2001 Kluwer Academic Publishers. Manufactured in The Netherlands.
Implicit Deregistration with Forced Registration for PCS
Mobility Management
AI-CHUN PANG and YI-BING LIN
Department of Computer Science and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan, ROC YUGUANG FANG
Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611, USA
Abstract. Registration/deregistration with a mobility database called Visitor Location Registers (VLRs) is required in a PCS network when a mobile phone
moves between registration areas. Several schemes were proposed to deregister a mobile phone after it moves out of a registration area. A simple scheme, called implicit deregistration, is studied in this paper, which does not specifically deregister any obsolete record in the VLR. If the VLR is full when a mobile phone arrives, a record in the VLR is deleted and the reclaimed storage is reassigned to the incoming mobile phone. It is possible that a valid record will be deleted. If so, the VLR record of a mobile phone may be deleted before a call to the mobile phone arrives. Our previous work assumed that the incoming call setup would be lost. In this paper, we propose forced registration to restore the VLR record before the call setup operation can proceed. With this modification, implicit deregistration totally eliminates the deregistration traffic at the cost of creating some forced registration traffic.
We derive the record-missing probability and the portion of the network traffic saved by implicit deregistration. Our study indicates that implicit deregistration with forced registration may significantly reduce the deregistration traffic if the user mobility is high and the number of mobile phones in a registration area is not very large.
Keywords: implicit deregistration, mobility management, personal communications, visitor location register
1. Introduction
In a personal communications services (PCS) network, reg-istration is the process by which mobile phones inform the network of their current locations (registration area or RA). When a mobile phone enters an RA (either when it is pow-ered on or when it moves between registration areas), it reg-isters at the Visitor Location Register (VLR) corresponding to the RA and the address of the new RA is reported to the Home Location Register (HLR) of the mobile phone [2,4, 11]. Note that a VLR may control several RAs. Without loss of generality, we assume that every VLR associates with ex-actly one RA. To locate a mobile phone, the HLR of the mobile phone is accessed to find the current VLR address of the mobile phone. Using the VLR address, the mobile phone is located. The VLR may be full when a mobile phone ar-rives, and the mobile phone cannot access the services pro-vided by the PCS network. When a mobile phone leaves an RA, or shuts off for a long period of time, the mobile phone should be deregistered from the RA so that the storage for the obsolete VLR record can be reused. However, both reg-istration and deregreg-istration may result in significant amount of network signaling traffic. Several approaches have been proposed to reduce the registration traffic. These approaches are described in [1,5] and the references therein. In this pa-per, we focus on reducing the deregistration traffic.
In IS-41 [2] and GSM MAP [4], the registration process ensures that a mobile phone’s registration in a new VLR causes deregistration in the previous VLR. This approach is referred to as explicit deregistration. Like registration, such deregistration may create significant traffic in the net-work [10]. In [9], we suggested that the deregistration
op-eration can be performed implicitly. In this implicit scheme, the record of a mobile phone is not deleted from the VLR when the mobile phone leaves that RA. Thus, the deregis-tration traffic is totally removed. If the VLR is full when a mobile phone P arrives in that RA, the implicit scheme deletes a record and the reclaimed storage is reassigned to P. We assume that the deleted record is selected randomly with equal probability. In doing so, a valid record may be re-placed, and call setup to the corresponding mobile phone cannot follow the normal procedure. In [9], we assume that the call is lost. In this paper we suppose that, instead of call blocking, a “forced registration” operation is performed to restore the VLR record before call setup. The forced reg-istration operation is similar to the VLR failure restoration operation [4] that restores the lost VLR records after a VLR database crash. In GSM, when a mobile phone makes a call, the call origination request is sent to the VLR. If the VLR record for the mobile phone was replaced due to the exer-cise of implicit deregistration, the request is rejected. Then the mobile phone is asked to initiate the location registration procedure. After the registration, the VLR record is recov-ered. Similarly, when a call termination is delivered to a mobile phone, the VLR record for the mobile phone may not exist. The serving VLR queries the HLR for the ser-vice information. Then, the target Mobile Switching Center (MSC) is asked to initiate paging of the mobile phone in all RAs of the serving VLR. If the paging is successful, the lo-cation information of the VLR record of the mobile phone is recovered. In the above call setup procedure, the network is “forced" to perform registration to recover the VLR record for a mobile phone.
Figure 1. The timing diagram.
In our approach, implicit deregistration totally eliminates the deregistration traffic at the cost of creating some forced registration traffic. Thus, it is important to study when a PCS network can significantly benefit from implicit dereg-istration when compared to explicit deregdereg-istration. Let µ1
be the deregistration traffic saved in implicit deregistration (compared with explicit deregistration) and µ2be the extra
traffic created due to forced registration. The portion β of the network traffic saved in implicit deregistration is expressed as
β= µ1− αµ2
µ1 , (1)
where α is the ratio of a deregistration cost to a registration cost. In a typical mobile phone network, 1 < α < 2 (e.g., FarEasTone GSM network). If β is negative, then it implies that the cost of implicit deregistration with forced registra-tion is higher than explicit registraregistra-tion.
In this paper, we use β to investigate the performance of implicit deregistration with forced registration.
2. The analytic model
This section proposes an analytic approach to model implicit deregistration with forced registration. Consider the timing diagram in figure 1. Suppose that the mobile phone P enters an RA at time t0, and leaves the RA at time t3. Then the
residence time of P in the RA is τ1= t3− t0. Suppose that
a call to P (either a call origination or a call termination) arrives at time t2(where t1< t2< t3), and the previous call
to P arrives at time t1. Then the inter-call-arrival time is τ2= t2−t1. It is clear that before t0, t1and t2, P’s VLR record may
not exist if implicit deregistration is exercised and the record is replaced. We note that at times t0, t1and t2, P always has a
record in the VLR (which is potentially restored by a forced registration). On the other hand, the VLR record of P may be replaced during the period[max(t0, t1), t2] (in figure 1,
max(t0, t1) = t1) if a registration or call setup for another
mobile phone selects P’s record for replacement during the period. Let the length of this period be τ = min(τ2, τ3),
where τ3 = t2− t0 represents the period between when P
enters the registration area and when the current call arrives at P. Thus, τ is the period during which the VLR record of P may be replaced when the current call to P arrives. To derive
β in (1), we make the following assumptions:
• The call arrivals to a mobile phone are a Poisson process with the call arrival rate λ; i.e., the inter-call-arrival
time τ2(the period between two consecutive call arrivals)
is exponentially distributed with mean 1/λ.
• The RA residence time τ1has a general density function fm(τ1) with mean 1/η.
Since the Poisson call arrivals are random observers to the RA residence times, following the residual life theorem [7], the density function rm(τ3) for τ3is
rm(τ3)= η Z ∞ t=τ3 fm(t) dt= η 1− Fm(τ3) , (2)
where Fm(·) is the distribution function of fm(·). The
den-sity function f (τ ) for τ = min[τ2, τ3] is f (τ )= Z ∞ τ2=τ λe−λτ2rm(τ ) dτ2+ Z ∞ τ3=τ λe−λτrm(τ3) dτ3 = e−λτr m(τ )+ λ 1− Rm(τ3) , (3)
where Rm(·) is the distribution function of rm(·). The
Laplace transform f∗(s) for the τ distribution is f∗(s)= Z ∞ τ=0 e−sτf (τ ) dτ =λs+ λ2+ ηs(1 − fm∗(λ+ s)) (s+ λ)2 . (4)
The number of mobile phones in an RA changes from time to time. Let the expected number be N . If N is sufficiently large, the mobile phone arrivals (i.e., registration traffic) can be approximated by a Poisson process with rate N η [8]. Also, the net call arrivals to the mobile phones in the VLR form a Poisson process with rate N λ.
Let M be the database size of the VLR. If the VLR record replacement is done (uniformly) randomly then
q = M− 1
M (5)
is the probability that the VLR record for a mobile phone P is not selected for replacement.
Let p be the record-missing probability that when a call to the mobile phone P arrives (i.e., the current call in figure 1), the VLR record rPof P does not exist. That is, rPis removed
in the period τ either by a registration operation (with rate
N η) or a call request to another mobile phone whose VLR
record does not exist (with the rate pN λ), and rPis selected
for replacement (with probability 1− q). Note that the rate for the call arrivals to the mobile phones in the VLR is given by N λ. Since the probability p is the record-missing prob-ability that when a call to the mobile phone P arrives, the VLR record of P does not exist. Thus, pN λ is the rate of call requests for which the corresponding VLR records do not exist. Based on the above discussion, the rate of opera-tions that may cause the replacement of a VLR record is
Furthermore, the number X of such operations in period τ is a Poisson random variable with the probability mass func-tion [7]
Pr[X = n] = (λ
∗τ )n
n! e
−λ∗τ.
Thus, the record-missing probability p can be expressed as
p= 1 − Z ∞ τ=0 ( ∞ X n=0 qnPr[X = n] ) f (τ ) dτ = 1 −X∞ n=0 (−qλ∗)n n! dnf∗(s) dsn s=λ∗ . (7)
Observing that f∗(s) is analytic in the right complex
plane, so by using power series expansion, we have
∞ X n=0 (−q)n(λ∗)n n! dnf∗(s) dsn s=λ∗ = f∗(λ∗(1−q)). (8)
From (4) and (8), (7) can be rewritten as
p= 1 − f∗ λ∗(1− q)
=A2+ λA − ηA[1 − fm∗(A+ λ)]
(A+ λ)2 , (9)
where A= λ∗(1− q).
We notice that from (6) and (5) we obtain
λ∗= η+ pλ 1− q N M , (10)
thus, (9) is not affected by N or M individually, instead, (9) is affected by the ratio N/M.
With (9) and (10), the unknown p can be computed by using the following iterative procedure.
Step 1. Select the initial value for p. Step 2. Compute λ∗based on (10).
Step 3. Let pold← p.
Step 4. Compute p based on (9).
Step 5. Let δ be a predefined small value. If|p−pold| < δp,
then exit. Otherwise, pold← p and go to step 2.
When the procedure exits at step 5, the p value will give the desired value. The above iterative procedure has been extensively used and validated by experiments [8]. Here we want to prove analytically that this iterative algorithm will converge to the unique solution. Let
g(p)= 1 − f∗ λ∗(1− q)= 1 − f∗ (η+ pλ)N M ,
which is the right-hand side of (9). We first show the exis-tence and uniqueness of a solution for the equation (9). No-tice that f∗(s) is the Laplace transform of a probability
den-sity function, which is analytic in the right complex plane. Let h(p) = p − g(p), we have h(0) = −g(0) = −1 +
f∗(ηN/M) < 0 and h(1)= 1−g(1) = f∗((η+λ)N/M) >
0, from the mean value theorem, there is a solution for (9), i.e., there exists a p∗in (0, 1) such that h(p∗)= 0. For any
positive number p in (0,+∞), we have (X(i)(s) denotes the ith derivative of the function X(s))
g(1)(p)= −f∗(1) (η+ pλ) N M λN M > 0 and g(2)(p)= −f∗(2) (η+ pλ) N M λN M 2 < 0,
where we have used the fact that f∗(1)(s) < 0 and
f∗(2)(s) > 0 on the positive real axis. We conclude that g(p) is a concave function, and it is also an increasing
func-tion in[0, 1]. Since g(0) > 0 and g(1) < 1, we conclude that there is only one point in (0, 1) at which p = g(p), thus, the solution of p= g(p) is unique in (0, 1). Next, we prove the convergence of the iterative algorithm. Let pn
de-note the nth iteration obtained from the iteration algorithm, then we have pn+1= g(pn), from which we have
pn+1− pn= g(pn)− g(pn−1) = g(1)(r n)(pn− pn−1) = · · · =g(1)(rn) · · ·g(1)(r1)(p1− p0), (11)
where r1, r2, . . . , rn are the intermediate values from the
mean value theorem. By noticing that g(1)(p) > 0 for any
value in (0, 1] and (11), we observe that if p1 > p0, pnis
nondecreasing; if p16 p0, pnis nonincreasing, thus, the
se-quence{pn} is a monotonic sequence. Besides, the sequence
is obviously bounded. Hence, the sequence will always con-verge and it will concon-verge to the unique equilibrium point.
With the above iterative procedure, we will demonstrate the effects of input parameters such as λ, η and α on the record-missing probability p and the portion β of the net-work traffic saved in implicit deregistration with forced reg-istration.
3. Numerical results
This section investigates the performance of implicit dereg-istration with forced regdereg-istration based on the analytic model developed in the previous section. We assume that the RA residence times have a Gamma density function with mean 1/η and variance v. The Gamma distribution is se-lected because it can approximate many other distributions as well as experimental data [6]. The Laplace transform for the Gamma RA residence time distribution is
fm∗(s)= 1 1+ ηvs 1/(η2v) . (12) From (12), (9) is rewritten as p=A
2+ λA − ηA[1 − (1 + ηvA + ηvλ)−1/(η2v)]
(A+ λ)2 , (13)
Figure 2. Effects of λ/η on p.
In appendix, we partially validate (13) for the case when the RA residence time distribution is exponentially distrib-uted.
From (13) and since µ1 = Nη and µ2 = pNλ, (1) can
be rewritten as β= 1 − αλ η ×
A2+ λA − ηA[1 − (1 + ηvA + ηvλ)−1/(η2v)] (A+ λ)2
,
(14) where A= λ∗(1− q).
Based on (13) and (14), we investigate the performance of implicit deregistration with forced registration as follows. 3.1. Effects of λ/η
Based on (13), figure 2 plots the record-missing probabil-ity p against λ/η (the expected number of calls to a mobile phone when the mobile phone is in an RA) with N/M= 0.1, 0.3, 0.5 and v= 0.1/η2, 1/η2, 10/η2. The figure indicates that p decreases significantly as λ/η increases, especially when N/M is large. That is, if the call arrival rate is low or the mobile phone’s mobility is high, it is more likely that when a call arrives, the corresponding VLR record has been replaced. The figure also shows an intuitive result that as
N/M increases, p increases (the more the mobile phones in
the VLR, the higher the record-missing probability). Based on (14), we compute the portion β of the network traffic saved in implicit deregistration with forced registra-tion. The results are plotted in figure 3, where N/M = 0.1, 0.3, 0.5, v= 0.1/η2, 1/η2, 10/η2, and α= 1. The fig-ure indicates that by exercising implicit deregistration, the
Figure 3. Effects of λ/η on β (α= 1).
portion of deregistration traffic can be significantly reduced if the expected number of mobile phones in an RA is not very large or the mobile phone’s mobility is high. Specifi-cally, for N < 0.5M, implicit deregistration is beneficial to a PCS network if λ/η < 4 (the number of calls to a mobile phone is less than 4 when the mobile phone is in an RA).
In practice, a PCS service provider can partition the VLR storage into two parts. The first part of the storage is used for mobile phones with low mobility where explicit deregistra-tion is exercised. The second part of the VLR accommodates the records for mobile phones with high mobility where im-plicit deregistration is employed.
Figure 3 also indicates that β is more sensitive to λ/η for a large N/M than a small one. In other words, when the number of mobile phones in the system is large, the ratio of the call arrival rate to the mobile phone’s mobility rate sig-nificantly affects the performance of implicit deregistration with forced registration.
3.2. Effects of v
Both figures 2 and 3 demonstrate how the variance v of the Gamma cell residence time distribution affects the system performance with a fixed mean 1/η. Figure 2 shows that as v increases, the record-missing probability p increases rapidly, especially for the case where λ/η 6 2.5. Fig-ure 3 shows that the portion β of the network traffic saved in implicit deregistration decreases as v increases. Moreover, when v 6 1/η2, the variance of the RA residence time dis-tribution only has insignificant effects on p and β. On the other hand, when v > 1/η2, p and β are significantly af-fected by v.
Figure 4. Effects of α on β.
3.3. Effects of α
Figure 4 plots β as a function of λ/η and α. It is appar-ent that with fixed mean and variance, β decreases as α in-creases. The non-intuitive result is that β is more sensitive to λ/η for a large α than a small one. We observe that when
N/M6 0.3, β is positive even if λ/η is large. However, for N/M = 0.5 and α = 2, β is negative for the cases where λ/η> 3. That is, if the number of mobile phones and the
ra-tio of the deregistrara-tion cost to the extra forced registrara-tion cost are large, a PCS network cannot benefit from implicit deregistration with forced registration.
4. Conclusions and future research
This paper proposed an analytic model to investigate the performance of implicit deregistration with forced registra-tion. The portion β of the network traffic saved when im-plicit deregistration is used is considered as the output mea-sure. The study indicated that the expected number of mo-bile phones, the momo-bile phone’s mobility and the ratio of a deregistration cost to a registration cost significantly affect the performance of β. Furthermore, we observed that β is significantly affected by the variance of the RA residence time distribution.
Besides the performance of β, several issues regarding implicit deregistration were discussed in [9]. Some addi-tional issues are discussed here:
• Effects of database failure. When a mobility database fails, the records are recovered through radio contacts from the mobile phones. Thus, the failure restoration procedure [3,4] for systems exercising implicit deregis-tration is the same as IS-41 or GSM MAP.
• Database revisit. A mobile phone may revisit a VLR that already has an (undeleted) entry for the mobile phone. A search operation is required to reuse the entry for the current visit of the mobile phone. Compared to the cost of transmission, the search cost can be ignored.
• Deregistration cost saving. The deregistration cost sav-ing depends on the message delivery distance between the HLR and the new and the old VLRs. The actual cost estimation may significantly vary from one PCS network to another. For example, the cost of a deregistration eration is roughly estimated as 50% of a registration op-eration in a GSM operator in Taiwan.
Our study suggested that a PCS service provider can par-tition the VLR storage into two parts. The first part of the storage is used for the mobile phones with low mobil-ity where explicit deregistration is exercised. The second part of the VLR accommodates the records for the mobile phones with high mobility where implicit deregistration is employed.
This paper only considered a simple uniform random re-placement policy for implicit deregistration. We anticipate that the scheme will be more effective if a sophisticated re-placement policy is used. Thus, the design and modeling of new heuristics for record replacement will be an important future research direction for implicit deregistration.
Appendix. Validating the p derivation for the case with exponential RA residence times
Consider a special case where τ1is exponentially distributed.
In this case (2) and (3) are rewritten as
rm(t)= fm(t)= ηe−ηtf (τ )= (λ + η)e−(λ+η)τ,
f∗(s)= λ+ η
s+ λ + η. (A.1)
Substitute (A.1) into (7), replace λ∗by (η+ pλ)N, and let
θ = N/M. We then have p= 1 − λ+ η λ+ η + λ∗(1− q) = θ η+ θλp (θ+ 1)η + λ + θλp = a+ bp c+ bp, (A.2)
where a= θη, b = θλ, and c = (θ + 1)η + λ. By replacing
v by 1/η2in (13), we have
p=A
2+ λA − ηA[1 − 1/(1 + A/η + λ/η)] (A+ λ)2 = θ η+ θλp (θ+ 1)η + λ + θλp = a+ bp c+ bp, (A.3) where A= λ∗(1− q).
It is clear that (A.3) is the same as (A.2). Thus, we par-tially validate (13) for the case when the RA residence time distribution is exponentially distributed.
Both equations (A.2) and (A.3) can be rewritten as
bp2+ (c − b)p − a = 0. (A.4) Since 06 p 6 1 and ab > 0, the only solution for (A.4) is p= −(c − b) + p (c− b)2+ 4ab 2b . (A.5) References
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Ai-Chun Pang received her B.S.C.S.I.E. and
M.S.C.S.I.E. degrees from National Chiao Tung University in 1996 and 1998, respectively. She is currently a Ph.D. candidate of the Department of Computer Science and Information Engineering, National Chiao Tung University. Her current re-search interests include personal communications services, computer telephony integration and voice over IP.
Yi-Bing Lin received his BSEE degree from
Na-tional Cheng Kung University in 1983, and his Ph.D. degree in computer science from the Univer-sity of Washington in 1990. From 1990 to 1995, he was with the Applied Research Area at Bell Com-munications Research (Bellcore), Morristown, NJ. In 1995, he was appointed as a Professor of De-partment of Computer Science and Information En-gineering (CSIE), National Chiao Tung University (NCTU). In 1996, he was appointed as a Deputy
Director of Microelectronics and Information Systems Research Center, NCTU. Since 1997, he has been elected as Chairman of CSIE, NCTU. His current research interests include design and analysis of personal commu-nications services network, mobile computing, distributed simulation, and performance modeling. Dr. Lin is an Associate Editor of IEEE Network, an Editor of IEEE Journal on Selected Areas in Communications: Wireless Series, an Editor of IEEE Personal Communications Magazine, an Editor of Computer Networks, an Area Editor of ACM Mobile Computing and Communication Review, a columnist of ACM Simulation Digest, an Editor of International Journal of Communications Systems, an Editor of Wireless Networks, an Editor of Computer Simulation Modeling and Analysis, an Editor of Journal of Information Science and Engineering, Program Chair for the 8th Workshop on Distributed and Parallel Simulation, General Chair for the 9th Workshop on Distributed and Parallel Simulation, Program Chair for the 2nd International Mobile Computing Conference, Guest Editor for the ACM/Baltzer MONET special issue on Personal Communications, a Guest Editor for IEEE Transactions on Computers special issue on Mobile Computing, and a Guest Editor for IEEE Communications Magazine spe-cial issue on Active, Programmable, and Mobile Code Networking. Lin re-ceived 1997 Outstanding Research Award from National Science Council, ROC, and Outstanding Youth Electrical Engineer Award from CIEE, ROC. E-mail: liny@csie.nctu.edu.tw
Yuguang Fang received the BS and MS degrees in
mathematics from Qufu Normal University, Qufu, Shandong, China, in 1984 and 1987, respectively, a Ph.D. degree in systems and control engineering from Department of Systems, Control and Indus-trial Engineering at Case Western Reserve Univer-sity, Cleveland, OH, in January 1994, and a Ph.D. degree in electrical engineering from Department of Electrical and Computer Engineering at Boston University, MA, in May 1997. Dr. Fang held a post-doctoral position in the Department of Electrical and Computer Engineering at Boston University from June 1994 to August 1995. From June 1997 to July 1998, he was a Visiting Assistant Professor in the Erik Jonsson School of Engineering and Computer Science at the University of Texas at Dallas. From July 1998 to May 2000, he was an Assistant Professor in the De-partment of Electrical and Computer Engineering at New Jersey Institute of Technology, Newark, NJ. Since June 2000, he is an Assistant Professor in the Department of Electrical and Computer Engineering at the Univer-sity of Florida, Gainesville, Florida. He has published over 40 papers in the professional journals and conferences. His recent research interests in-clude wireless networks and mobile communications, personal communi-cation services (PCS), call admission control and resource allocommuni-cations. Dr. Fang is an Editor for IEEE Journal on Selected Areas in Communications: Wireless Communications Series, an Area Editor for ACM Mobile Com-puting and Communications Review, and the Editor for Scanning the Liter-ature for IEEE Personal Communications Magazine. He is the Program Vice-Chair for the 2000 IEEE Wireless Communications and Network-ing Conference (WCNC’2000). He was the Program Vice-Chair for the Seventh International Conference on Computer Communications and Net-working (IC3N’98), a member of Technical Program Committee for the In-ternational Conference on Computer Communications (INFOCOM’98 and INFOCOM’00), a member of the Technical Program Committee for the First International Workshop for Wireless and Mobile Multimedia (WOW-MoM’98), and a member of the Technical Program Committee for the 1999 IEEE Wireless Communication and Networking Conference (WCNC’99). E-mail: fang@njit.edu
