812:82 4.68
a3;1 a3;2 a3;3 a3;4
F , B 0.19 22:07 846.03 10; 214:00 F , N 0.071 8:21 315.18 3807:80 F,B 0.01 1:18 49.17 455:01 F,N 0.01 1:02 39.75 433:22 L, B 1:00 60.00 2305:10 3019.10 L, N 0:01 7.00 343:10 5838.10 L,B 0:91 94.97 3004:60 30,450.00 L,N 0:88 100.28 3614:10 42,299.00
(a)
x = (F ,N)(b)
x = (L* ,B)Figure 7.Nx.j/ and Orx.j/ .j D f.T =30min/ C 1g/.
Table III. The errors between Or
x.T / and Nx.T /.
x(non-burst) F , N F,N L, N L,N
error 1.09% 0.25% 5.95% 6.79%
x(burst) F , B F,B L, B L,B
error 13.24% 15.34% 8.11% 14.43%
non-burst traffic happens every minute, which implies that more samples are collected for the non-burst traffic, and therefore can be approximated more accurately.
4. CONCLUSION
This paper derived the arrival distribution functions for SMS. We considered SMS arrival distributions for dif-ferent traffic types and observation days. Specifically, we modeled the short message arrivals as non-homogeneous Poisson processes. Then we compute the arrival rate func-tions based on the measured data from CHT’s commercial operation. On the basis of the SMS arrival distributions derived from our model, the mobile operators have better
understanding about the volumes of short messages in different times and days, which can be used to design more flexible short message charging rates. For example, peer-to-peer SMS has a higher charging rate at busy hour (17:00) and business users have a lower charging rate if their applications do not send SMS at [0,30] min each hour. We observed that the SMS arrival curves have two major turnover points around the breakfast and the lunch times. Therefore, we partitioned a day into three time zones, and approximated each zone by a cubic polyno-mial arrival rate function. For non-burst traffic, the errors between the derived arrival rate functions and the mea-sured data are between 0.25% and 6.79%. For burst traf-fic, the errors are between 8.11% and 15.43%. Our study
458 Wirel. Commun. Mob. Comput. 2014; 14:450–459 © 2012 John Wiley & Sons, Ltd.
indicated that the errors of the arrival rate functions for burst traffic are acceptable for network planning purposes of commercial SMS operation. Although the user behav-ior of other telecom operators may be different from that of Chunghwa Telecom, they can apply our model with their measured data to predict the potential SMS volume in their commercial operation, and then modify their network configurations to achieve SMS traffic load balancing.
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AUTHORS’ BIOGRAPHIES
Hui-Nien Hung received the Ph.D. degree in statistics from the Uni-versity of Chicago, Chicago, Illinois in 1996. He is currently a pro-fessor at the Institute of Statis-tics, National Chiao Tung University, Hsinchu, Taiwan. His research inter-ests include applied probability, bio-statistics, statistical inference, statistical computing, and industrial statistics.
Yi-Bing Lin is Vice President and Life Chair professor of the College of Computer Science, National Chiao Tung University (NCTU), and a Vis-iting professor of King Saud Univer-sity. He is also with the Institute of Information Science and the Research Center for Information Technology Innovation, Academia Sinica, Nankang, Taipei, Taiwan, R.O.C. Lin has authored books on Wireless and Mobile Network Architecture (Wiley, 2001), Wireless and Mobile All-IP Networks (John Wiley, 2005), and Charging for Mobile All-IP Telecommunications (Wiley, 2008). Lin has received numerous research awards including the 2005 NSC Distinguished Researcher and 2006 Academic Award of Ministry of Education. Lin is an ACM Fellow, an AAAS Fellow, an IEEE Fellow and an IET Fellow.
Chao-Liang Luo is currently work-ing toward the Ph.D. degree at the Department of Computer Sci-ence and Engineering, National Chiao Tung University. In 2001, he joined the Telecommunication Laboratories, Chunghwa Telecom Co., Ltd., and was involved in the implementation of value-added services in mobile networks. In 2005, he was with the short message service team. Since then, he has been involved in the design of the Next Gener-ation Network (NGN), mobile packet switched data and multimedia services, and the study of mobile network evolution. His research interests include the design and analysis of personal communications services network, 3G networks, wireless Internet, mobile computing, and performance modeling.
