Index Terms—Call holding time, cell residence times, location modeling, mobility, personal communications services (PCS).
I. INTRODUCTION
P
ERSONAL Communications Services (PCS) networks are poised to provide integrated services such as voice, data and multimedia to mobile users anywhere, anytime [1], [11], in an uninterrupted and seamless way, using advanced micro-cellular and handoff concepts [8]. In such a network, the ser-vice area is populated with base stations which provide the radio links for communications. The radio coverage of each base sta-tion is called a cell. The base stasta-tion is responsible for locating a mobile user or a portable through paging or some other loca-tion tracking strategies [17], [21], and delivers calls from and to the portable. The service of a PCS network is also divided into registration areas (RAs), each of which consists of an aggrega-tion of cells, forming a contiguous geographical region. For a call from or to a roaming user, the location of the roaming user has to be determined for the call delivery. Two-level hierarchies which maintain a system of a home database (called home tion register or HLR) and a visited database (called visitor loca-tion register or VLR) are commonly used for mobility manage-ment. When a user subscribes to a service from a PCS network, the user will first register at HLR where the user’s information profile is stored. When the user requests a service in a visited RA, it will contact the VLR associated with the RA, the VLR will contact the HLR of the user for authentication, the user’s record will be temporarily stored in the VLR. The VLR acts as an agent for the roaming user in the RA it is visiting.Manuscript received October 29, 1998; revised January 14, 2000. This work was supported in part by the U.S. Army Research Office under Contract DAAG55-97-1-0312 and DAAG55-97-1-0382. The work of Y. Fang was also supported, in part, by the New Jersey Institute of Technology under Grant SBR421980 and the New Jersey Center for Multimedia Research.
Y. Fang is with the Department of Electrical and Computer Engi-neering, University of Florida, Gainesville, FL 32611–6130 USA (e-mail: fang@ece.ufl.edu).
I. Chlamtac is with the Erik Jonsson School of Engineering and Computer Science, The University of Texas at Dallas, Richardson, TX 75083-0688 USA. Y.-B. Lin is with the Department of Computer Sciences and Information Engi-neering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: liny@csie.nctu.edu.tw).
Publisher Item Identifier S 0018-9545(00)04848-9.
be needed to find the best location update and paging scheme. In order to carry out this task, an appropriate movement model for a portable needs to be constructed. From [13], we observe that one critical quantity in the cost analysis is the probability of the number of RA crossings, that is, the probability that a portable moves RA between the two consecutive served calls (i.e., during the interservice time). For example, in IS-41, each RA crossing will incur at least one signaling message (for reg-istration), the average number of signaling messages for a call life, which can be found from the probability distribution of RA crossings, will be used in the tradeoff analysis [13]. How-ever, the cost analysis carried out in [13] is valid only for the cases when the interservice time is exponentially distributed, moreover, the busy-line effect is not considered. In general, this assumption is not valid, which was also observed in the same paper. The difficulty in carrying out the same cost analysis for general situation lies in the lack of analytical result for the prob-ability of the number of RA crossings under general interser-vice time. As long as we find the computational procedure for the probability of the number of RA crossings, the same cost analysis in [13] can be carried out in a similar fashion. We also observe that the probability distribution of RA crossings is also signifying the portable movements in PCS networks.
In this paper, under the assumptions that the interservice times and the RA residence times are generally distributed, we derive some analytical results for the probability of the number of RA crossings. The results presented in this paper are very useful for cost analysis in finding a best tradeoff between location updating and paging for tracking mobile users in PCS networks.
II. PROBABILITY OF THENUMBER OFRA CROSSINGS In this section, we study the patterns of the incoming calls and the portable movement. Assume that the incoming calls to a portable form a Poisson process, the time the portable stays in an RA (called the RA residence time) has a general distribu-tion. We will derive the probability that a portable moves across RAs between two phone calls. The time between the start of a call served and the start of the following call served by the portable is called the interservice time. The interservice time is of interest because it can be used to characterize the mo-bility of the portable. It is possible that a new call arrives while 0018–9545/00$10.00 © 2000 IEEE
Fig. 1. The time diagram forK RA crossings.
the previous call served is still in progress [13]. In this case, the portable cannot initiate/accept the new call. In this analysis, we ignore call waiting service, and this new call is rejected. Thus, the interarrival (inter-call) times are different from the interser-vice times. This phenomenon is called the busy line effect. Al-though the incoming calls form a Poisson process (i.e., the in-terarrival times are exponentially distributed), the interservice times may not be exponentially distributed. By ignoring the busy line effect, Lin [13] is able to give analysis for the model. In this section, we assume that the interservice times are generally dis-tributed and derive an analytic expression for
Before we give the analytical result for we first demon-strate the use of in cost analysis. We take the IS-41 system for illustration purpose. In IS-41, each RA crossing incurs at least one signaling message, i.e., the registration message. The signaling cost for atypical call will be directly proportional to the average number of signaling message, which can be found from the following formula:
-Thus, we have to find the probability distribution In [13], simple formula can be found for - [and other quantities which are functions of ] under the exponential assumption on the interservice time. However, when the interservice time is not exponentially distributed, which is the case in practice, we must found viable computational procedure to calculate This is the motivation of the current paper. Next, we present an analytical result for
Let denote the RA residence times and de-note the residual life of the previous call served in the initiating RA (i.e., the time interval between when the call is served and when the portable exits the RA). Let denote the interservice time between two consecutive served calls to a portable (i.e., the time interval between the instant the previous call is served and the instant the next call is served). Notice that the consecu-tive served calls may not be necessarily the consecuconsecu-tive arriving calls because some calls may be blocked when the portable is busy. This implies that the interservice time is different from the interarrival time. Fig. 1 shows the time diagram for RA
crossings. Suppose that the portable is in an RA when the previous call arrives and is served, it then moves RA’s during the interservice time, and resides in the th RA for a period
Let be independent and identically distributed (iid) with a general density function let be generally distributed with density function and let be the density function of Let and be the Laplace transforms of and respectively. Let
and From the random observer
property [10], we have
(1) where is the distribution function of It is obvious that the probability is given by
Pr (2)
Pr
(3) We first calculate Since the Laplace transform of is from (2), the inverse Laplace transform and the independence of and we have
Pr
Let and be the density function and the Laplace transform of From the independence of
we have
Thus, the density function is given by
Also, the Laplace transform of Pr (the distribution
func-tion) is We have
Pr
Pr
Taking this into (3), we obtain Pr
Pr
(5) It is obvious that the integrand without term in (4) and (5) is analytic on the right half open complex plane. If has no branch point and has only finite possible isolated poles in the right half plane (which is equivalent to saying that has only finite number of isolated poles in the left half plane), then the Residue Theorem can be applied to (4) and (5) using a semi-circular contour in the right half plane. Indeed, if we use to denote the set of poles of in the right half complex plane, then from (4) and (5), and the Residue Theorem [12], we obtain the following.
choosing the contour enclosed by the semi-circle at center and with radius sufficiently large, then we can apply the Residue Theorem to complete the proof.
If the interservice times are exponentially distributed with
pa-rameter then which has a unique
pole, and From Theorem 1, we can easily obtain
where is the call-to-mobility ratio. These equations have been obtained in [13] using a different approach.
One general distribution which is used often in many applica-tions is the Gamma distribution [2] whose density function and its Laplace transform are given as follows:
(7) where is the shape parameter, is the scale parameter, and is the Gamma function. When is a positive integer, the Gamma becomes the Erlang distribution
(8) where The gamma distribution applies to many appli-cations. When it becomes the exponential distribution; When is sufficiently large, the distribution is asymptotically normal around [2].
Let us assume that the interservice times are iid with Erlang distribution as in (8), since its mean is hence we have
hence
Let
(9)
From Theorem 1, we can easily obtain
(11)
(12)
where denotes the th derivative of function
In order to compute those quantities in (11) and (12), we need to find the th derivative of certain functions, which may be complicated in general. Fortunately, we can have some recur-sive algorithms to compute them. We need the following iden-tity:
(13)
For fixed let To compute we can use
the following (applying (13) with and
To compute we use (applying (13) with and
(see equation at the bottom of the page) where
It is known [9] that the weighted summation of Erlang dis-tributions (the hyper-Erlang disdis-tributions) can approximate any distribution. For the cases when the interservice time is dis-tributed according to the weighted summation of Erlang distri-bution, using Theorem 1, we can also find simple analytical results. It has been shown [20] that the SOHYP (the sum of hyper-exponential distributions) is also a very general distribu-tion to model the LA residence time (i.e., the dwell time), we ob-serve that Theorem 1 is also applicable when the LA residence time is SOHYP distributed. In fact, as along as the Laplace transform of the density function of the interservice time is rational function, our results can be applied to find the proba-bility
III. BUSY-LINEEFFECT
As we mentioned earlier, the busy-line effect is the phenom-enon when a new call to the portable arrives it finds the portable is busy, and hence it is rejected. In Lin [13], this effect is ne-glected for the purpose of analysis. Next, we show that our re-sults can conveniently be used to study this effect.
As before, assume that the call arrivals to the portable form a Poisson process, i.e., the inter-call times are independent and exponentially distributed. Let be the probability that a call to the portable finds the portable busy (i.e., busy-line). This prob-ability will depend on the call arrival traffic and call holding times. In this analysis, we will use this probability to represent the total effect of call arrival traffic and call holding times. Let denote the density function of Erlang distribution with shape parameter and scale parameter [see (8)]. It is well known [10] that this Erlang distribution is the distribu-tion of the summadistribu-tion of independent exponential distribu-tion with parameter A call to the portable finds the portable busy with probability and finds the portable free and is served immediately with probability hence the conditional prob-ability density function of the interservice time when calls find the portable busy while the th call finds the portable free is given by
Hence the probability density function of the interservice time is given by
(14)
Applying Theorem 1 with (11) and (12), we obtain the following result.
Theorem 2: If is the probability of busy-line, then
(15)
Notice that when the busy line effect is neglected, i.e., Theorem 2 reduces to the case when the interservice time is ex-ponentially distributed. We can not find simpler forms for the above formulae, however, the above formulae provide starting point for approximation. Considering that is usually quite small, we can use finite number of terms in (15) for approxi-mation.
Fig. 2. Probability (K): (a) and (b) for the cases when the interservice time is exponentially distributed and RA residence time is Gamma distributed; (c) and (d) for the case when the interservice time is Erlang distributed and RA residence time is Gamma distributed; = 0.1, 0.5, 1, 2, 6, 10, m = 2.
Fig. 3. Comparison of probability (K) when the interservice time is exponentially distributed (solid line) and Erlang distributed (dashed line): call-to-mobility ratio is small (<1), m = 2.
IV. DISCUSSIONS ANDCOMMENTS
In this section, we present a few examples to discuss our results. We assume that the interservice time is Erlang dis-tributed and the RA residence times are Gamma distributed. Let have the following Erlang density
whose mean is and variance is When
it becomes the exponential distribution. Let have the Gamma density function
whose mean is and variance is By
Fig. 4. Comparison of probability (K) when the interservice time is exponentially distributed (solid line) and Erlang distributed (dashed line): call-to-mobility ratio is large (>1), m = 2.
Fig. 5. Busy-line effect on the probability (K) of K RA crossing, = 1.5. time, in the same token, varying is equivalent to varying the variance of RA residence time. We study the effects on when the variances of the interservice time and the RA residence time vary.
Fig. 2 shows the probability when the RA residence time is Gamma distributed with various values of variance while
the interservice time is exponentially distributed and Erlang dis-tributed, respectively. It can be easily observed that when the variance of the RA residence time has significant effect on the probability for small when call-to-mobility is small
and that hardly has any effect on the probability when call-to-mobility is large The Erlang
V. CONCLUSION
In this paper, we propose a new model for the portable move-ment in PCS networks. Previous work [13] investigated this issue by ignoring the busy-line effect. We relax this restriction by accommodating interservice times with general distribution. We apply the Residue Theorem to obtain analytical results for the probability of number of the RA crossings. Our new model can be applied to investigate the impact on busy-line effect on many location tracking strategies [13], [14], [21].
ACKNOWLEDGMENT
The authors would like to express their sincere gratitude to Dr. Stephen Rappaport and the anonymous reviewers for their detailed and constructive suggestions which greatly improve the quality of this paper.
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Yuguang Fang (S’96–M’98–SM’99) received the
B.S. and M.S. degrees in mathematics from Qufu Normal University, Qufu, Shandong, China, in 1984 and 1987, respectively, the Ph.D degree in systems and control engineering from Department of Systems, Control and Industrial Engineering at Case Western Reserve University, Cleveland, OH, in January 1994, and the Ph.D. degree in electrical engineering from the Department of Electrical and Computer Engineering at Boston University, MA, in May 1997.
From 1987 to 1988, he held research and teaching positions in both Depart-ment of Mathematics and the Institute of Automation at Qufu Normal Univer-sity. From September 1989 to December 1993, he was a Teaching/Research As-sistant in the Department of Systems, Control and Industrial Engineering at Case Western Reserve University, where he held a Research Associate position from January 1994 to May 1994. He held a Postdoctoral position in Department of Electrical and Computer Engineering at Boston University from June 1994 to August 1995. From September 1995 to May 1997, he was a Research Assis-tant in Department of Electrical and Computer Engineering at Boston Univer-sity, MA. From June 1997 to July 1998, he was a Visiting Assistant Professor in Department of Electrical Engineering at the University of Texas at Dallas. From July 1998 to May 2000, he was an Assistant Professor in the Department of Electrical and Computer Engineering at New Jersey Institute of Technology, Newark, NJ. Since June 2000, he has been an Assistant Professor in the De-partment of Electrical and Computer Engineering at the University of Florida, Gainesville, FL. He has published more than 40 papers in the professional jour-nals and refereed conferences. His recent research interests include wireless net-works and mobile communications, personal communication services (PCS), call admission control and resource allocations.
Dr. Fang is an Editor for IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS: WIRELESSCOMMUNICATIONSSERIES, an Editor for IEEE
Transactions on Communications, and an Area Editor for ACM Mobile Com-puting and Communications Review. He is the Program Vice-Chair for the 2000 IEEE Wireless Communications and Networking Conference (WCNC’2000),
the Program Vice-Chair for the Seventh International Conference on Computer
Communications and Networking (IC N’98), a member of Technical Program
Committee for the International Conference on Computer Communications
(INFOCOM’98 and INFOCOM’00), a member of the Technical Program
Com-mittee 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).
Imrich Chlamtac (M’86–SM’86–F’93) received the
Ph.D. degree in computer science from the University of Minnesota, Duluth, in 1979.
Since 1997, he has held the Distinguished Chair in Telecommunications at the University of Texas at Dallas (UTD) and is the Director of CATSS, the Center for Advanced Telecommunications Systems and Services. Prior to joining UTD, he was a Professor of Electrical Engineering, a member of the Photonic Center at Boston University, and President of BCN Inc., a company dealing with network design. He holds the honorary titles of Senior Professor at Tel Aviv University, and of University Professor at the Technical University of Budapest. He has published more than 250 in refereed journals and conferences, chapters in books and encyclopedias, as well as the first textbook on LAN’s (1981) and delivered dozens of Distinguished Lectures, Plenary, and Keynote addresses.
Dr. Chlamtac is a Fellow of the ACM. He serves as the founding Editor in Chief of the ACM/URSI/Baltzer Wireless Networks (WINET) and Mobile
Net-works and Applications (MONET) journals, and the new SPIE/Baltzer Optical Networks Magazine. Previously, he was on the editorial boards of most major
publications in telecommunications. He was the General Chair of leading ACM and IEEE conferences and workshops, including ACM Sigcomm, ACM/IEEE MobiCom, and IEEE CCW. He is the founder and Steering Committee Chair of the ACM/IEEE MobiCom and the Founding Chairman of ACM Sigmobile.
Yi-Bing Lin (S’80–M’96–SM’96) received the
B.S.E.E. degree from National Cheng Kung Uni-versity, Hsinchu, Taiwan, R.O.C., in 1983, and the Ph.D. degree in computer science from the University of Washington, St. Louis, MO, in 1990.
From 1990 to 1995, he was with the Applied Research Area at Bell Communications Research (Bellcore), Morristown, NJ. In 1995, he was appointed as a Professor of the Department of Com-puter Science and Information Engineering (CSIE), National Chiao Tung University (NCTU). In 1996, he was appointed as Deputy Director of Microelectronics and Information Systems Research Center, NCTU. In 1997, he was been elected as Chairman of CSIE, 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 IEEE NEURALNETWORKS, an Editor of IEEE JOURNAL ONSELECTEDAREAS INCOMMUNICATIONS-WIRELESSSERIES, an Editor of IEEE PERSONALCOMMUNICATIONSMAGAZINE, an Editor of
Com-puter 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 ACM/Baltzer 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 ONCOMPUTERSSPECIALISSUE ONMOBILECOMPUTING, and a Guest Editor for IEEE COMMUNICATIONSMAGAZINESPECIAL ISSUE ON ACTIVE,PROGRAMMABLE, ANDMOBILECODENETWORKING. He received the 1997 Outstanding Research Award from National Science Council, ROC, and the Outstanding Youth Electrical Engineer Award from CIEE, ROC.
