cost analysis of short message retransmissions

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Cost Analysis of Short Message Retransmissions

Sok-Ian Sou , Yi-Bing Lin , Fellow, IEEE, and Chao-Liang Luo

May 1, 2009

Abstract

Short Message Service (SMS) is the most popular mobile data service today. In Taiwan, a sub-

scriber sends more than 200 short messages per year on average. The huge demand for SMS

signicantly increases network trafc, and it is essential that mobile operators should provide

efcient SMS delivery mechanism. In this paper, we study the short message retransmission

policies and derive some facts about these policies. Then we propose an analytic model to in-

vestigate the short message retransmission performance. The analytic model is validated against

simulation experiments. We also collect SMS statistics from a commercial mobile telecommu-

nications network. Our study indicates that the performance trends for the analytic/simulation

models and the measured data are consistent.

Keywords : Mobile telecommunications network, Short Message Service (SMS), retransmis-

sion policy, delivery delay

Sok-Ian Sou is Assistant Professor of Department of Electrical Engineering, National Cheng Kung University,Tainan, Taiwan, ROC. Email: [email protected]

Yi-Bing Lin is Chair Professor and Dean of College of Computer Science, National Chiao Tung University(NCTU), Hsinchu, Taiwan, ROC. Email: [email protected]

Chao-Liang Luo is with the Department of Computer Science, NCTU, and the Multimedia Applications Lab-oratory, Chunghwa Telcom Co., Ltd., ROC. Email: [email protected]

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1 Introduction

Short Message Service (SMS) provides a non real-time transfer of messages with low-capacity

and low-time performance. Because of its simplicity, SMS is the most popular mobile data

service today. Many mobile data applications have been developed based on the SMS technol-

ogy [5, 6, 8, 9, 10, 11, 13].

Fig. 1 illustrates the SMS network architecture for Universal Mobile Telecommunications

System (UMTS) [1, 2, 3]. In this architecture, when a User Equipment (UE) (called the orig-

inating UE; see Fig. 1 (a)) sends a short message to another UE (called the terminating UE;

see Fig. 1 (b)), the short message is rst sent to the Originating Mobile Switching Center (MO-

MSC; Fig. 1 (d)) through the originating UMTS Terrestrial Radio Access Network (UTRAN;

Fig. 1 (c)). The MO-MSC forwards the message to the Inter-Working MSC (IWMSC; Fig. 1

(e)). The IWMSC passes this message to a Short Message-Service Center (SM-SC; Fig. 1 (f)).

Following the UMTS roaming protocol, the Gateway MSC (GMSC; Fig. 1 (g)) forwards the

message to the Terminating MSC (MT-MSC; Fig. 1 (h)). Finally, the short message is sent to

the terminating UE via the terminating UTRAN. Due to user behavior and mobile network un-

availability (e.g., the user moves to a weak signal area such as a tunnel, an elevator, a basement,

and so on), a short message may not be successfully delivered at the rst time. If a short mes-

sage transmission fails, the SM-SC retransmits the short message to the terminating UE after a

waiting period. Retransmission may repeat several times until the short message is successfully

delivered or the SM-SC gives up delivering the short message.

SMS retransmission may result in huge mobile network signaling trafc and long elapsed

times of short message delivery. Therefore, it is essential to exercise an efcient SMS retrans-

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OriginatingUE

MO-MSC

SM-SC

IWMSC

TerminatingUE

GMSC

a

b

c de

h g

f Originating UTRAN

MT-MSC

Terminating UTRAN

Figure 1: The SMS delivery architecture

mission policy to determine when and how many times to retransmit a short message to the

terminating UE. To address this issue, we propose analytic and simulation models to investigate

the performance of SMS retransmission in terms of the expected number of retransmissions and

the message delivery delay. We also collect measured data from a commercial UMTS system

to further analyze the performance trends on SMS retransmission policies.

2 Some Facts on Short Message Retransmission

This section denes SMS retransmission policies, and derives some facts on short message

retransmission.

Denition 1. An SMS retransmission policy is an order set s = {a i |0 i I }, where I is

the maximum number of retransmissions and a i is the time of issuing the i-th retransmission.

By convention, a0 = 0 is the rst time when the short message is sent to the terminating UE. If

the short message delivery fails at the (i 1)-th retransmission, then it is resent after a waiting

period t r,i = a i a i 1, for i 1.

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Time

A shortmessage arrival

...

The short message issuccessfully

retransmitted

T (s )

0 a 1 a 2 a 3 a 4b 1=a 1

connected connected connecteddisconnected disconnected disconnected

policy s

policy s b 2=a 4

retransmissions retransmission

t r ,1 t r ,2 t r ,3 t r ,4

Figure 2: Timing diagram for the connected and disconnected periods ( represents a short

message retransmission)

Example 1. If s ={0, 5 mins, 10 mins, 20 mins, 1 hr, 3 hrs, 6 hrs, 12 hrs, 24 hrs }, then I = 8 ,

and t r, 3 = 10 minutes (=20-10). We note that after I retransmissions, the short message will

not be retransmitted, and the delivery fails.

Denition 2. Consider a short message delivered under policy s . Denote T (s ) as the last time

when the short message is retransmitted, and N (s ) as the number of retransmissions.

In Example 1, if a short message is successfully delivered at hour 1, then T (s )=1 hr and

N (s )=4.

Denition 3. A policy s is a subset of another policy s (i.e., s s ) if for all b s , we have

bs .

To make a fair comparison of the retransmission policies, we consider the same SMS net-

work availability condition concept (i.e., different policies see the same connected/disconnected

periods of a UE). In Fig. 2, the time line shows the connected and disconnected periods of a

terminating UE, where the short message is successfully delivered in the connected periods and

cannot be delivered in the disconnected periods. Let s = {0, a1, a2, . . .} and s = {0, b1, b2, . . .},

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where s s , b1 = a1, b2 = a4, and so on. In this gure, T (s ) = a4 and N (s ) = 4 for policy s .

For policy s , T (s ) = b2 and N (s ) = 2 .

Under the same SMS network availability condition, we derive the following facts:

Fact 1. If s s , then for every short message, we have

T (s ) T (s )

Proof: Due to the same SMS network availability condition, if a short message is delivered

under s , then it can also be successfully delivered at b under policy s . Furthermore, it is possible

that the message is successfully delivered at some time a s , where a b. Therefore T (s ) =

a b = T (s ).

Denition 4. Consider two policies s s . For any time point a s , dene the subset

(a) = {c|c < a, c s , c /s }. Denote |(a)| as the size of (a).

Fact 2. For s s , if T (s ) = T (s ), then N (s ) N (s ).

Proof: Since |(T (s )) | 0, s retransmits same or more times than s before T (s ), and N (s )

N (s ).

Fact 3. For s s and T (s ) < T (s ), let b j = T (s ). There exists bi s such that bi T (s ) 0 and from

(1), we have

Pr[N (s ) = n] = (1 Pr[N (s ) = 0])

j =1Pr[N (s ) = n |J = j ]Pr[J = j ]

=md

mc + md

j =1Pr[N (s ) = n |J = j ]Pr[J = j ] (2)

The SM-SC performs the i-th short message retransmission after an Exponential random

time t r,i with mean E [t r,i ] = 1/ . Denote N (s , t ) as the number of retransmissions occurring

in period t . Then Pr[N (s , t ) = n] follows the Poisson distribution

Pr[N (s , t ) = n] =(t )n

n!e t (3)

In (2), the probability mass function Pr[J = j ] can be derived as follows: For j > 1, we rst

compute the probability that no SMS retransmission occurs in the rst j 1 connected periods

tc,k , where 1 k j 1, and then compute the probability that an SMS retransmission occurs

in the j -th connected period. For j = 1 , we only need to compute the probability that an SMS

retransmission occurs in the tc,1 period. From (3), Pr[J = j ] is expressed as

Pr[J = j ] = j 1

k=1

t c,k =0Pr[N (s , t c,k ) = 0]f c(tc,k )dtc,k

1

t c,j =0Pr[N (s , t c,j ) = 0]f c(tc,j )dtc,j

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= j 1

k=1

t c,k =0e t c,k f c(tc,k )dtc,k 1

t c,j =0e t c,j f c(tc,j )dt c,j

= [f c

()] j 1

[1 f c

()] (4)

In the right-hand side of Eq. (2), the term Pr[N (s ) = n |J = j ] is derived below. As shown

in Fig. 3, td,1 = c,1 0 is the interval between the time when the short message arrives and

when the UE is connected to the network again. Let r d(td,1) and r d(s) be the density function

and the Laplace transform of td,1, respectively. From the renewal theory [4], td,1 is the residual

time of td,1. Therefore, we have

r d(s) =1

sm d[1 f d (s)] (5)

For J = 1 , Pr[N (s ) = n |J = 1] is the probability that n 1 message retransmissions occur

in the td,1 period. From (3), Pr[N (s ) = n |J = 1] is computed as

Pr[N (s ) = n |J = 1] =

t d, 1 =0Pr[N (s , td,1) = n 1]r d(td,1)dtd,1

=

t d, 1 =0

( td,1)n 1

(n 1)!e t d, 1 r d(td,1)dtd,1

=( )n 1

(n 1)! dn 1r d(s)dsn 1 s=

(6)

For J = j > 1 and 0 l n 1, we rst consider that there are l message retransmissions

occurring in the j -th disconnected period td,j , and n l 1 retransmissions occur in the rst

j 1 disconnected periods. From (3), Pr[N (s ) = n |J = j ] is expressed as

Pr[N (s ) = n |J = j ] =n 1

l=0

td,j =0Pr[N (s , td,j ) = l]f d(td,j )dtd,j Pr[N (s ) = n l|J = j 1]

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=n 1

l=0

td,j =0

(t d,j )l

l!e t d,j f d(td,j )dtd,j Pr[N (s ) = n l|J = j 1]

=n 1

l=0

( )l

l! dlf d

(s)ds l s= Pr[N (

s ) = n l|J = j 1] (7)

Eqs. (6) and (7) can be used to iteratively compute Pr[N (s ) = n |J = j ]. Finally, Pr[N (s ) =

n] is computed by substituting Eqs. (4) and (7) into (2). For N (s ) = 1 , Eq. (7) is simplied to

Pr[N (s ) = 1 |J = j ] = f d ()Pr[N (s ) = 1 |J = j 1] = r

d()f

d () j 1 (8)

Substituting Eqs. (5) and (8) into (2) to yield

Pr[N (s ) = 1] =md

mc + md

j =1Pr[N (s ) = 1 |J = j ]Pr[J = j ]

=md

mc + md

j =1

1m d

[1 f d ()]f

d () j 1 [f c

()] j 1[1 f c()]

= 1(mc + md)

[1 f c

()][1 f d

()]1 f c

()f d ()(9)

For N = 2 , Eq. (7) is rewritten as

Pr[N (s ) = 2 |J = j ] =1

l=0

( )l

l! dlf d

(s)ds l s=

Pr[N (s ) = n l|J = j 1]

= [f d ()] j 1 dr

d(s)ds s= + ( j 1)r d()[f d ()] j 2 df

d (s)ds s=

(10)

Substituting Eqs. (5) and (10) into (2), Pr[N (s ) = 2] is derived as

Pr[N (s ) = 2] =md

mc + md

j =1Pr[N (s ) = 2 |J = j ]Pr[J = j ]

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= m d[1 f c

()]mc + md

dr d(s)ds s=

j =1[f c ()f d

()] j 1

+ r

d()f d ()

df

d (s)ds s=

j =1( j 1)[f c ()f d

()] j 1

=1

mc + md1 f c ()

1 f c ()f

d ()

1 f d ()

+

df d (s)ds s=

1 f c ()(1 f

d ())1 f c ()f

d ()

(11)

If tc,j and td,j are Exponentially distributed, then (2) can be simplied to

Pr[N (s ) = n] =mc(1 + m c)n 1

mc + mdmd

mc + md + m cmd

n, n > 0 (12)

and E [N (s )] is expressed as

E [N (s )] =

n =1n Pr[N (s ) = n]

=mc

mc + md

n =1n

mnd (1 + m c)n 1

(mc + md + m cmd)n

=mdmc

mc + md(1 + m c)mc + md

(13)

The expected delivery delay can be expressed as

E [T (s )] = E [t r,i ]E [N (s )] (14)

If t r,i is Exponentially distributed with mean 1/ , then from (13) and (14), we have

E [T (s )] =md

m cmc + md(1 + m c)

mc + md(15)

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Based on (1) and (12), the probability pf (s ) that a short message is not successfully delivered

within I retransmissions can be derived as

pf (s ) = 1 I

n =0Pr[N (s ) = n] (16)

If tc,j and td,j are Exponentially distributed, (16) is rewritten as

pf (s ) =md

mc + md

I

n =1

mc(1 + m c)n 1

mc + mdmd

mc + md + m cmd

n

=1

mc + mdmd

cmc1 + m c

1 cI

1 c(17)

where c = m d (1+ m c )m c + m d + m c m d .

The above analytic model is validated against simulation experiments. Details of the sim-

ulation model are described in Appendix B. Table 1 lists E [N (s )], E [T (s )] and pf (s ) values

when the connected periods and the disconnected periods have Exponential distributions. Table

2 lists the Pr[N (s ) = n] values when the connected periods and the disconnected periods have

Gamma distributions. Both tables indicate that the discrepancies between the analytic results

(specically, Eqs. (1), (9), (12), (13), (15) and (17)), and the simulation data are within 2%.

4 Numerical Examples

This section uses numerical examples to investigate the performance of the SMS retransmission

policies. We show how the retransmission rate , the expected connected (disconnected) periods

mc(md) and the variances vc(vd) affect the expected number E [N (s )] of the SMS retransmis-

sions (using (13)) and the expected delivery delay E [T (s )] (using (15)). For demonstration

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Table 1: Comparison of the analytic and the simulation results (where E [t r,i ] = 1/ , tc,j and

td,j are Exponential distributed, and mc = md)

E [N ( s )] E [T ( s )] pf ( s ) (I = 5 )(Unit: 1/m c ) Ana. Sim. Diff. * Ana. Sim. Diff. Ana. Sim. Diff.

0.1 1.0500 1.0495 0.05 % 10.5000 10.4945 0.05 % 1.970 % 1.974 % 0.21 %

0.25 1.1250 1.1239 0.10 % 4.5000 4.4966 0.08 % 2.650 % 2.656 % 0.21 %0.5 1.2500 1.2498 0.01 % 2.5000 2.4991 0.04 % 3.890 % 3.901 % 0.28 %0.75 1.3750 1.3747 0.02 % 1.8333 1.8324 0.05 % 5.220 % 5.218 % -0.03 %

1 1.5000 1.4984 0.11 % 1.5000 1.4988 0.08 % 6.580 % 6.569 % -0.17 %2.5 2.2500 2.2538 -0.17 % 0.9000 0.9015 -0.17 % 14.230 % 14.206 % -0.17 %5 3.5000 3.4984 0.05 % 0.7000 0.6998 0.03 % 23.130 % 23.192 % 0. 27%

7.5 4.7500 4.7252 0.52 % 0.6333 0.6323 0.16 % 28.670 % 28.681 % 0.04 %10 6.0000 5.9363 1.06 % 0.6000 0.6014 -0.23 % 32.360 % 32.329 % -0.10 %

* Ana: Analysis data; Sim: Simulation data; Diff: Difference

Table 2: Comparison of the analytic and the simulation results (where E [t r,i ] = 1/ , tc,j and

td,j are Gamma-distributed, and mc = md = 12 )

vc = vd Pr[ N ( s ) = 0] Pr[ N (s ) = 1] Pr[ N (s ) = 2](Unit: 1/ 2 ) Ana. Sim. Diff. * Ana. Sim. Diff. Ana. Sim. Diff.

0.01 0.50000 0.50001 -0.003 % 0.24260 0.24267 -0.027 % 0.12730 0.12724 0.051 %0.1 0.50000 0 .50006 -0.012 % 0.22400 0.22399 0.003 % 0.12510 0.12508 0.014 %1 0.50000 0.50002 -0.003 % 0.13650 0.13647 0.020 % 0.09560 0.09558 0.024 %

10 0.50000 0.49828 0.344 % 0.03800 0.03804 -0.103 % 0.03210 0.03200 0.302 %100 0.50000 0.50029 -0.058 % 0.00660 0.00659 0.227 % 0.00600 0.00597 0.517 %

* Ana: Analysis data; Sim: Simulation data; Diff: Difference

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purposes, we consider Gamma-distributed tc,j and td,j . Gamma distribution is selected because

of its exibility in setting various parameters and can be used to t the rst two moments of the

sample data. From [7], the Laplace transforms of f c(.) and f d(.) are

f c (s) =1

1 + ( vc/m c)s

m 2c /v c

and f d (s) =1

1 + ( vd/m d)s

m 2d /v d

(18)

In (18), m2c /v c and m2d/v d are the shape parameters, vc/m c and vd/m d are the scale param-

eters. The effects of the input parameters are described as follows:

Effects of the t r,i distribution. Fig. 4 plots E [N (s )] and E [T (s )] for Exponential and Fixed

t r,i distribution, where vc = m2c and vd = m2d . This gure indicates that the performance

of Fixed t r,i (the dashed curves) is slightly better than that of Exponential t r,i (the solid

curves).

Effects of the retransmission rate . Fig. 4 shows that E [N (s )] is an increasing function of ,

which means that |(T (s )) | > j i in Fact 3. E [T (s )] is a decreasing function of (Fact

1). When < 1/m d , E [N (s )] is not signicantly affected by the change of , and E [T (s )]

is signicantly decreases as increases. Conversely, when > 1/m d , increasing

signicantly increases E [N (s )], but only has insignicantly effect on E [T (s )]. Therefore,

when is small, increasing can signicantly improve the E [T (s )] at the cost of slightly

degrading E [N (s )]. When is large, decreasing can signicantly improve E [N (s )] at

the cost of slightly degrading E [T (s )]. Fig. 4 indicates that 0.5/m d < < 5/m d is the

range that both E [N (s )] and E [T (s )] do not degrade signicantly when changes.

Effects of the availability ratio mc/m d . Fig. 5 plots E [N (s )] and E [T (s )] against mc/m d ,

where vc = m2c , vd = m2d and t r,i = 1 / is xed. This gure shows trivial results that

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10 1 10 0 10 10

0.5

1

1.5

2

(Unit: 1/ m d )

E [ N ( s ) ]

m c /m

d =5

m c /m

d =9m

c /m

d =13

(a) Effects on E [N (s )]

101

100

101

0

0.5

1

1.5

2

(Unit: 1/ m d )

E [ T ( s ) ] ( U n i

t : 1 / m

d )

m c /m

d =5

m c /m

d =9

m c

/m d =13

(b) Effects on E [T (s )]

Figure 4: Effects of and the t r,i distribution ( vc = m2c and vd = m2d; Solid lines: Exponential

t r,i ; Dashed lines: Fixed t r,i )

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101

100

101

102

0

2

4

6

8

10

12

m c /m

d

E [ N ( s ) ]

=0.5/ m d

=1/ m d

=5/ m d

(a) Effect on E [N (s )]

101

100

101

102

0

2

4

6

8

10

12

14

16

18

20

22

m c /m

d

E [ T ( s ) ] ( U n i

t : 1 / m

d )

=0.5/ m d

=1/ m d

=5/ m d

(b) Effect on E [T (s )]

Figure 5: Effects of mc/m d and (vc = m2c , vd = m2d and t r,i = 1 / is xed)

both E [N (s )] and E [T (s )] decrease when mc/m d increases. The non-trivial observation

is that when mc/m d < 7.5, the E [N (s )] and E [T (s )] performances are signicantly de-

graded as mc/m d decreases. In a commercial SMS network, the base station deployment

should meet the requirement mc/m d > 10 to achieve good SMS performance.

Effects of variances vc and vd . Fig. 6 plots E [N (s )] and E [T (s )] as functions of the variance

vc, where mc/m d = 9 , vd = m2d and t r,i = 1 / is xed. When vc is large (e.g.,

vc > 2.5m2c ), both E [N (s )] and E [T (s )] increase as the variance vc increases. This

phenomenon is explained as follows: As the variance vc increases, the number of short

connected periods are far more than the number of long connected periods. Therefore,

in an arbitrary connected/disconnected cycle, the disconnected time is more likely longer

than the connected time [12]. When the SM-SC sends the short message to the terminat-

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101

100

101

0

0.2

0.4

0.6

0.8

1

v c (Unit: m c

2 )

E [ N ( s ) ]

=0.5/ m d =1/ m d =5/ m d


101

100

101

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

v c (Unit: m c

2 )

E [ T ( s ) ] ( U n i

t : 1 / m

d )

=0.5/ m d =1/ m d =5/ m d


Figure 6: Effects of vc (mc/m d = 9 and t r,i = 1 / is xed)

ing UE at 0, if the UE is disconnected from the network, message retransmissions are

more likely to fall in the disconnected periods than the connected periods. When vc is

larger than 2.5m2c , the performances of both E [N (s )] and E [T (s )] signicantly degrade

as vc increases. When vc < m 2c , E [N (s )] and E [T (s )] are not affected by the change of

vc. Furthermore, for larger , the effect of vc on E [N (s )] is more signicant. On the other

hand, the effects of changing vc on E [T (s )] are similar for different values.

Effects of vd are similar to vc, and the results are not presented. From Fig. 6, it is important

that a mobile operator maintains low vc and vd values for good SMS performance.

Effects of I on pf (s ). Fig. 7 plots the pf (s ) curves against retransmission rates and the max-

imum number I of retransmissions. It is trivial that pf (s ) decreases as I increases from

5 to 10 . We observe that pf (s ) decreases as increases. This effect is consistent

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1 2 3 4 5 6 7

103

102

10 1

100

(Unit: 1/ m d )

p f

( s ) ( % )

I = 5I = 10

Figure 7: Effects of I on pf (s ) (mc/m d = 9 and t r,i = 1 / is xed)

with Fact 4 (i.e., if s s , then pf (s ) pf (s )). This gure also indicates that when

is sufcient large (e.g., > 3/m d for I = 5 ), increasing does not improve pf (s )

performance.

Performance trends observed from the measured data. From the commercial UMTS sys-

tem of Chunghwa Telecom (CHT), we obtained the output measures Pr[N (s ) = n],

E [N (s )], and E [T (s )] for several retransmission policies. Dene s k as a policy where

a short message is retransmitted for every k minutes. For k = 5, 10, 20 and 30, it is clear

thats

30

s10

s5 and

s20

s10

s5 . We have collected the statistics for more than

400,000 SMS deliveries (100,000 deliveries for each policy). Fig. 8 shows E [N (s k )] and

E [T (s k )] in the curves. To compare the measured data and the analytic results, we

consider a policy s e with Exponential t r,i , where E [t r,i ] is the same as s k for various k

values. We use (13) and (15) to analytically compute E [N (s e )] and E [T (s e )], and the

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results are shown in the curves in Fig. 8.

When is small, the analytic results are the lower bounds for the measured data. The

reason why both results do not exactly match is due to the fact that we are not able to

measure the connected/disconnected distributions. However, the gure shows that the

trends are consistent, which follows Fact 1. Fig. 8 (a) shows that |(T (s )) | > j i, which

implies that short periods of outage seldom occur in CHTs network (see the comments

on Fact 3).

Fig. 9 plots the probability mass function Pr[N (s ) = n]. From the measurements of the

CHTs commercial SMS network, Pr[N (s 20 ) = 0] 0.9, which means that mc/m d 9

(see (1)). We also note that Pr[N (s 20 ) = 1] 0.042. For policy s e (with Exponential

retransmission delays) where mc/m d = 9 and Pr[N (s e ) = 1] = 0 .042, we obtain m c =

11.43 from (12).

With the computed m c value, we can derive Pr[N (s e ) = n] for N (s ) > 1. Fig. 9 shows

that Pr[N (s e ) = n] has the same trend as Pr[N (s 20 ) = n]. Therefore, our study indicates

that the performance trends for the analytic/simulation models and the measurements

from the CHT commercial SMS network are consistent. Note that exact match of analytic

and measured data is not possible at this point because we are not able to produce the

same SMS network availability condition for the commercial SMS data collection, and as

previously mentioned, the exact connected and disconnected periods cannot be measured

in commercial operations. A useful conclusion is that Eqs. (13) and (15) can be used to

quickly and roughly estimate the SMS network performance for network planning.

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100

101

0

0.2

0.4

0.6

0.8

1

1.2

(Unit: 1/ m d )

E [ N ( s ) ]

Analytic dataMeasured data


100

101

0

0.05

0.1

0.15

0.2

0.25

0.3

(Unit: 1/ m d )

E [ T ( s ) ] ( U n i t : 1 / m

d )

Analytic dataMeasured data


Figure 8: Performance trends of the analytic data and the measured data ( mc/m d = 9 ; the

parameters for s e are vc = vd = m2d and E [t r,i ] = 1/ )

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1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

P r

[ N ( s ) = n

]

s=se (analytic data)

s=s20 (measured data)

Figure 9: Performance trends of the analytic data and the measured data

5 Conclusions

In this paper, we studied the short message retransmission policies and derived some facts on

short message retransmissions. Then we proposed an analytic model to investigate the short

message retransmission performance in terms of the expected number of short message retrans-

missions E [N (s )] and the message delivery delay E [T (s )]. The analytic model was validated

against simulation experiments. We showed how the retransmission rate, the expected val-

ues and the variances of the connected/disconnected period distributions affect E [N (s )] and

E [T (s

)]. The following observations are made in our study:

The performance of xed retransmission period is slightly better than that of Exponen-

tial retransmission approach. Therefore, it is appropriate that an operator chooses xed

retransmission period value in the delivery policy.

The retransmission rate should be less than ve times of the handset disconnected period

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and more than half times of that period so that the SMS performance is not signicantly

affected when the retransmission rate changes.

In a commercial SMS network, the operator should maintain at least 9 to 1 ratio for the

connected/disconnected periods to achieve good SMS performance.

For each message delivery, it sufces to set the maximum number of retransmissions to

be less than 10.

Our study indicates that by selecting appropriate retransmission policy (in particular, the re-

transmission frequency), the SMS delivery cost can be signicantly reduced. We also collected

measured data from a commercial mobile telecom network, and observed that the performance

trends of the commercial operation are consistent with our analytic/simulation models. Based

on our study, a mobile operator can predict the performance trends and select the appropriate

parameter values for the SMS retransmission policy in various network conditions.

Acknowledgment

The authors would like to thank the editor and the anonymous reviewers. Their valuable com-

ments have signicantly improved the quality of this paper. Their efforts are highly appreciated.

A Notation

f c(.) : the density function of tc,j .

f c (.) : the Laplace transform of f c(.).

f d(.) : the density function of td,j .

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f d (.) : the Laplace transform of f d(.).

I : the maximum number of short message retransmissions.

J : the number of connected periods occurring in period ( 0, 4) in Fig. 3.

mc : the expected value of tc,j .

md : the expected value of td,j .

N (s ) : the number of retransmissions of a short message delivery for policy s .

N (s , t ) : the number of message retransmissions occurring in period t for policy s .

1/ : the expected value of the retransmission period t r,i .

pf (s ) : the probability of failed SMS delivery for policy s .

Pr[N (s ) = n] : the probability that a short message is retransmitted for n times in policy

s , where n 0.

r d(.) : the density function of td,1.

r d(.) : the Laplace transform of td,1.

tc,j : the j -th connected period, where j 1.

td,j : the j -th disconnected period, where j 1.

td,1 : the residual time of td,1.

t r,i : the period between the (i 1)-th (re)transmission and the i-th retransmission of a

short message delivery, where i 1.

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T (s ) : the short message delivery time for policy s or the elapsed time between when the

SM-SC starts and when it stops delivering a short message.

vc : the variance of tc,j .

vd : the variance of td,j .

B Simulation Model for SMS Retransmission

In this paper, we developed a discrete event simulation model for short message retransmission.

In this simulation model, three types of events are dened:

UE AVAILABILITY STATUS CHANGE (UE ASC) : the switching of the UE availabil-

ity status. These events occur at c,j and d,j in Fig. 3.

NEW ARRIVAL : the arrival of a short message (occurring at 0 in Fig. 3).

RETRANSMISSION : retransmission of a short message (occurring at 1, 2, 3 and 4

in Fig. 3).

To simulate the status of the UE availability, a variable UE STATUS is maintained in the

simulation. When UE STATUS is set to On, the UE is connected to the mobile network (in

the tc,j periods) and can receive the short message successfully. When UE STATUS is set to

Off, the UE is disconnected from the mobile network (in the td,j periods) and the message

transmission fails. Several random number generators are used to produce the connected period

tc, the disconnected period td, and the retransmission waiting period t r . In an experiment, we

simulate N short messages (excluding retransmissions) delivered to the terminating UE. The

short message inter-arrival time is ta . The reader should note that ta and t r are different.

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In a UE ASC event, the following elds are maintained:

the type eld identies the UE ASC event.

the ts eld records the timestamp of the event (when the event occurs).

Besides the type and the ts elds, the following elds are also maintained in a NEW ARRIVAL

and a RETRANSMISSION events:

the start time eld records the time when the short message is rst transmitted (i.e., 0

in Fig. 3).

the count eld records the number of retransmissions. Note that it is always true that

count=0 in a NEW ARRIVAL event and count > 0 in a RETRANSMISSION event.

The events are inserted in an event list, and are deleted/processed from the event list in the

non-decreasing timestamp order. In each experiment, N = 10 , 000, 000 short messages are

simulated to ensure that the results are stable. The output measures of the simulation are T and

M , where

T is the sum of message delivery delays of the N short messages simulated in the exper-

iment.

M is the number of retransmissions for the N short messages simulated in the experiment.

These output measures are used to compute E [N (s )] and E [T (s )] as follows:

E [N (s )] =M N

and E [T (s )] =T N

(19)

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The simulation owchart is shown in Fig. 10. Step 1 initializes the variables N , T , M and

UE STATUS. Step 2 generates the rst UE ASC and NEW ARRIVAL events, and inserts these

events into the event list. In Steps 3 and 4, the next event e in the event list is processed based

on its type.

If e.type== UE ASC , Step 5 checks if UE STATUS is On. If so, Step 6 sets UE STATUS

to Off, and generates the disconnected period td and the next UE ASC event e1, where the

next switching time is e1.ts= e.ts+ td . If UE STATUS==Off at Step 5, Step 7 sets UE STATUS

to On, generates the connected period tc and the next UE ASC event e1, where the nextswitching time is e1.ts= e.ts+ tc. The simulation proceeds to Step 12.

If e.type== NEW ARRIVAL at Step 4, the simulation proceeds to Step 8. Step 8 generates

the next NEW ARRIVAL event e1 with the inter-arrival time ta , where e1.ts= e.ts+ ta , e1.count=0

and e1.start time= e1.ts. At Step 9, if UE STATUS is On, the message is successfully deliv-

ered to the UE, and Step 10 updates the variables N , T and M . Otherwise (UE STATUS is

Off at Step 9), the UE is disconnected from the network, and Step 11 generates the next

RETRANSMISSION event e1 with the waiting period t r for the next message retransmis-

sion, where e1.ts= e.ts+ t r , e1.count= e.count+1 and e1.start time= e.start time. The simulation

proceeds to Step 12.

If e.type== RETRANSMISSION at Step 4, the simulation proceeds to execute Steps 9-11.

At Step 12, if N = 10 , 000, 000 short messages are successfully delivered, the simulation

terminates and the output measures are calculated at Step 13.

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Generate t a ;Generate the next

event e 1;e 1.ts= e .ts+ t a ;e 1.count=0;e 1.start_time= e 1.ts;

StartEnd

Generate the first and events;

N< 10,000,000

E[ N (s)]= M / N ;E[T (s)]= T / N ;

N =0; T =0; M =0;UE_STATUS="On";

Process the next event e from the event list;Yes

e .type?

No

2

3

1

UE_STATUS="Off";Generate t d ;Generate the next

event e 1;e 1.ts= e .ts+ t d ;

45

7 76

12

13

8

Yes

No

UE_STATUS="On";Generate t c ;Generate the next

event e 1;e 1.ts= e .ts+ t c ;

7 77

Yes No

9

11

10 Generate t r ;Generate the next

event e 1;e 1.ts= e .ts+ t r ;e 1.count= e .count+1;e 1.start_time= e .start_time;

N = N +1;T =T +(e .ts-e .start_time);

M = M +e .count;

UE_STATUS=="On"?

UE_STATUS=="On"?

Figure 10: Simulation ow chart

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cost analysis of short message retransmissions

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