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Modeling location management in wireless networks with generally distributed parameters
Computer Communications 29 (2006) 2386–2395 www.elsevier.com/locate/comcom
Modeling location management in wireless networks with generally distributed parameters Yan Zhang a
a,*,1
, Jun Zheng b, Lili Zhang c, Yifan Chen c, Maode Ma
c
Wireless Communication Laboratory, NICT Singapore, National Institute of Information and Communications Technology, Singapore b Department of Computer Science, Queens College, City University of New York, USA c School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Received 18 September 2005; received in revised form 1 February 2006; accepted 14 February 2006 Available online 10 March 2006
Abstract In the wireless mobile networks, location update (LU) and paging are two basic operations in the framework of mobility management. The trade-off analysis between LU and paging is traditionally performed under the assumption that the key variables follow exponential distribution. However, recent work showed that such assumptions may be incorrect. In this paper, by introducing the concept of LU inter-arrival time and additionally relaxing the assumptions for the inter-service time, cell residence time and LU inter-arrival time, we present an analytical technique to develop the closed-form formula for the signaling cost applicable to different location management schemes. Specifically, the result is applied to the static scheme and the dynamic movement-based scheme to demonstrate the general applicability. Illustrative numerical examples are given to compare the location management scheme performance based on the cell characteristics. 2006 Elsevier B.V. All rights reserved. Keywords: Dynamic location management; Location update; Paging; Inter-service time; Cell residence time; Location area; Wireless networks
1. Introduction Location management is an important component in the present as well as the next generation wireless mobile networks to support the seamless mobility requirement. There are two operations in the framework of location management: updating the Mobile Terminal’s (MT) location and paging the MT. Accordingly, the signaling cost is comprised of the location update cost and paging cost. The two-tier Home Location Register/Visitor Location Register (HLR/VLR) architecture and static location management scheme are employed in the second-generation mobile network [1]; and it has been found that this mechanism generates high signaling traffics. To reduce the associated signaling cost, a variety of dynamic location *
Corresponding author. Tel.: +65 67711011; fax: +65 67795753. E-mail addresses: [email protected] (Y. Zhang), [email protected] (J. Zheng). 1 Present address: NICT Singapore, National Institute of Information and Communications Technology, Singapore. 0140-3664/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.02.014
managements have been proposed. These include timerbased scheme [5], movement-based scheme [5], distancebased scheme [5,24], and pointer forwarding scheme [14]. The difference between these strategies lies in the threshold or the event triggering the location update process. For instance, in timer-based scheme, updates are made at a fixed timer interval. In movement-based scheme, updates are made whenever the number of cell-boundary crossings (or equivalently termed as movement) since the last update exceeds a specified threshold. There are a number of critical tele-parameters in studying the network performance as well as evaluating the location management cost. For the sake of analytical simplicity and tractability, the exponential distribution has been popularly assumed for these parameters. However, the recent challenge on tele-parameters modeling reveals that the exponential assumption is not realistic. One instance is the modeling of call holding time in both wireline and wireless systems. In the study [6], the author stated that the exponential assumption may not be valid for modern
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
telephone services. Chlebus [7] reported that call duration in mobile telephony follows the same pattern as shown in [6] for fixed telephony. The long-tailed nature of the call holding time was found in [9]. For wireless mobile system, Erlang call holding time is employed to study the call completion characteristics [21]. The sum of hyper-Exponential (SOHYP) distribution is applied to model the call holding time [23]. In [8,17], real-time measurements from LANs, variable-bit-rate (VBR) video sources, and the World Wide Web have shown that the packet data traffic exhibits a behavior of self-similar nature. In the study [26], the authors investigated the tele-parameters sensitivity problem on the wireless network performance including call holding time, cell residence time, and handoff dwell time. Similarly, the call inter-arrival time is the most important variable characterizing the call arrival process targeting to an MT. Normally, exponential call inter-arrival time has been widely assumed. Owing to the demand of multimedia services, the call inter-arrival time to an MT may not follow the exponential distribution. In addition, since no signaling exchange for location registration during an active conversation, the location management performance should be evaluated between two served calls. However, due to the busy-line effect [10], the time between two served calls may not follow the identical distribution function as the call inter-arrival time. We follow the term inter-service time introduced in [10,11], terming the time duration between two served calls. Specifically, Fang [10,11] introduced the concept of inter-service time to reflect the busy-line effect and motivated the general modeling of location management. The result indicated that the distribution function of inter-service time plays a significant role in analyzing the signaling load. Now, we will present two motivations in our paper for further study in general modeling for location management. In terms of general modeling, the tele-traffic parameter generalization is one aspect. On the other hand, it is observed that different analytical approaches should be employed to develop the cost function for different location management schemes. Hence, it will be helpful in case we could develop an analytical framework applicable to different location management schemes. To achieve this, we introduce the concept location update (LU) inter-arrival time defining the time duration between two LU consecutive operations. This variable is flexible enough to incorporate various location management strategies. The details in deriving the cost function based on LU inter-arrival time and specifying this variable will be presented in following sections. Another motivation is to perform an easy comparison between different location management mechanisms. In two-dimension network, the static scheme is analyzed by assuming the location area as the basic unit. The cost of static scheme is formulated by the location area characteristics, e.g., the mean, variance and probability density function (pdf) of the location area residence time. In contrast, the movement-based scheme is derived with the cell as the basic unit [3]. The cost function of
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movement-based scheme is formulated by the characteristics of cell, e.g., the mean, variance and pdf of cell residence time. Since the relationship between the location area residence time and the cell residence time has not explicitly identified, the comparison between these two schemes is almost infeasible due to the different basic units. To address this challenge, we found that the location area residence time is not an input parameter; but relevent to the mobility behavior, network architecture and system configurations. Based on this, we develop an analytical model to calculate the location area residence time; and consequently bridge the location area residence time and cell residence time. This connection facilitates the comparison between the static scheme and movement-based scheme while such comparison is unavailable in the previous studies. In summary, we first introduce the concept LU inter-arrive time to incorporate different location management strategies. With generally distributed cell residence time, inter-service and LU inter-arrival time, an analytical technique is proposed to derive the closed form formula for the signaling traffic cost; and we then apply the result in the static and dynamic scheme. To facilitate the easy comparison between static and dynamic policies, we derive the location area residence time based on the mobility pattern, cell residence time characteristics and network architectures. Finally, we perform an extensive numerical investigation on the sensitivity of key tele-traffic parameters. The potential application of the proposed analytical technique is also discussed. 2. System model and formula derivation The location management cost is evaluated via the signaling traffic load between two calls, i.e., inter-service time duration. Denote tc as the inter-service time with mean 1/kc, probability density function (pdf) ftc ðtÞ and Laplace Transform (LT) of pdf ftc ðsÞ. It is known that one particular benefit of the hyper-Erlang distribution is its general approximation property, i.e., a good approximation to many other distributions as well as measured data. In particular, its general approximation property has been applied in different scenarios. Yeo and Jun [25] employed the hyper-Erlang distribution to model the call holding time. Ashtianti et al. [4] proposed the hyper-Erlang model for the cell residence time. Recently, [20] model the serving GPRS support node (SGSN) residence time as hyperErlang distribution to evaluate authentication signaling traffic in UMTS network. Without loss of generality, we assume that tc follows the hyper-Erlang distributed tc with pdf [15] ftc ðtÞ ¼
H X i¼1
n
bi
ðni ai Þ i tni 1 ni ai t e ; ðni 1Þ!
H X
bi ¼ 1;
ð1Þ
i¼1
where H and ni are positive integers. ai are constant. The mean of tc is given by
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H 1 X bi ¼ : kc ai i¼1
ð2Þ
hðkÞ ¼ PrðK ¼ kÞ ¼ PrðT k < tc < T kþ1 Þ Z 1 ftc ðtÞPrðT k < t < T kþ1 Þ dt ¼ 0
The LT of pdf is expressed as ni H X ni ai ftc ðsÞ ¼ bi : s þ ni ai i¼1
ð3Þ
Since different location management strategies may be operating in the system, the number of LU during tc is dependent on the specific strategy. Hence, to incorporate different location update schemes, we introduce the generic concept of LU inter-arrival time terming the time duration between two LU requests. The time duration depends on different location management schemes, diverse network architectures and dynamic system configurations. In addition, for different location management strategies, the LU inter-arrival time specifies the period between two consecutive LU requests in the particular mechanism. Denote tLU as the LU inter-arrival time with mean 1/gLU, variance VLU, pdf ftLU ðtÞ, and LT of pdf ftLU ðsÞ. Let t1 denote the period between the call arrival and the instant of the subsequently first LU. The pdf and according LT of t1 are denoted as ft1 ðtÞ and ft1 ðsÞ, respectively. Fig. 1 shows that there are k LU requests during inter-service time tc. Define " # k X tLU;i 1kP2 ; ð4Þ T k ¼ t1 1kP1 þ i¼2
where the indicator function 1e is equal to 1 if the event e is true, zero otherwise. The LT of Tk is given by h ik1 : ð5Þ fTk ðsÞ ¼ ft1 ðsÞ ftLU ðsÞ Denote the variable K as the number of LU during the time duration tc. Let h (k) represent the probability that there are K = k LU during the time duration tc. The LU cost Cu is expressed as the average number of LU operations between inter-service time tc. 1 X C u ¼ du khðkÞ; ð6Þ k¼1
where du represents the unit cost in performing an LU. Now, we will proceed to compute the probability h (k). For k P 1,
¼
Z
1
ftc ðtÞ
0
¼
Z
Z
t
fT k ðxÞ
Z
0
1
Z
1
ftLU ðyÞ dy dx dt
tx t
ftc ðtÞ fT k ðxÞ½1 F tLU ðt xÞ dx dt 0 0 Z 1 Z t ¼ F T k ðxÞ fT k ðxÞF tLU ðt xÞ dx ftc ðtÞ dt 0 0 Z 1 ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞftc ðtÞ dt; ¼
where ~ represents the convolution operation. Substituting this result into (6), we have ) Z 1 (X 1 C u ¼ du k ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞ ftc ðtÞ dt: ð8Þ 0
k¼1
Define the item between {Æ} in the above Eq. (8) as / (t), i.e. /ðtÞ ¼
1 X
k ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞ:
ð9Þ
k¼1
The LT of / (t) is given by 1 X fT ðsÞ ft ðsÞ k k fTk ðsÞ LU s s k¼1 2 h ih ik1 3 1 X 6ft1 ðsÞ 1 ftLU ðsÞ ftLU ðsÞ 7 k4 ¼ 5 s k¼1
/ ðsÞ ¼
h i 1 h ik1 ft1 ðsÞ 1 ftLU ðsÞ X ¼ k ftLU ðsÞ s k¼1 h i ft1 ðsÞ 1 ftLU ðsÞ 1 h ¼ i2 s 1 ftLU ðsÞ ft1 ðsÞ i: ¼ h s 1 ftLU ðsÞ
ð10Þ
Call arrival
Call arrival inter-service time tc
time
… t1
tLU,2
ð7Þ
0
tLU,k-1
Fig. 1. LU operation between inter-service time.
tLU,k
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
Further simplifying (8), we obtain the location update cost as Z 1 /ðtÞftc ðtÞ dt C u ¼ du 0 " # Z 1 n H X ðni ai Þ i tni 1 ni ai t e /ðtÞ bi ¼ du dt ðni 1Þ! 0 i¼1 Z 1 H X ðni ai Þni ¼ du bi /ðtÞtni 1 eni ai t dt ðni 1Þ! 0 i¼1 ¼ du
H X i¼1
ni
bi
ðni ai Þ ðni 1Þ / ðsÞjs¼ni ai ; ðni 1Þ!
Remark that since the concept LU inter-arrival time is generic and the formula is derived based on the relaxed assumptions, the result is applicable in the variety of location management policies. In particular, to evaluate the location update cost in a specific location management policy, we only need to find the LT of tLU and t1 in this scheme. Next, we will present the particular result by substituting the typical inter-service time into the closed-form formula (11). The goal is to demonstrate the flexibility and generalization of the formula, and also to develop the typical results for ready utilization. 2.1. Exponential tc For exponentially distributed inter-service time tc with mean 1/kc, the LT of inter-service time is expressed kc ftc ðsÞ ¼ sþk . This is the situation with Poisson call arrival c process to an MT. The location update cost becomes ft1 ðkc Þ : 1 ftLU ðkc Þ
ð13Þ
2.2. Hyper-exponential tc It has been proofed that hyper-exponential distribution is able to sufficiently approximate the distribution functions whose coefficient of variance is greater than 1. In addition, due to its simplicity, it has been employed to model the call holding time in the literature [22,23]; and also to approximate the heavy-tailed distribution [12]. For the hyper-exponential inter-service time with pdf [13] ftc ðtÞ ¼
H X i¼1
bi ai eai t ;
H X
bi ¼ 1;
H X i¼1
bi ft1 ðai Þ : 1 ftLU ðai Þ
ð15Þ
2.3. Erlang tc For n-stage Erlang inter-service time [13] with mean 1/kc, variance V c ¼ 1=ðnk2c Þ, and pdf n
ftc ðtÞ ¼
ðnkc Þ tn1 nkc t e ; ðn 1Þ!
where n is positive integer: ð16Þ
ð11Þ
where /(n) (s) represents the nth order derivative of the function /* (s) with respect to s; and we have employed the following result for the generic function g (t) with its LT g*(s). Z 1 n n d g ðsÞ gðtÞtn ext dt ¼ ð1Þ j : ð12Þ dsn s¼x 0
C u ¼ du
C u ¼ du
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ð14Þ
i¼1
where H is a positive integer and ai are constant. The location update cost becomes
In such case, the location update cost is given by 0 1ðn1Þ ft1 ðsÞ ðnkc Þn @ h iA js¼nkc : C u ¼ du ðn 1Þ! s 1 f ðsÞ
ð17Þ
tLU
In the following Section, we will focus on developing the pdf or its LT of tLU and t1 in movement-based scheme and static scheme, respectively, in order to demonstrate the applicability of the developed result. As reported in [11], the movement-based scheme is the best choice with respect to the signaling traffic load and implementation complexity. For the similar reason, we analyze the movement-based scheme and compare the existed result for verification. 3. LU inter-arrival time 3.1. LU inter-arrival time in movement-based scheme Let tcrt denote cell residence time with mean 1/l, variance V, pdf ftcrt ðtÞ, and its LT ftcrt ðsÞ. We term the residual cell residence time trcrt as the time duration from an intermediate moment in the cell to the instant the MT leaving the cell. Referring to the residual life theorem [15], we express l½1f ðsÞ
tcrt . the LT of trcrt as ftr ðsÞ ¼ s crt In movement-based scheme, the MT performs an LU operation every M movement, i.e., every M cell boundary crossing. As a consequence, LU inter-arrival time is equal to the summation of M cell residence time and its LT is given by M ftLU ðsÞ ¼ ftcrt ðsÞ : ð18Þ
Referring to the definition, t1 denotes the period from a call arrival to the subsequent first LU operation. Then, t1 is equal to the summation of residual cell residence time and M 1 cell residence time; and its LT is given by M1 ft1 ðsÞ ¼ ftr ðsÞ ftcrt ðsÞ : ð19Þ crt
Substituting (18) and (19) into (11), we obtain the LU cost in movement-based scheme C u ¼ du
H X i¼1
n
bi
ðni ai Þ i ðni 1Þ / ðsÞjs¼ni ai ; ðni 1Þ!
ð20Þ
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where
Under the situation with n-stage Erlang distributed tc, we have 0 M1 1ðn1Þ n l 1 f ðsÞ f ðsÞ ðnkc Þ tcrt h tcrt @ C T ¼ du M i A ðn 1Þ! 2 s 1 f ðsÞ
M1 ftr ðsÞ ftcrt ðsÞ crt / ðsÞ ¼ h M i s 1 ftcrt ðsÞ M1 l 1 ftcrt ðsÞ ftcrt ðsÞ h ¼ : M i s2 1 ftcrt ðsÞ
tcrt
ð21Þ
This is same as the formula in [11], in which an alternative analytical technique is employed. For a trade-off case study, we consider the selective page strategy in [3]. C p ¼ dp ½1 þ 3MðM 1Þ
ð22Þ
with dp representing the unit cost for each paging. Note that we can choose the unit costs du and dp to indicate the significance of the signaling with respect to LU and paging. The total cost CT is the summation of location update cost and the paging cost. CT ¼ Cu þ Cp ¼ du
H X i¼1
M1
n
l½1 ftcrt ðsÞ½ftcrt ðsÞ
ðni ai Þ i bi ðni 1Þ!
!ðni 1Þ
M
s2 ½1 ½ftcrt ðsÞ
js¼ni ai þ dp ½1 þ 3MðM 1Þ: Specifically, in case of exponential tc, we obtain M1 l 1 ftcrt ðkc Þ ftcrt ðkc Þ C u ¼ du ; M kc 1 f ðkc Þ
ð23Þ
ð24Þ
tcrt
C p ¼ dp ½1 þ 3MðM 1Þ:
ð25Þ
In this situation, [18] made a solid mathematical proof that there exists an optimal LU threshold M* to minimize the total signal cost. Specifically, after computing the first derivative and the second derivative of Cu in (24) and Cp in (25) with respect to the parameter M, we can find that Cu is an decreasing and convex function; and Cp is an increasing and convex function. As a consequence, CT is an convex function and there exists an unique LU threshold to minimize the total signaling cost. It is suggested to refer to the elegant algorithm in [18] to determine the optimal threshold. In case of hyper-exponential tc, we have C u ¼ du
M1 H X bi l ½1 ftcrt ðai Þ½ftcrt ðai Þ ; M ai 1 ½ft ðai Þ i¼1
ð26Þ
crt
C p ¼ dp ½1 þ 3MðM 1Þ:
ð27Þ
Following the similar mathematical proof in case of exponential tc and the study [18], we are able to find that CT with hyper-exponential tc is also an convex function and there exists an unique LU threshold to minimize the total signaling cost. In addition, as stated in Section 2.2, any distributions with coefficient of variance Cv larger than 1 can be sufficiently approximated by hyper-exponential distribution. As a consequence, for any distributed inter-service time with Cv > 1, there exists an LU threshold M* to minimize CT.
js¼nkc þ dp ½1 þ 3MðM 1Þ:
ð28Þ
3.2. LU inter-arrival time in static scheme In the static scheme [1], the whole network is statically partitioned into several consecutive location area (LA). Each LA includes a group of cells as shown in Fig. 2. The innermost cell ‘‘0’’ is called the center cell. Cells labeled ‘‘1’’ form the first ring around cell ‘‘0’’, cells labeled ‘‘2’’ form the second ring around cell ‘‘0’’ and so forth. The outermost cells are in the ring K, e.g., K = 3 in Fig. 2. We define LA residence time tLA as the time duration an MT stays in an LA. Hence, in the static strategy, an MT performs a location registration every LA residence time; and the LU inter-arrival time tLU is equal to the LA residence time tLA. Hence, in static scheme, we may use these two terminologies interchangeable. In the static scheme, we gLU ½1ft ðsÞ LU express the LT of t1 as ft1 ðsÞ ¼ . Submitting this s and LT of tLU into (11), we can compute the LU cost in static mechanism. g C u ¼ du LU : ð29Þ kc Observing (20) and (29), we can clearly see that it is impossible to compare the cost in movement-based scheme and static scheme because of the different basic units. The formula (20) is derived based on the cell residence time. In contrast, the formula (29) is related to the LA residence time. In the previous studies, for the sake of analytical simplicity, LA residence time is traditionally assumed to be an input and independent from the MT’s mobility behavior and network architecture; and additionally follow an exponential distribution. In the study [16], the exponential assumption is relaxed to be a general function. However, LA residence time is still assumed to be independent from the system dynamics and regarded as an input parameter. In this section, we will show that LA residence time is not an input parameter, but dependent on the network infrastructure and mobility pattern; and additionally develop the LA residence time based on the concept of virtual ring. Since our derived LA residence time has close relationship with the cell residence time, we can compare the static scheme and movement-based scheme on a common basis. In Fig. 2, let state k represent the situation an MT stays in the ring-k (0 6 k 6 K). In the static location management, whenever an MT leaves an old LA (LAo) and enters a new LA (LAn), an LU is triggered. More specifically, an MT’s behavior in terms of state transit is as follows. The MT moves from the outmost ring-K in LAo. After a number of cell boundary crossing in the coverage of LAo, the MT leaves LAo and enters the ring-(K + 1), and then trig-
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Ring-3 Ring-2 Ring-1
0
Ring-0
Location Area
Fig. 2. Location area definition.
ger LU message. Here, the ring-(K + 1) is a virtual ring in the LAo and actually the ring-K in the LAn. Hence, for the MT, the state (K + 1) is the absorbing state. Let qi,j be the first-step transit probability from ring-i to ring-j. Employing random walking model, we have qi;i1 ¼
2i 1 ; 1 6 i 6 K; 6i
qi;i ¼
1 3
ð30Þ
From (31) to (34), we can efficiently calculate the probability pK,K+1 (m) for different m. Thereafter, LA residence time is equal to the summation of m cell residence time with the probability pK,K+1 (m). Taking into account the normalization constraint, the LT of tLU pdf is given by m P1 m¼1 p K;Kþ1 ðmÞ ftcrt ðsÞ P1 : ð35Þ ftLU ðsÞ ¼ m¼1 p K;Kþ1 ðmÞ
with the boundary condition q0,1 = 1 and qK+1,K+1 = 1. We now need to know how many cells have been traversed when the MT moves from ring-K and first arrives at ring-(K + 1). Accordingly, we denote the probability by pi,j (m) that there are m cell boundary crossings, given that an MT starts from the ring-i and first passages the ring-j. For "i, j, a recursive relationship is developed as follows:
The average LU inter-arrival time 1/gLU is given by P1 1 1 m¼1 mpK;Kþ1 ðmÞ P1 : ð36Þ ¼ gLU l m¼1 p K;Kþ1 ðmÞ
ð31Þ pi;j ð1Þ ¼ qi;j ; m ¼ 1; X pi;j ðmÞ ¼ qi;k pk;j ðm 1Þ ¼ qi;i1 pi1;j ðm 1Þ 1ði16¼jÞ
ð37Þ
k6¼j
þ qi;i pi;j ðm 1Þ 1ði6¼jÞ þ qi;iþ1 piþ1;j ðm 1Þ 1ðiþ16¼jÞ ; m > 1;
ð32Þ
where qi,j is given by (30). Hence, pK,K+1 (m) represents the probability that there are m cell boundary crossings when the MT moves from ring-K and first arrives at ring-(K + 1). Its recursive algorithm is specified as pK;Kþ1 ð1Þ ¼ qK;Kþ1 ; X pK;Kþ1 ðmÞ ¼ qK;k pk;Kþ1 ðm 1Þ; k6¼Kþ1
ð33Þ m > 1:
ð34Þ
Hence, the LU cost is given by P1 pK;Kþ1 ðmÞ l : C u ¼ du P1m¼1 kc m¼1 mpK;Kþ1 ðmÞ
The number of cells in an LA is 3K (K + 1) + 1. In static scheme, the number of paged cells is the total number of cells in an LA. Hence, the total cost is given by P1 pK;Kþ1 ðmÞ l þ dp ½3KðK þ 1Þ þ 1: ð38Þ C T ¼ du P1m¼1 kc m¼1 mpK;Kþ1 ðmÞ As discussed, in two-dimension network, the static scheme is traditionally analyzed with the LA as a basic unit and the movement-based scheme is derived with the cell as the basic unit. This makes the infeasible comparison between the two schemes. It is clear that our formula (23) for movement-based scheme and the (38) for static scheme
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are developed based on the same basic unit, i.e., cell residence time characteristics. Hence, we can make an thorough comparison between these two schemes under different scenarios. 4. Numerical results The sensitivity of inter-service time in movement-based scheme has been extensively discussed and validated in the study [11]. In the literatures, the LU cost for static scheme is formulated based on the minimum unit as LA. This handling makes the absence of comparison between static scheme and movement-based scheme on a common basis. In this section, we will concentrate on the comparison between static and movement-based scheme under different scenarios. The cell residence time follows Gamma distribution with l = 1.0 and variance V [27]. Then, the LT of tcrt pdf is given by c lc 1 ð39Þ ftcrt ðsÞ ¼ ; c¼ 2 : s þ lc lV If not specified, the variance V = 1.5/l2. Define callmobility-ratio (CMR) as CMR = kc/l. Without loss of generality, the per-cell paging cost is set as dp = 1 and the unit LU cost is set as du = 10. Fig. 3 shows the total cost comparison between static scheme and the movement-based scheme. In this example, the exponential inter-service time is assumed. The results indicate that the cost decreases with larger CMR. This can be explained as follows. Greater CMR shows more frequent call activities, which lead to less LU operations during two calls and, consequently, smaller cost. It is observed that the cost in movement-based scheme is always larger than the cost in static scheme. This is due to the possible random roaming in static scheme before
moving out of an LA. Fig. 4 shows the total cost comparison between static scheme and the movement-based scheme. In this example, 4-stage Erlang distributed inter-service time is assumed. Similarly, the movementbased scheme generates higher signaling cost than the static scheme. Fig. 5 shows the total cost in terms of CMR under different du. In the figures, the curves without symbol represent the total cost in static scheme with different LA size K. The solid line with ‘•’ shows the cost in movement-based scheme with the optimal M, where each optimal threshold is able to minimize CT under the specific scenario. Under the condition of optimized M, the comparison result is dependent on the particular situation. The results indicate that, in case CMR smaller than 0.08 when K = 3, movement-based scheme generates higher signaling cost than the static scheme. For instance, the increased load can reach up to 20.8%. In case CMR larger than 0.08, movement-based scheme outperforms the static scheme by generating less signaling load. 5. Conclusions In this paper, we perform a general modeling and analysis for the location management by relaxing the tele-traffic parameters. To achieve this aim, we introduce the term location update inter-arrival time applicable in a generic location management scheme; and then specify its characteristics in static and dynamic schemes. A closedform formula is derived for the location management signaling cost function with the generalized LU inter-arrival time, cell residence time and inter-service time. The results indicate that the developed formulae enable the easy comparison between static and movement-based schemes.
350
300 Static(K=2) Movement (M=3)
Static(K=3) Movement (M=4)
300
250
250 200
CT
CT
200 150
150 100 100 50
50
0 –2 10
0
10 CMR
0 –2 10
0
10 CMR
Fig. 3. Total cost comparison between static scheme and movement-based scheme with exponential inter-service time.
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395 350
300
300 Static(K=3) Movement (M=4)
Static(K=2) Movement (M=3) 250
250 200
CT
CT
200 150
150 100 100 50
50
0 –2 10
0
10 CMR
0 –2 10
0
10 CMR
Fig. 4. Total cost comparison between static scheme and movement-based scheme with 4-stage Erlang inter-service time.
Fig. 5. Total cost comparison between static scheme and movement-based scheme with 4-stage Erlang inter-service time.
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Acknowledgement The authors thank the anonymous reviewers for their very helpful suggestions and comments. References [1] ETSI, Digital Cellular Telecommunications System (Phase 2+): Mobile Application Part (MAP) Specification (GSM 09.02 version 7.51 Release). 1998. [3] I.F. Akyildiz, J. Ho, Y. Lin, Movement-based location update and selective paging for PCS networks, IEEE/ACM Trans. Network. 4 (1996) 629–638. [4] F. Ashtiani, J.A. Salehi, M.R. Aref, Mobility modeling and analytical solution for spatial traffic distribution in wireless multimedia networks, IEEE J. Select. Areas Commun. 21 (10) (2003) 1699–1709. [5] A. Bar-Noy, I. Kessler, M. Sidi, Mobile users: to update or not to update? ACM/Baltzer Wireless Networks 1 (1995) 175–195. [6] V.-A. Bolotin, Modeling call holding time distributions for CCS network design and performance analysis, IEEE J. Select. Areas Commun. 12 (3) (1994) 433–438. [7] E. Chlebus, Empirical validation of call holding time distribution in cellular communications systems, Teletraffic Contributions for the Information Age (Proc. 15th ITC), vol. 2b, Elsevier, Amsterdam, the Netherlands, 1999, pp. 1179–1188. [8] M.E. Crovella, A. Bestavros, Sel-similarity in world wide web traffic: evidence and possible causes, IEEE/ACM Trans. Network. 5 (6) (1997) 835–846. [9] D.E. Duffy, A.A. McIntosh, M. Rosenstein, W. Willinger, Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks, IEEE J. Select. Areas Commun. 12 (1994) 544–551. [10] Y.G. Fang, General modeling and performance analysis for location management in wireless mobile networks, IEEE Trans. Comput. 51 (10) (2002) 1169–1181. [11] Y.G. Fang, Movement-based location management and tradeoff analysis for wireless mobile networks, IEEE Trans. Comput. 52 (6) (2003) 791–803. [12] A. Feldmann, W. Whitt, Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Perform. Evaluat. 31 (8) (1998) 963–976. [13] N.A.J. Hastings, J.B. Peacock, Statistical Distributions, Wiley, New York, 1975. [14] R. Jain, Y.-B. Lin, ‘‘An auxiliary user location strategy employing forwarding pointers to reduce network impacts of PCS’’, IEEE ICC’95, pp. 740–744, June,1995, WA, US. [15] F.P. Kelly, Reversibility and Stochastic Networks, Wiley, New York, 1979. [16] S.J. Kwon, M.Y. Chung, D.D. Sung, Mobility management schemes for support of UPT in mobile networks, IEEE ICC’2000 2 (2000) 675–679. [17] W. Leland, M. Taqqu, W. Willinger, D. Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Trans. Network. 2 (1) (1994) 1–15. [18] J. Li, H. Kameda, K. Li, Optimal dynamic mobility management for PCS networks, IEEE/ACM Trans. Network. 8 (3) (2000) 319–327. [20] Y.B. Lin, Y.K. Chen, Reducing authentication signaling traffic in third-generation mobile network, IEEE Trans. Wireless Commun. 2 (3) (2003) 493–501. [21] Y.B. Lin, I. Chlamtac, Effects of Erlang call holding times on PCS call completion, IEEE Trans. Veh. Technol. 48 (3) (1999) 815–823. [22] M.A. Marsan, G. Ginella, R. Maglione, M. Meo, Performance analysis of hierarchical cellular networks with generally distributed call holding times and dwell times, IEEE Trans. Wireless Commun. 3 (2004) 248. [23] P.V. Orlik, S.S. Rappaport, A model for teletraffic performance and channel holding time characterization in wireless cellular communication with general session and dwell time distributions, IEEE J. Select. Areas Commun. 16 (1998) 788803.
[24] V. Wong, V. Leung, An adaptive distance-based location update algorithm for next generation PCS networks, IEEE J. Sel. Areas Commun. 19 (10) (2001) 1942–1952. [25] K. Yeo, C.-H. Jun, Modeling and analysis of hierarchical cellular networks with general distributions of call and cell residence times, IEEE Trans. Veh. Technol. 51 (6) (2002) 1361–1374. [26] Y. Zhang, B.-H. Soong, Handoff dwell time distribution effect on mobile network performance, IEEE Trans. Veh. Technol. 54 (4) (2005). [27] M.M. Zonoozi, P. Dassanayake, User mobility modeling and characterization of mobility patterns, IEEE J. Select. Areas Commun. 15 (1997) 1239–1252. Yan Zhang received the Ph.D. degree in School of Electrical & Electronics Engineering, Nanyang Technological University, Singapore. Since Aug. 2004, he has been working with the NICT Singapore, National Institute of Information and Communications Technology. He is currently serving the Book Series Editor for the book series on ‘‘Wireless Networks and Mobile Communications’’ (Auerbach Publications, CRC Press, Taylor and Francis Group). He has served as coeditor for several books. His research interests include resource, mobility and security management and performance evaluation in cellular networks and mobile ad hoc networks.
Jun Zheng received the B.S. and M.S. degrees in Electrical Engineering from Chongqing University, China, in 1993, 1996, respectively, the M.S.E. degree in Biomedical Engineering from Wright State University, Dayton, Ohio, in 2001, and the Ph.D. degree in Computer Engineering from University of Nevada, Las Vegas in 2005. Currently he is an assistant professor in the Department of Computer Science at Queens College - City University of New York. His research interests are mobility and resource management in wireless and mobile networks, media access control, performance evaluation, network security, computer architectures, fault-tolerant computing, and image processing.
Lili Zhang received her B.S. degree in electronic engineering from Shanghai Jiaotong University, Shanghai, China in 1999, the M.S. degree in communication engineering from China Academy of Telecommunication Technologies, Beijing, China in 2002, and the Ph.D. degree in communication engineering from school of electrical and electronic engineering, Nanyang Technological University, Singapore in 2005. She is currently a Research Associate in the Intelligent System Center, Nanyang Technological University, Singapore. Her research interests include multi-channel medium access control design, topology control and cross-layer design for mobile ad hoc networks and wireless sensor networks.
Yifan Chen received the B.Eng. and Ph.D. degrees in electrical and electronic engineering from Nanyang Technological University (NTU), Singapore, in 2002 and 2005, respectively. He is presently with the Biomedical Engineering Research Centre, NTU, as a research scientist. His current research interests involve microwave imaging of biological structures, statistical modeling of mobile radio channels, ultra wideband source localization, and multiple-antenna system performance analysis.
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395 Maode Ma received his B.E. degree in computer engineering from Tsinghua University in 1982, M.E. degree in computer engineering from Tianjin University in 1991 and Ph.D. degree in computer science from Hong Kong University of Science and Technology in 1999. Now, Dr. Ma is an assistant professor in the School of Electrical and Electronic Engineering at Nanyang Technological University in Singapore. Dr. Ma has published more than 50 academic papers in the areas
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of optical networks, wireless networks, etc. His current research interests are wireless networks, optical networks, performance analysis of computer networks, etc. He currently serves as an associate editor for IEEE Communications Letters and an editor for IEEE Communications Surveys & Tutorials.
Modeling location management in wireless networks with generally distributed parameters Yan Zhang a
a,*,1
, Jun Zheng b, Lili Zhang c, Yifan Chen c, Maode Ma
c
Wireless Communication Laboratory, NICT Singapore, National Institute of Information and Communications Technology, Singapore b Department of Computer Science, Queens College, City University of New York, USA c School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore Received 18 September 2005; received in revised form 1 February 2006; accepted 14 February 2006 Available online 10 March 2006
Abstract In the wireless mobile networks, location update (LU) and paging are two basic operations in the framework of mobility management. The trade-off analysis between LU and paging is traditionally performed under the assumption that the key variables follow exponential distribution. However, recent work showed that such assumptions may be incorrect. In this paper, by introducing the concept of LU inter-arrival time and additionally relaxing the assumptions for the inter-service time, cell residence time and LU inter-arrival time, we present an analytical technique to develop the closed-form formula for the signaling cost applicable to different location management schemes. Specifically, the result is applied to the static scheme and the dynamic movement-based scheme to demonstrate the general applicability. Illustrative numerical examples are given to compare the location management scheme performance based on the cell characteristics. 2006 Elsevier B.V. All rights reserved. Keywords: Dynamic location management; Location update; Paging; Inter-service time; Cell residence time; Location area; Wireless networks
1. Introduction Location management is an important component in the present as well as the next generation wireless mobile networks to support the seamless mobility requirement. There are two operations in the framework of location management: updating the Mobile Terminal’s (MT) location and paging the MT. Accordingly, the signaling cost is comprised of the location update cost and paging cost. The two-tier Home Location Register/Visitor Location Register (HLR/VLR) architecture and static location management scheme are employed in the second-generation mobile network [1]; and it has been found that this mechanism generates high signaling traffics. To reduce the associated signaling cost, a variety of dynamic location *
Corresponding author. Tel.: +65 67711011; fax: +65 67795753. E-mail addresses: [email protected] (Y. Zhang), [email protected] (J. Zheng). 1 Present address: NICT Singapore, National Institute of Information and Communications Technology, Singapore. 0140-3664/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.comcom.2006.02.014
managements have been proposed. These include timerbased scheme [5], movement-based scheme [5], distancebased scheme [5,24], and pointer forwarding scheme [14]. The difference between these strategies lies in the threshold or the event triggering the location update process. For instance, in timer-based scheme, updates are made at a fixed timer interval. In movement-based scheme, updates are made whenever the number of cell-boundary crossings (or equivalently termed as movement) since the last update exceeds a specified threshold. There are a number of critical tele-parameters in studying the network performance as well as evaluating the location management cost. For the sake of analytical simplicity and tractability, the exponential distribution has been popularly assumed for these parameters. However, the recent challenge on tele-parameters modeling reveals that the exponential assumption is not realistic. One instance is the modeling of call holding time in both wireline and wireless systems. In the study [6], the author stated that the exponential assumption may not be valid for modern
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
telephone services. Chlebus [7] reported that call duration in mobile telephony follows the same pattern as shown in [6] for fixed telephony. The long-tailed nature of the call holding time was found in [9]. For wireless mobile system, Erlang call holding time is employed to study the call completion characteristics [21]. The sum of hyper-Exponential (SOHYP) distribution is applied to model the call holding time [23]. In [8,17], real-time measurements from LANs, variable-bit-rate (VBR) video sources, and the World Wide Web have shown that the packet data traffic exhibits a behavior of self-similar nature. In the study [26], the authors investigated the tele-parameters sensitivity problem on the wireless network performance including call holding time, cell residence time, and handoff dwell time. Similarly, the call inter-arrival time is the most important variable characterizing the call arrival process targeting to an MT. Normally, exponential call inter-arrival time has been widely assumed. Owing to the demand of multimedia services, the call inter-arrival time to an MT may not follow the exponential distribution. In addition, since no signaling exchange for location registration during an active conversation, the location management performance should be evaluated between two served calls. However, due to the busy-line effect [10], the time between two served calls may not follow the identical distribution function as the call inter-arrival time. We follow the term inter-service time introduced in [10,11], terming the time duration between two served calls. Specifically, Fang [10,11] introduced the concept of inter-service time to reflect the busy-line effect and motivated the general modeling of location management. The result indicated that the distribution function of inter-service time plays a significant role in analyzing the signaling load. Now, we will present two motivations in our paper for further study in general modeling for location management. In terms of general modeling, the tele-traffic parameter generalization is one aspect. On the other hand, it is observed that different analytical approaches should be employed to develop the cost function for different location management schemes. Hence, it will be helpful in case we could develop an analytical framework applicable to different location management schemes. To achieve this, we introduce the concept location update (LU) inter-arrival time defining the time duration between two LU consecutive operations. This variable is flexible enough to incorporate various location management strategies. The details in deriving the cost function based on LU inter-arrival time and specifying this variable will be presented in following sections. Another motivation is to perform an easy comparison between different location management mechanisms. In two-dimension network, the static scheme is analyzed by assuming the location area as the basic unit. The cost of static scheme is formulated by the location area characteristics, e.g., the mean, variance and probability density function (pdf) of the location area residence time. In contrast, the movement-based scheme is derived with the cell as the basic unit [3]. The cost function of
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movement-based scheme is formulated by the characteristics of cell, e.g., the mean, variance and pdf of cell residence time. Since the relationship between the location area residence time and the cell residence time has not explicitly identified, the comparison between these two schemes is almost infeasible due to the different basic units. To address this challenge, we found that the location area residence time is not an input parameter; but relevent to the mobility behavior, network architecture and system configurations. Based on this, we develop an analytical model to calculate the location area residence time; and consequently bridge the location area residence time and cell residence time. This connection facilitates the comparison between the static scheme and movement-based scheme while such comparison is unavailable in the previous studies. In summary, we first introduce the concept LU inter-arrive time to incorporate different location management strategies. With generally distributed cell residence time, inter-service and LU inter-arrival time, an analytical technique is proposed to derive the closed form formula for the signaling traffic cost; and we then apply the result in the static and dynamic scheme. To facilitate the easy comparison between static and dynamic policies, we derive the location area residence time based on the mobility pattern, cell residence time characteristics and network architectures. Finally, we perform an extensive numerical investigation on the sensitivity of key tele-traffic parameters. The potential application of the proposed analytical technique is also discussed. 2. System model and formula derivation The location management cost is evaluated via the signaling traffic load between two calls, i.e., inter-service time duration. Denote tc as the inter-service time with mean 1/kc, probability density function (pdf) ftc ðtÞ and Laplace Transform (LT) of pdf ftc ðsÞ. It is known that one particular benefit of the hyper-Erlang distribution is its general approximation property, i.e., a good approximation to many other distributions as well as measured data. In particular, its general approximation property has been applied in different scenarios. Yeo and Jun [25] employed the hyper-Erlang distribution to model the call holding time. Ashtianti et al. [4] proposed the hyper-Erlang model for the cell residence time. Recently, [20] model the serving GPRS support node (SGSN) residence time as hyperErlang distribution to evaluate authentication signaling traffic in UMTS network. Without loss of generality, we assume that tc follows the hyper-Erlang distributed tc with pdf [15] ftc ðtÞ ¼
H X i¼1
n
bi
ðni ai Þ i tni 1 ni ai t e ; ðni 1Þ!
H X
bi ¼ 1;
ð1Þ
i¼1
where H and ni are positive integers. ai are constant. The mean of tc is given by
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H 1 X bi ¼ : kc ai i¼1
ð2Þ
hðkÞ ¼ PrðK ¼ kÞ ¼ PrðT k < tc < T kþ1 Þ Z 1 ftc ðtÞPrðT k < t < T kþ1 Þ dt ¼ 0
The LT of pdf is expressed as ni H X ni ai ftc ðsÞ ¼ bi : s þ ni ai i¼1
ð3Þ
Since different location management strategies may be operating in the system, the number of LU during tc is dependent on the specific strategy. Hence, to incorporate different location update schemes, we introduce the generic concept of LU inter-arrival time terming the time duration between two LU requests. The time duration depends on different location management schemes, diverse network architectures and dynamic system configurations. In addition, for different location management strategies, the LU inter-arrival time specifies the period between two consecutive LU requests in the particular mechanism. Denote tLU as the LU inter-arrival time with mean 1/gLU, variance VLU, pdf ftLU ðtÞ, and LT of pdf ftLU ðsÞ. Let t1 denote the period between the call arrival and the instant of the subsequently first LU. The pdf and according LT of t1 are denoted as ft1 ðtÞ and ft1 ðsÞ, respectively. Fig. 1 shows that there are k LU requests during inter-service time tc. Define " # k X tLU;i 1kP2 ; ð4Þ T k ¼ t1 1kP1 þ i¼2
where the indicator function 1e is equal to 1 if the event e is true, zero otherwise. The LT of Tk is given by h ik1 : ð5Þ fTk ðsÞ ¼ ft1 ðsÞ ftLU ðsÞ Denote the variable K as the number of LU during the time duration tc. Let h (k) represent the probability that there are K = k LU during the time duration tc. The LU cost Cu is expressed as the average number of LU operations between inter-service time tc. 1 X C u ¼ du khðkÞ; ð6Þ k¼1
where du represents the unit cost in performing an LU. Now, we will proceed to compute the probability h (k). For k P 1,
¼
Z
1
ftc ðtÞ
0
¼
Z
Z
t
fT k ðxÞ
Z
0
1
Z
1
ftLU ðyÞ dy dx dt
tx t
ftc ðtÞ fT k ðxÞ½1 F tLU ðt xÞ dx dt 0 0 Z 1 Z t ¼ F T k ðxÞ fT k ðxÞF tLU ðt xÞ dx ftc ðtÞ dt 0 0 Z 1 ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞftc ðtÞ dt; ¼
where ~ represents the convolution operation. Substituting this result into (6), we have ) Z 1 (X 1 C u ¼ du k ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞ ftc ðtÞ dt: ð8Þ 0
k¼1
Define the item between {Æ} in the above Eq. (8) as / (t), i.e. /ðtÞ ¼
1 X
k ½F T k ðtÞ fT k ðtÞ~F tLU ðtÞ:
ð9Þ
k¼1
The LT of / (t) is given by 1 X fT ðsÞ ft ðsÞ k k fTk ðsÞ LU s s k¼1 2 h ih ik1 3 1 X 6ft1 ðsÞ 1 ftLU ðsÞ ftLU ðsÞ 7 k4 ¼ 5 s k¼1
/ ðsÞ ¼
h i 1 h ik1 ft1 ðsÞ 1 ftLU ðsÞ X ¼ k ftLU ðsÞ s k¼1 h i ft1 ðsÞ 1 ftLU ðsÞ 1 h ¼ i2 s 1 ftLU ðsÞ ft1 ðsÞ i: ¼ h s 1 ftLU ðsÞ
ð10Þ
Call arrival
Call arrival inter-service time tc
time
… t1
tLU,2
ð7Þ
0
tLU,k-1
Fig. 1. LU operation between inter-service time.
tLU,k
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
Further simplifying (8), we obtain the location update cost as Z 1 /ðtÞftc ðtÞ dt C u ¼ du 0 " # Z 1 n H X ðni ai Þ i tni 1 ni ai t e /ðtÞ bi ¼ du dt ðni 1Þ! 0 i¼1 Z 1 H X ðni ai Þni ¼ du bi /ðtÞtni 1 eni ai t dt ðni 1Þ! 0 i¼1 ¼ du
H X i¼1
ni
bi
ðni ai Þ ðni 1Þ / ðsÞjs¼ni ai ; ðni 1Þ!
Remark that since the concept LU inter-arrival time is generic and the formula is derived based on the relaxed assumptions, the result is applicable in the variety of location management policies. In particular, to evaluate the location update cost in a specific location management policy, we only need to find the LT of tLU and t1 in this scheme. Next, we will present the particular result by substituting the typical inter-service time into the closed-form formula (11). The goal is to demonstrate the flexibility and generalization of the formula, and also to develop the typical results for ready utilization. 2.1. Exponential tc For exponentially distributed inter-service time tc with mean 1/kc, the LT of inter-service time is expressed kc ftc ðsÞ ¼ sþk . This is the situation with Poisson call arrival c process to an MT. The location update cost becomes ft1 ðkc Þ : 1 ftLU ðkc Þ
ð13Þ
2.2. Hyper-exponential tc It has been proofed that hyper-exponential distribution is able to sufficiently approximate the distribution functions whose coefficient of variance is greater than 1. In addition, due to its simplicity, it has been employed to model the call holding time in the literature [22,23]; and also to approximate the heavy-tailed distribution [12]. For the hyper-exponential inter-service time with pdf [13] ftc ðtÞ ¼
H X i¼1
bi ai eai t ;
H X
bi ¼ 1;
H X i¼1
bi ft1 ðai Þ : 1 ftLU ðai Þ
ð15Þ
2.3. Erlang tc For n-stage Erlang inter-service time [13] with mean 1/kc, variance V c ¼ 1=ðnk2c Þ, and pdf n
ftc ðtÞ ¼
ðnkc Þ tn1 nkc t e ; ðn 1Þ!
where n is positive integer: ð16Þ
ð11Þ
where /(n) (s) represents the nth order derivative of the function /* (s) with respect to s; and we have employed the following result for the generic function g (t) with its LT g*(s). Z 1 n n d g ðsÞ gðtÞtn ext dt ¼ ð1Þ j : ð12Þ dsn s¼x 0
C u ¼ du
C u ¼ du
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ð14Þ
i¼1
where H is a positive integer and ai are constant. The location update cost becomes
In such case, the location update cost is given by 0 1ðn1Þ ft1 ðsÞ ðnkc Þn @ h iA js¼nkc : C u ¼ du ðn 1Þ! s 1 f ðsÞ
ð17Þ
tLU
In the following Section, we will focus on developing the pdf or its LT of tLU and t1 in movement-based scheme and static scheme, respectively, in order to demonstrate the applicability of the developed result. As reported in [11], the movement-based scheme is the best choice with respect to the signaling traffic load and implementation complexity. For the similar reason, we analyze the movement-based scheme and compare the existed result for verification. 3. LU inter-arrival time 3.1. LU inter-arrival time in movement-based scheme Let tcrt denote cell residence time with mean 1/l, variance V, pdf ftcrt ðtÞ, and its LT ftcrt ðsÞ. We term the residual cell residence time trcrt as the time duration from an intermediate moment in the cell to the instant the MT leaving the cell. Referring to the residual life theorem [15], we express l½1f ðsÞ
tcrt . the LT of trcrt as ftr ðsÞ ¼ s crt In movement-based scheme, the MT performs an LU operation every M movement, i.e., every M cell boundary crossing. As a consequence, LU inter-arrival time is equal to the summation of M cell residence time and its LT is given by M ftLU ðsÞ ¼ ftcrt ðsÞ : ð18Þ
Referring to the definition, t1 denotes the period from a call arrival to the subsequent first LU operation. Then, t1 is equal to the summation of residual cell residence time and M 1 cell residence time; and its LT is given by M1 ft1 ðsÞ ¼ ftr ðsÞ ftcrt ðsÞ : ð19Þ crt
Substituting (18) and (19) into (11), we obtain the LU cost in movement-based scheme C u ¼ du
H X i¼1
n
bi
ðni ai Þ i ðni 1Þ / ðsÞjs¼ni ai ; ðni 1Þ!
ð20Þ
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Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
where
Under the situation with n-stage Erlang distributed tc, we have 0 M1 1ðn1Þ n l 1 f ðsÞ f ðsÞ ðnkc Þ tcrt h tcrt @ C T ¼ du M i A ðn 1Þ! 2 s 1 f ðsÞ
M1 ftr ðsÞ ftcrt ðsÞ crt / ðsÞ ¼ h M i s 1 ftcrt ðsÞ M1 l 1 ftcrt ðsÞ ftcrt ðsÞ h ¼ : M i s2 1 ftcrt ðsÞ
tcrt
ð21Þ
This is same as the formula in [11], in which an alternative analytical technique is employed. For a trade-off case study, we consider the selective page strategy in [3]. C p ¼ dp ½1 þ 3MðM 1Þ
ð22Þ
with dp representing the unit cost for each paging. Note that we can choose the unit costs du and dp to indicate the significance of the signaling with respect to LU and paging. The total cost CT is the summation of location update cost and the paging cost. CT ¼ Cu þ Cp ¼ du
H X i¼1
M1
n
l½1 ftcrt ðsÞ½ftcrt ðsÞ
ðni ai Þ i bi ðni 1Þ!
!ðni 1Þ
M
s2 ½1 ½ftcrt ðsÞ
js¼ni ai þ dp ½1 þ 3MðM 1Þ: Specifically, in case of exponential tc, we obtain M1 l 1 ftcrt ðkc Þ ftcrt ðkc Þ C u ¼ du ; M kc 1 f ðkc Þ
ð23Þ
ð24Þ
tcrt
C p ¼ dp ½1 þ 3MðM 1Þ:
ð25Þ
In this situation, [18] made a solid mathematical proof that there exists an optimal LU threshold M* to minimize the total signal cost. Specifically, after computing the first derivative and the second derivative of Cu in (24) and Cp in (25) with respect to the parameter M, we can find that Cu is an decreasing and convex function; and Cp is an increasing and convex function. As a consequence, CT is an convex function and there exists an unique LU threshold to minimize the total signaling cost. It is suggested to refer to the elegant algorithm in [18] to determine the optimal threshold. In case of hyper-exponential tc, we have C u ¼ du
M1 H X bi l ½1 ftcrt ðai Þ½ftcrt ðai Þ ; M ai 1 ½ft ðai Þ i¼1
ð26Þ
crt
C p ¼ dp ½1 þ 3MðM 1Þ:
ð27Þ
Following the similar mathematical proof in case of exponential tc and the study [18], we are able to find that CT with hyper-exponential tc is also an convex function and there exists an unique LU threshold to minimize the total signaling cost. In addition, as stated in Section 2.2, any distributions with coefficient of variance Cv larger than 1 can be sufficiently approximated by hyper-exponential distribution. As a consequence, for any distributed inter-service time with Cv > 1, there exists an LU threshold M* to minimize CT.
js¼nkc þ dp ½1 þ 3MðM 1Þ:
ð28Þ
3.2. LU inter-arrival time in static scheme In the static scheme [1], the whole network is statically partitioned into several consecutive location area (LA). Each LA includes a group of cells as shown in Fig. 2. The innermost cell ‘‘0’’ is called the center cell. Cells labeled ‘‘1’’ form the first ring around cell ‘‘0’’, cells labeled ‘‘2’’ form the second ring around cell ‘‘0’’ and so forth. The outermost cells are in the ring K, e.g., K = 3 in Fig. 2. We define LA residence time tLA as the time duration an MT stays in an LA. Hence, in the static strategy, an MT performs a location registration every LA residence time; and the LU inter-arrival time tLU is equal to the LA residence time tLA. Hence, in static scheme, we may use these two terminologies interchangeable. In the static scheme, we gLU ½1ft ðsÞ LU express the LT of t1 as ft1 ðsÞ ¼ . Submitting this s and LT of tLU into (11), we can compute the LU cost in static mechanism. g C u ¼ du LU : ð29Þ kc Observing (20) and (29), we can clearly see that it is impossible to compare the cost in movement-based scheme and static scheme because of the different basic units. The formula (20) is derived based on the cell residence time. In contrast, the formula (29) is related to the LA residence time. In the previous studies, for the sake of analytical simplicity, LA residence time is traditionally assumed to be an input and independent from the MT’s mobility behavior and network architecture; and additionally follow an exponential distribution. In the study [16], the exponential assumption is relaxed to be a general function. However, LA residence time is still assumed to be independent from the system dynamics and regarded as an input parameter. In this section, we will show that LA residence time is not an input parameter, but dependent on the network infrastructure and mobility pattern; and additionally develop the LA residence time based on the concept of virtual ring. Since our derived LA residence time has close relationship with the cell residence time, we can compare the static scheme and movement-based scheme on a common basis. In Fig. 2, let state k represent the situation an MT stays in the ring-k (0 6 k 6 K). In the static location management, whenever an MT leaves an old LA (LAo) and enters a new LA (LAn), an LU is triggered. More specifically, an MT’s behavior in terms of state transit is as follows. The MT moves from the outmost ring-K in LAo. After a number of cell boundary crossing in the coverage of LAo, the MT leaves LAo and enters the ring-(K + 1), and then trig-
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395
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Ring-3 Ring-2 Ring-1
0
Ring-0
Location Area
Fig. 2. Location area definition.
ger LU message. Here, the ring-(K + 1) is a virtual ring in the LAo and actually the ring-K in the LAn. Hence, for the MT, the state (K + 1) is the absorbing state. Let qi,j be the first-step transit probability from ring-i to ring-j. Employing random walking model, we have qi;i1 ¼
2i 1 ; 1 6 i 6 K; 6i
qi;i ¼
1 3
ð30Þ
From (31) to (34), we can efficiently calculate the probability pK,K+1 (m) for different m. Thereafter, LA residence time is equal to the summation of m cell residence time with the probability pK,K+1 (m). Taking into account the normalization constraint, the LT of tLU pdf is given by m P1 m¼1 p K;Kþ1 ðmÞ ftcrt ðsÞ P1 : ð35Þ ftLU ðsÞ ¼ m¼1 p K;Kþ1 ðmÞ
with the boundary condition q0,1 = 1 and qK+1,K+1 = 1. We now need to know how many cells have been traversed when the MT moves from ring-K and first arrives at ring-(K + 1). Accordingly, we denote the probability by pi,j (m) that there are m cell boundary crossings, given that an MT starts from the ring-i and first passages the ring-j. For "i, j, a recursive relationship is developed as follows:
The average LU inter-arrival time 1/gLU is given by P1 1 1 m¼1 mpK;Kþ1 ðmÞ P1 : ð36Þ ¼ gLU l m¼1 p K;Kþ1 ðmÞ
ð31Þ pi;j ð1Þ ¼ qi;j ; m ¼ 1; X pi;j ðmÞ ¼ qi;k pk;j ðm 1Þ ¼ qi;i1 pi1;j ðm 1Þ 1ði16¼jÞ
ð37Þ
k6¼j
þ qi;i pi;j ðm 1Þ 1ði6¼jÞ þ qi;iþ1 piþ1;j ðm 1Þ 1ðiþ16¼jÞ ; m > 1;
ð32Þ
where qi,j is given by (30). Hence, pK,K+1 (m) represents the probability that there are m cell boundary crossings when the MT moves from ring-K and first arrives at ring-(K + 1). Its recursive algorithm is specified as pK;Kþ1 ð1Þ ¼ qK;Kþ1 ; X pK;Kþ1 ðmÞ ¼ qK;k pk;Kþ1 ðm 1Þ; k6¼Kþ1
ð33Þ m > 1:
ð34Þ
Hence, the LU cost is given by P1 pK;Kþ1 ðmÞ l : C u ¼ du P1m¼1 kc m¼1 mpK;Kþ1 ðmÞ
The number of cells in an LA is 3K (K + 1) + 1. In static scheme, the number of paged cells is the total number of cells in an LA. Hence, the total cost is given by P1 pK;Kþ1 ðmÞ l þ dp ½3KðK þ 1Þ þ 1: ð38Þ C T ¼ du P1m¼1 kc m¼1 mpK;Kþ1 ðmÞ As discussed, in two-dimension network, the static scheme is traditionally analyzed with the LA as a basic unit and the movement-based scheme is derived with the cell as the basic unit. This makes the infeasible comparison between the two schemes. It is clear that our formula (23) for movement-based scheme and the (38) for static scheme
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are developed based on the same basic unit, i.e., cell residence time characteristics. Hence, we can make an thorough comparison between these two schemes under different scenarios. 4. Numerical results The sensitivity of inter-service time in movement-based scheme has been extensively discussed and validated in the study [11]. In the literatures, the LU cost for static scheme is formulated based on the minimum unit as LA. This handling makes the absence of comparison between static scheme and movement-based scheme on a common basis. In this section, we will concentrate on the comparison between static and movement-based scheme under different scenarios. The cell residence time follows Gamma distribution with l = 1.0 and variance V [27]. Then, the LT of tcrt pdf is given by c lc 1 ð39Þ ftcrt ðsÞ ¼ ; c¼ 2 : s þ lc lV If not specified, the variance V = 1.5/l2. Define callmobility-ratio (CMR) as CMR = kc/l. Without loss of generality, the per-cell paging cost is set as dp = 1 and the unit LU cost is set as du = 10. Fig. 3 shows the total cost comparison between static scheme and the movement-based scheme. In this example, the exponential inter-service time is assumed. The results indicate that the cost decreases with larger CMR. This can be explained as follows. Greater CMR shows more frequent call activities, which lead to less LU operations during two calls and, consequently, smaller cost. It is observed that the cost in movement-based scheme is always larger than the cost in static scheme. This is due to the possible random roaming in static scheme before
moving out of an LA. Fig. 4 shows the total cost comparison between static scheme and the movement-based scheme. In this example, 4-stage Erlang distributed inter-service time is assumed. Similarly, the movementbased scheme generates higher signaling cost than the static scheme. Fig. 5 shows the total cost in terms of CMR under different du. In the figures, the curves without symbol represent the total cost in static scheme with different LA size K. The solid line with ‘•’ shows the cost in movement-based scheme with the optimal M, where each optimal threshold is able to minimize CT under the specific scenario. Under the condition of optimized M, the comparison result is dependent on the particular situation. The results indicate that, in case CMR smaller than 0.08 when K = 3, movement-based scheme generates higher signaling cost than the static scheme. For instance, the increased load can reach up to 20.8%. In case CMR larger than 0.08, movement-based scheme outperforms the static scheme by generating less signaling load. 5. Conclusions In this paper, we perform a general modeling and analysis for the location management by relaxing the tele-traffic parameters. To achieve this aim, we introduce the term location update inter-arrival time applicable in a generic location management scheme; and then specify its characteristics in static and dynamic schemes. A closedform formula is derived for the location management signaling cost function with the generalized LU inter-arrival time, cell residence time and inter-service time. The results indicate that the developed formulae enable the easy comparison between static and movement-based schemes.
350
300 Static(K=2) Movement (M=3)
Static(K=3) Movement (M=4)
300
250
250 200
CT
CT
200 150
150 100 100 50
50
0 –2 10
0
10 CMR
0 –2 10
0
10 CMR
Fig. 3. Total cost comparison between static scheme and movement-based scheme with exponential inter-service time.
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395 350
300
300 Static(K=3) Movement (M=4)
Static(K=2) Movement (M=3) 250
250 200
CT
CT
200 150
150 100 100 50
50
0 –2 10
0
10 CMR
0 –2 10
0
10 CMR
Fig. 4. Total cost comparison between static scheme and movement-based scheme with 4-stage Erlang inter-service time.
Fig. 5. Total cost comparison between static scheme and movement-based scheme with 4-stage Erlang inter-service time.
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Acknowledgement The authors thank the anonymous reviewers for their very helpful suggestions and comments. References [1] ETSI, Digital Cellular Telecommunications System (Phase 2+): Mobile Application Part (MAP) Specification (GSM 09.02 version 7.51 Release). 1998. [3] I.F. Akyildiz, J. Ho, Y. Lin, Movement-based location update and selective paging for PCS networks, IEEE/ACM Trans. Network. 4 (1996) 629–638. [4] F. Ashtiani, J.A. Salehi, M.R. Aref, Mobility modeling and analytical solution for spatial traffic distribution in wireless multimedia networks, IEEE J. Select. Areas Commun. 21 (10) (2003) 1699–1709. [5] A. Bar-Noy, I. Kessler, M. Sidi, Mobile users: to update or not to update? ACM/Baltzer Wireless Networks 1 (1995) 175–195. [6] V.-A. Bolotin, Modeling call holding time distributions for CCS network design and performance analysis, IEEE J. Select. Areas Commun. 12 (3) (1994) 433–438. [7] E. Chlebus, Empirical validation of call holding time distribution in cellular communications systems, Teletraffic Contributions for the Information Age (Proc. 15th ITC), vol. 2b, Elsevier, Amsterdam, the Netherlands, 1999, pp. 1179–1188. [8] M.E. Crovella, A. Bestavros, Sel-similarity in world wide web traffic: evidence and possible causes, IEEE/ACM Trans. Network. 5 (6) (1997) 835–846. [9] D.E. Duffy, A.A. McIntosh, M. Rosenstein, W. Willinger, Statistical analysis of CCSN/SS7 traffic data from working CCS subnetworks, IEEE J. Select. Areas Commun. 12 (1994) 544–551. [10] Y.G. Fang, General modeling and performance analysis for location management in wireless mobile networks, IEEE Trans. Comput. 51 (10) (2002) 1169–1181. [11] Y.G. Fang, Movement-based location management and tradeoff analysis for wireless mobile networks, IEEE Trans. Comput. 52 (6) (2003) 791–803. [12] A. Feldmann, W. Whitt, Fitting mixtures of exponentials to long-tail distributions to analyze network performance models, Perform. Evaluat. 31 (8) (1998) 963–976. [13] N.A.J. Hastings, J.B. Peacock, Statistical Distributions, Wiley, New York, 1975. [14] R. Jain, Y.-B. Lin, ‘‘An auxiliary user location strategy employing forwarding pointers to reduce network impacts of PCS’’, IEEE ICC’95, pp. 740–744, June,1995, WA, US. [15] F.P. Kelly, Reversibility and Stochastic Networks, Wiley, New York, 1979. [16] S.J. Kwon, M.Y. Chung, D.D. Sung, Mobility management schemes for support of UPT in mobile networks, IEEE ICC’2000 2 (2000) 675–679. [17] W. Leland, M. Taqqu, W. Willinger, D. Wilson, On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Trans. Network. 2 (1) (1994) 1–15. [18] J. Li, H. Kameda, K. Li, Optimal dynamic mobility management for PCS networks, IEEE/ACM Trans. Network. 8 (3) (2000) 319–327. [20] Y.B. Lin, Y.K. Chen, Reducing authentication signaling traffic in third-generation mobile network, IEEE Trans. Wireless Commun. 2 (3) (2003) 493–501. [21] Y.B. Lin, I. Chlamtac, Effects of Erlang call holding times on PCS call completion, IEEE Trans. Veh. Technol. 48 (3) (1999) 815–823. [22] M.A. Marsan, G. Ginella, R. Maglione, M. Meo, Performance analysis of hierarchical cellular networks with generally distributed call holding times and dwell times, IEEE Trans. Wireless Commun. 3 (2004) 248. [23] P.V. Orlik, S.S. Rappaport, A model for teletraffic performance and channel holding time characterization in wireless cellular communication with general session and dwell time distributions, IEEE J. Select. Areas Commun. 16 (1998) 788803.
[24] V. Wong, V. Leung, An adaptive distance-based location update algorithm for next generation PCS networks, IEEE J. Sel. Areas Commun. 19 (10) (2001) 1942–1952. [25] K. Yeo, C.-H. Jun, Modeling and analysis of hierarchical cellular networks with general distributions of call and cell residence times, IEEE Trans. Veh. Technol. 51 (6) (2002) 1361–1374. [26] Y. Zhang, B.-H. Soong, Handoff dwell time distribution effect on mobile network performance, IEEE Trans. Veh. Technol. 54 (4) (2005). [27] M.M. Zonoozi, P. Dassanayake, User mobility modeling and characterization of mobility patterns, IEEE J. Select. Areas Commun. 15 (1997) 1239–1252. Yan Zhang received the Ph.D. degree in School of Electrical & Electronics Engineering, Nanyang Technological University, Singapore. Since Aug. 2004, he has been working with the NICT Singapore, National Institute of Information and Communications Technology. He is currently serving the Book Series Editor for the book series on ‘‘Wireless Networks and Mobile Communications’’ (Auerbach Publications, CRC Press, Taylor and Francis Group). He has served as coeditor for several books. His research interests include resource, mobility and security management and performance evaluation in cellular networks and mobile ad hoc networks.
Jun Zheng received the B.S. and M.S. degrees in Electrical Engineering from Chongqing University, China, in 1993, 1996, respectively, the M.S.E. degree in Biomedical Engineering from Wright State University, Dayton, Ohio, in 2001, and the Ph.D. degree in Computer Engineering from University of Nevada, Las Vegas in 2005. Currently he is an assistant professor in the Department of Computer Science at Queens College - City University of New York. His research interests are mobility and resource management in wireless and mobile networks, media access control, performance evaluation, network security, computer architectures, fault-tolerant computing, and image processing.
Lili Zhang received her B.S. degree in electronic engineering from Shanghai Jiaotong University, Shanghai, China in 1999, the M.S. degree in communication engineering from China Academy of Telecommunication Technologies, Beijing, China in 2002, and the Ph.D. degree in communication engineering from school of electrical and electronic engineering, Nanyang Technological University, Singapore in 2005. She is currently a Research Associate in the Intelligent System Center, Nanyang Technological University, Singapore. Her research interests include multi-channel medium access control design, topology control and cross-layer design for mobile ad hoc networks and wireless sensor networks.
Yifan Chen received the B.Eng. and Ph.D. degrees in electrical and electronic engineering from Nanyang Technological University (NTU), Singapore, in 2002 and 2005, respectively. He is presently with the Biomedical Engineering Research Centre, NTU, as a research scientist. His current research interests involve microwave imaging of biological structures, statistical modeling of mobile radio channels, ultra wideband source localization, and multiple-antenna system performance analysis.
Y. Zhang et al. / Computer Communications 29 (2006) 2386–2395 Maode Ma received his B.E. degree in computer engineering from Tsinghua University in 1982, M.E. degree in computer engineering from Tianjin University in 1991 and Ph.D. degree in computer science from Hong Kong University of Science and Technology in 1999. Now, Dr. Ma is an assistant professor in the School of Electrical and Electronic Engineering at Nanyang Technological University in Singapore. Dr. Ma has published more than 50 academic papers in the areas
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of optical networks, wireless networks, etc. His current research interests are wireless networks, optical networks, performance analysis of computer networks, etc. He currently serves as an associate editor for IEEE Communications Letters and an editor for IEEE Communications Surveys & Tutorials.