(explicit, implicit, and timeout (TO) deregistration) for personal communication service (PCS) networks to determine the network conditions under which each strategy gives the best performance. Two performance measures are considered: 1) the probability
that a portable cannot register (and receive service) and 2)
the number of deregistration messages sent in a strategy. For the same database size, is smaller for explicit deregistration (ED) than it is for TO or implicit deregistration (ID). On the other hand, ID does not create any deregistration message traffic. With an appropriate TO period, the deregistration message traffic for TO deregistration is much smaller than the traffic for ED. Suppose that there areN portables in a registration area (RA) on the average. To ensure that < 1003, our study indicates that if the database size is larger than 4N, then the implicit scheme should be selected (to eliminate deregistration traffic). If the database size is smaller than 1.5N, then the explicit scheme should be selected. Otherwise, the TO scheme should be selected to achieve the best performance.
Index Terms— Deregistration, home-location register, mobil-ity management, personal communications services, registration, visitor-location register.
I. INTRODUCTION
T
HIS PAPER studies three deregistration strategies for personal communication service (PCS) networks. In a PCS network, registration is the process by which portables in-form the network of their current location (registration area or RA). We assume that a location database (i.e., visitor-location register or VLR) is assigned to exact one RA (although a VLR may cover several RA’s in the existing PCS systems). A portable registers its location when it is powered on and when it moves between RA’s. If the database is full when a portable arrives, the portable cannot access the services provided by the PCS network. When a portable leaves an RA or shuts off for a long period of time, the portable should be deregistered from the RA so that any resource previously assigned to the portable can be deallocated.In IS-41 [1], [3], the registration process ensures that a portable registration in a new RA causes deregistration in the previous RA. This approach is referred to as explicit deregistration (ED). This approach to deregistration may create significant traffic in the network [8]. Also, ED does not provide a means of deregistering portables that are shut off, broken, or otherwise disabled for a significant period of time. Bellcore personal access communications systems (PACS’s)
Manuscript received July 29, 1996; revised November 21, 1996. This work was supported in part by the National Science Council under Contract NSC 86-2213-E-009-074.
The author is with the Department and Institute of Computer Science and Information Engineering, National Chiao-Tung University, Taiwan, R.O.C. (e-mail: liny@csie.nctu.edu.tw).
Publisher Item Identifier S 0018-9545(98)00715-4.
certain time period elapses without the portable reregistering. This scheme is referred to as timeout (TO) deregistration [11]. Another possibility is to perform deregistration implicitly [2], [7]. Suppose that the database is full when a portable arrives at an RA. The implicit scheme selects a record based on some replacement strategy. This record is deleted and is then reassigned to Note that the record being replaced may be valid, in which case the corresponding portable is forced to deregister. Thus, the size of the registration database (i.e., the amount of resources) must be sufficiently large so as to ensure that the probability of a valid registration record being replaced is extremely low (say 10 ). Lin and Noerpel [7] proposed an analytical model to determine the database size for an implicit scheme that selects the oldest record for replacement. This paper proposes analytical models to study the explicit scheme, implicit scheme with a new replacement strategy, and TO scheme.
II. EXPLICIT DEREGISTRATION
In the explicit scheme, a registration record is deleted when the corresponding portable moves out of the RA. Thus, the database is full if and only if the number of portables in the RA is larger than the size of the database. To derive the probability that a portable cannot register at a particular RA, we first derive the distribution for the number of portables in an RA. Let be the expected number of portables in an RA. Suppose that the residence time of a portable in an RA has a general distribution with the density function and mean 1 In the steady state, the rate at which portables move into an RA equals the rate at which portables move out of the RA. In other words, the rate at which portables move into an RA is The arrival of portables can be viewed as being generated from input streams, which have the same general distribution with arrival rate If is reasonably large in an RA, the net input stream is approximated as a Poisson process with arrival rate Thus, the distribution for the portable population can be modeled by an queue with arrival rate and mean residence time 1 Let be the steady-state probability that there are portables in the RA. This model was validated against simulation experiments by Lin and Chen [6]. By the standard technique [4]
(1) Fig. 1(a) plots the population distribution when , , and , respectively. Skew distributions are observed for small values. We note that the typical number of portables
(a) (b)
Fig. 1. The performance of the explicit scheme. (a) The population distribution. (b) The probability that a portable cannot register in the explicit scheme.
in a RA is much larger than 150. The numbers 50, 100, and 150 are selected only for the demonstration purpose.
Suppose that the size of the registration database is Let be the probability that the registration database is full when a portable arrives (and thus the portable cannot register). Then
Fig. 1(b) plots for different values. The figure indicates that for (where ), the explicit scheme can accommodate almost all arriving portables (i.e., ). Note that the rate of the deregistration messages sent in the network is per RA. The deregistration messages may significantly contribute to the PCS network traffic.
III. IMPLICIT DEREGISTRATION
In the implicit scheme, no deregistration message is sent upon the movement of a portable. The obsolete record is kept in the database. When the database is full, the scheme reclaims a record (for the incoming portable) based on some strategy. A possible replacement strategy is described below.
A. Strategy ID
At time , a portable is said to be inactive for a time period if has not interacted (sending or receiving messages) with the RA since Define a threshold If is inactive for a time period , we may expect that has already left the RA. On the other hand, if there is a phone call for or registers (i.e., moves into the RA) within the period , then the implicit scheme assumes that is still in the RA. The implicit deregistration (ID) strategy works as follows. When arrives at an RA, let and be the oldest and the second oldest portables in the RA (i.e., for all the
portables in the RA, we have ). When
arrives, the inactive time periods for and are and , respectively. If , ID assumes that is not in the RA, and is selected for replacement. Otherwise, if then ID assumes that is not in the RA and is selected for replacement. If (or ) and (or ), then is selected for replacement. In ID, more than two portables may be considered for replacement. For
demonstration purposes, here we consider only the oldest two portables. The following notation is introduced.
1) the time period between ’s arrival and ’s arrival [c.f. Fig. 2(a)].
2) the time period between ’s arrival and ’s arrival [c.f. Fig. 3(a)].
3) the time period between ’s arrival and ’s arrival [c.f. Fig. 3(a)]. Note that
4) the residence time of [c.f. Fig. 2(a)]. 5) the residence time of [c.f. Fig. 3(a)]. 6) the number of portables that arrive in the period
(excluding ). Note that
7) the number of portables that arrive in the period (excluding ). Note that
8) the number of portables that arrive in the period (excluding ). Note that
9) the time interval between the last phone call to before ’s arrival and the time when arrives [c.f. Fig. 2(b)]. Note that there is no phone call to in the time period
10) the time interval between the last phone call to before ’s arrival and the time when arrives [c.f. Fig. 3(b)].
11) the time interval between the last phone call to and the time when moves out [c.f. Fig. 2(b)]. 12) the time interval between the last phone call to
and the time when moves out [c.f. Fig. 3(b)]. Since the portable arrivals to an RA form a Poisson process, has an Erlang distribution with the density function
Similarly, and have the Erlang density functions and , respectively. If we assume exponential portable residence times, then and have an identical density function
Let the intercall arrival times to a portable be exponentially distributed with the density function
(a) (b)
Fig. 2. The timing diagram for ID Case 1. (a)t > X; 1< t 0 X: (b) t > X; t 0 X < 1< t; t 0 ( 10 x1) > X:
Since the movements of a portable are a Poisson process, a portable is a random observer of the call interarrival times when it moves out of the RA. From the random observer property of the Poisson process and the memoryless property of the exponential distribution, both and have the same density function Similarly, the arrival of is a random observer of the call arrivals to and , and and also have the same density function
Let be the probability that and is not
in the RA. Let be the probability that (or ), , and is not in the RA. Let
be the probability that (or ), (or ),
and is not in the RA. Then
is the probability that the portable (either or ) selected by ID is not in the RA.
The probability is derived in the following three cases. 1) Case 1: ID assumes that is not in the RA, and is not in the RA.
That is, ID assumes that has moved out of the RA when moves in, which implies that the inactive period is longer than the threshold when arrives, and
Since is not in the RA when
arrives, either [c.f. Fig. 2(a)] or ,
and [c.f. Fig. 2(b)]. The probability
for Fig. 2(a) is
(2)
(3) (3) is derived from (2) based on the fact that
Since implies , the probability for
Fig. 2(b) is
(4) Thus,
2) Case 2: ID assumes that is in the RA, is not in the RA, and is not in the RA. Since ID assumes that is not in the RA when moves in, it implies that the inactive period is longer than the threshold , and
(5) As in the situations described in Case 1, either
[c.f. Fig. 3(a) and (c)] or , and
[c.f. Fig. 3(b) and (d)]. Since ID assumes that
is in the RA and , it implies that and
There are two cases.
3) Case 2a: and [c.f. Fig. 3(a) and (b)].
4) Case 2b: and
[c.f. Fig. 3(c) and (d)].
From (5) and Cases 2a and 2b, there are four combinations for Case 2, as illustrated in Fig. 3. The probability for Fig. 3(a) is
(a) (b)
(c) (d)
Fig. 3. The timing diagram for ID Case 2. (a) t1> 0; t2> X; 1> t1 + t2; 1< X;0 < 2< t2 0 X: (b) t1> 0;t2> X; 1> t1 + t2;
1< X;t2 0 X < 2< t2;x2> X + 2 0 t2: (c) t1> 0;t2> X; t1 + t2 0 X < 1< t1 + t2;x1< 1 + X 0 t1 0 t2; 2< t2 0 X: (d)
t1> 0;t2> X; t1+ t20 X < 1< t1+ t2;x1< 1+ X 0 t10 t2; t20 X < 2< t2x2> X + 20 t2:
(6)
The probability for Fig. 3(b) is
(7)
The probability for Fig. 3(c) is
(8)
(c) (d) Fig. 4. The timing diagram for ID Case 3.
(9)
Thus,
5) Case 3: ID assumes that both and are in the RA, but is not in the RA. Note that ID assumes that is in the RA, which implies that or
There are three possibilities.
[c.f. Fig. 4(a) and (b)]. Since ID assumes that is in the RA, we have or There are two cases.
6) Case 3a: , as shown in Fig. 4(a).
The probability is
(10)
7) Case 3b: Since , we have
and , as shown in Fig. 4(b). The probability is
(11) and [c.f. Fig. 4(c)]. Since and ID assumes that is in the RA, the situation is the same as in Case 3b (i.e., and
(a) (b) Fig. 5. Performance of ID. (a) The effect of X on ID ( = 2): (b) The effect of on ID (X = 1:5=):
(12)
and as
shown in Fig. 4(d). Since and ID assumes that is in the RA, the situation is the same as Case 3b
(i.e., and ), and
(13) Thus,
Suppose that the size of the database is For the oldest portable , it is apparent that Similarly, for
the second oldest portable 1 Since
1 1
a lower bound for is
1 1
Fig. 5(a) illustrates the effect of on The figure indicates that the maximum value for occurs when
(for The figure indicates that erring on the side of an value that is too large will degrade performance less than erring on the side of an value that is too small. It is apparent that the performance of ID improves as increases. Fig. 5(b) illustrates that is an increasing function of
Let be the probability that a portable (either or ) cannot register (i.e., is forced to deregister) when arrives. An upper bound for is
Fig. 9(a) indicates that for IV. TIMEOUT DEREGISTRATION
In the TO scheme, a portable sends a reregistration message to the RA for every time period The TO scheme is better than the explicit scheme if the reregistration traffic (in TO) is less than the deregistration traffic [in ED]. This section derives the number of reregistration messages sent in the TO scheme. Let be the expected number of reregistration messages sent before a portable leaves an RA. Let be the portable residence time distribution (with mean 1 Then
(14) For the exponential residence time distribution, (14) is rewritten as
For the uniform residence time distribution in 0 2
Fig. 6(a) plots against The figure indicates that if for the exponential residence times and for the uniform residence times. In the explicit scheme, a deregistration message is sent when a portable moves out of the RA. On the other hand, in the TO scheme, the number of reregistration messages sent by a portable is Thus, the deregistration traffic in the explicit scheme is times the reregistration traffic in the TO scheme. For
(a) (b)
Fig. 6. The expected number of reregistration messages. (a) The registration message overhead in the TO scheme. (b) Comparison of the de(re)registration message overhead for the explicit scheme and the TO scheme.
(a) (b)
Fig. 7. The portable distribution seen by the TO scheme. (a) The impact ofT: (b) The impact of the residual time distributions.
example, if the deregistration traffic generated by the explicit scheme is about four–five times the traffic generated by the TO scheme assuming exponential residence times [c.f. Fig. 6 (b)]. In other words, if is sufficiently large, then the TO scheme significantly reduces the network traffic due to deregistration (compared with the explicit scheme).
The portable residence time seen by the TO scheme is different from the true portable residence time (the TO scheme only differentiates on multiples of ). Since
, the expected residence time seen by the TO scheme is
For the exponential residence time distribution
(15) From (15) and the model described in Section II, the steady-state probability that the TO scheme sees portables in the RA is
The distribution is plotted in Fig. 7 for different values. Fig. 7(a) indicates that the portable seen by the TO scheme increases as increases. Fig. 7(b) indicates that the number of portables seen by the TO registration scheme is closer to the true number for the uniform residence times than for the exponential residence times. Let be the probability that the TO scheme sees a full registration database in an RA when a portable arrives. Then
Suppose that is not allowed to register if the TO scheme sees a full database at ’s arrival. Then is the probability that a portable (i.e., ) cannot register (and receive services). Fig. 8 plots against It is clear that is a de-creasing function of the database size and is an increasing function of [c.f. Fig. 8(a)]. It is interesting to note that for the same ratio, the value for a small is
smaller than the for a large when The
opposite is true when [c.f. Fig. 8(b)]. Fig. 9(a)
compares with For (where ,
Note that for 2.5 However,
when 2 , is much larger than
Other replacement strategies can be used if the TO scheme sees a full database when a portable arrives. Let us consider
(a) (b)
Fig. 8. The probability that the TO scheme sees a full registration database when a portable arrives. (a) The impact of T (N = 100): (b) The impact of N (T = 1:8):
(a) (b)
Fig. 9. Performance for different deregistration schemes(N = 100): (a) The values for different schemes. (b) The performance for the TO scheme with different replacement strategies (T = 1:8=):
the replacement strategies used in ID in the previous section. Let be the probability that cannot register in the TO scheme with the ID replacement strategy. Then
Fig. 9(b) plots The figure indicates that with the ID replacement strategy, the performance of the TO scheme is significantly improved.
V. CONCLUSIONS
This paper has studied three deregistration strategies for PCS networks. Two output measures were considered: the number of messages sent in the deregistration strategies and the probability that a portable cannot register (and receive service). Assume 100 portables in an RA on the average. To satisfy the constraint that , the size of the database required in the explicit scheme is , which is smaller than the database size for the implicit scheme and the TO scheme On the other hand, the number of deregistration messages sent in the explicit scheme is four–five times the number of messages sent in the TO scheme (with the registration period In the implicit
scheme, neither deregistration nor reregistration messages are sent. Our study indicates that if the database size is expected to be large, then the implicit scheme should be used to eliminate the deregistration message traffic. If the database size has to be small, on the other hand, then the explicit scheme should be used to achieve a low value. If the database size is between 2.5–4N, then the TO scheme with the ID replacement strategy should be used to ensure a reasonably small value and a low level of reregistration message traffic.
In summary, ID and ED are mutually exclusive. TO dereg-istration is a useful tool to clean up regdereg-istration databases and can be combined with either one of the ID or ED approaches. In PACS [5], [10], polling reregistration was introduced so that the system can poll the portables to see if the portables are still in the RA [9]. A combination of TO deregistration, ID, and polling reregistration might be best in all circumstances. Performance modeling of such a combination will be one of our future research directions.
ACKNOWLEDGMENT
The author would like to thank J. R. Cruz, S. Y. Hwang, A. R. Noerpel, and the reviewers for their valuable comments and assistance in improving the quality of this paper.
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University, Taiwan, R.O.C., where he was appointed Full Professor in 1995. In 1996, he was appointed Deputy Director of the Microelectronics and Information Systems Research Center, NCTU. His current research interests include design and analysis of personal communications services network, mobile computing, distributed simulation, and performance modeling.
Dr. Lin is an Associate Editor of the ACM Transactions on Modeling and
Computer Simulation, a Subject Area Editor of the Journal of Parallel and Distributed Computing, an Associate Editor of the International Journal in Computer Simulation, an Associate Editor of IEEE Networks, an Associate
Editor of SIMULATION, an Area Editor of ACM Mobile Computing and
Communication Review, a Columnist of ACM Simulation Digest, a Member
of the editorial boards of International Journal of Communications and
Computer Simulation Modeling and Analysis, and Guest Editor for the ACM/Baltzer WINET Special Issue on Personal Communications and the
IEEE TRANSACTIONS ON COMPUTERS Special Issue on Mobile Computing. He was 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, and Publicity Chair of ACM Sigmobile.
