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Billing Strategies and Performance

Analysis for PCS Networks

Yuguang Fang,

Member, IEEE,

Imrich Chlamtac,

Fellow, IEEE,

and Yi-Bing Lin,

Senior Member, IEEE

Abstract— It is predicted that by the year 2000, U.S. cellular

carriers will invest billions of dollars for cellular billing and customer care. Two of the most desirable attributes of the cellular billing systems are the flexibility of upgrade and ability to inform the billing experts quickly about the status of the system to minimize any possible fraud and improve customer service. Although the second attribute can be realized by reporting the billing customer record in real time, this tends to congest the signaling channel, which is not desirable. In this paper, we propose several schemes for the provision of a quick billing status report which maintains low-signaling traffic. The performance of these schemes is derived analytically. We expect that these results may provide a guideline for the future design of on-line billing systems in personal communication services (PCS’s) networks.

Index Terms—Billing, cell residence times, PCS networks,

real-time updating.

I. INTRODUCTION

B

ILLING WORLD [11] predicts that by the year 2000, U.S. cellular carriers will invest U.S. $1.6 billion for cellular billing and customer care. According to the managers from the top 20 cellular carriers, two of the most desirable attributes of the cellular billing systems are:

• flexibility of upgrade;

• capability to inform the in-house billing experts quickly about the status of the system.

Telecommunication services are “culture sensitive,” and the ways of service charging will significantly affect customers’ behavior. For example, in many U.S. cellular services, cellular subscribers are charged for the cellular usage, whether they are the calling or the called parties. Thus, cellular customers will tend to share their cellular phone numbers with only a small group of people to avoid “junk calls.” On the other hand, the calling party is always charged for the cellular usage in some countries, for instance, in Taiwan. Therefore, cellular subscribers (especially people using cellular phones Manuscript received November 30, 1996; revised June 18, 1998. The work of Y. Fang and I. Chlamtac was supported in part by the U.S. Army Research Office under Contracts DAAG55-97-1-0312 and DAAG55-97-1-0382. The work of Y.-B. Lin was supported in part by the National Science Council, Taiwan, R.O.C., and Far EasTone Telecommunications Co. Ltd.

Y. Fang is with the Department of Electrical and Computer Engineer-ing, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: fang@megahertz.njit.edu).

I. Chlamtac is with the Erik Jonsson School of Engineering and Computer Science, University of Texas at Dallas, Richardson, TX 75083 USA.

Y.-B. Lin is with the Department of Computer Sciences and Information Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C. (e-mail: liny@csie.nctu.edu.tw).

Publisher Item Identifier S 0018-9545(99)01049-X.

for business) tend to distribute their cellular phone numbers as widely as possible to enhance their business opportunities. To maximize profits, a cellular carrier will therefore need to offer a variety of billing plans for same services and may need change the plans from time to time to adapt to changing “customer culture.” Thus, flexibility with upgrade is considered a highly important attribute of the billing system.

Another important attribute is the provision of quick billing status report, which is essential for the monitoring and diagnos-ing the billdiagnos-ing system. One way to achieve this attribute is to report the customer billing records in real time. Unfortunately, this feature is not supported in most existing cellular billing systems. By comparison, in the public switched telephone net-work (PSTN) (wireline netnet-work), real-time billing information is possible. Typically, billing information will be delivered in the signaling system no. 7 (SS7) messages [9] during the call setup/release processes. The billing information is produced from the automatic message accounting (AMA) records. During the call setup/release processes, the monitor system tracks SS7 messages of the call and generates a call detail record (CDR) in AMA format when the call is completed. The CDR record will be stored in Bellcore accounting format (BAF), and the data can be transfered to the rating and billing systems.

The difficulty of providing real-time cellular customer billing records is due to the fact that the cellular users may roam from their “home systems” (HS’s) to the “visited systems” (VS’s). When a cellular user is in a VS, the billing records for all call activities are kept in the VS. In the existing cellular roaming management/call control protocols [7], there is no interaction between the VS and the HS at the end of a call. Typically the billing information is kept in the VS as a “roam-type” cellular intercarrier billing exchange record (CIBER). The roam CIBER’s will be batched and periodically sent to a clearinghouse electronically or via mail in a tape format and later forwarded by the clearinghouse to the customer’s HS. The whole process may take from five days to more than two weeks. To speed up the billing information transmission, a cellular billing transmission standard called EIA/TIA IS-124 has been developed by working group four of TIA’s TR 45.2 committee [1]. IS-124 will allow real-time billing information exchange, which will help control fraud by reducing the lag time created by the use of overnighted tape messages. Version A of IS-124 will also accommodate both US AMPS and international GSM carriers, which is desirable for heterogeneous personal communication services (PCS’s) integration [8].

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Fig. 1. The timing diagram for the checkpointing model based on the number of calls.

An important performance issue of cellular billing informa-tion transmission is the frequency of the billing informainforma-tion exchange. In the ideal case, records would be transmitted for every phone call to achieve the real-time operation. However, real-time transmission would significantly increase the cellular signaling traffic and seriously overload the signaling network of the PSTN. In order to achieve quick billing status report, a tradeoff is therefore needed between the frequency of the billing information transmission and the signaling traffic.

In this paper, we propose three strategies for controlling the billing traffic for future PCS networks, which can be easily implemented in the billing systems of current cellular systems. The first strategy is to update the billing information when a fixed number of calls have been handled. The second strategy is to report the customer billing record in fixed real-time intervals. The third strategy is a combination of these two strategies. We provide performance analysis of these strategies. Our study provides the guidelines to select the appropriate frequency of transmissions for different network engineering requirements.

II. THE BILLINGCHECKPOINTING MODELS

The purpose of real-time billing information transmission is to ensure that when one (either a billing expert of the HS or a querying customer) checks the customer billing records, the information is up to date. The time of checking is referred to as the billing information retrieval point or retrieval point. Very often the billing record observer may tolerate certain degree of information outstandingness. We define an observation at a retrieval point as -call outstanding if the most recent call records have not been received by the HS at the retrieval point. We can also define an observation at a retrieval point as -time outstanding if the time elapsed at the retrieval point from the last record update is no more than time units. It is very important to know the distributions of the number of the outstanding calls and the outstanding time. Since the billing system makes decisions based on the billing information received from the VS’s, the degree of “outstandingness” of the billing information affects the accuracy of the billing decisions. Thus, it is desirable to know the average number of outstanding calls and the average total calling time of outstanding calls for the performance evaluation of billing systems.

We propose three strategies for the provision of quick billing status report while keeping the signaling traffic due to the on-line billing update in a certain tolerable level.

A. Checkpointing Model Based on the Number of Calls Assume that the billing information of a roamer is sent back to the HS for every calls. We refer to as the checkpointing interval, where The real-time requirement in the cellular billing system specification may be “the probability of more than -call outstanding observation should be less than ” Thus, based on the mobility and call activities of a user, an appropriate value can be selected to satisfy the above requirement.

Note that when the user leaves the VS, the HS will send a deregistration or cancellation message to the VS informing it that the user left the VS, and the VS will acknowledge the deregistration ([12]). We assume that the not-yet checkpointed billing records will be sent back to the HS through the acknowledgment, and no extra billing transmission message will be created.

Suppose that the roamer enters a VS at time zero. The roamer resides at the VS for a period as shown in Fig. 1. We call as the VS residence time. Suppose that the roamer’s billing information is retrieved at the HS at time , i.e., an expert or a customer of the HS queries the billing information at time If there are phone calls to the roamer during , then the billing retrieval is -call outstanding if for some Let have a general distribution with the Laplace transform and the mean the calls to the roamer be a Poisson process with arrival rate and the billing retrieval be a random observer. We note that a billing record for a call is created when the call is completed. Thus, “call arrival time” in this context means that the time when the call record is created. Let be the probability that a billing retrieval is -call outstanding. We use in the subsequent development, to denote the th order derivative of function at point Then is derived in Appendix I [see (18)] as

Let denote the number of outstanding calls, i.e., the number of arriving calls during the interval between the last billing update and the retrieval point. These calls are not available at the retrieval point at HS. Let be the total calling time of these outstanding calls, and let the arriving calls have the expected call holding time Then the expected number of

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outstanding calls is given by

Let denote the call holding time of an arriving call, then the total calling time of outstanding calls is given by

From Wald’s equation ([14]), we obtain

Remark: If is interpreted as the average call hold-ing times for possible fraudulent calls, then the maximum fraudulent calling time is

Furthermore, if the VS residence time has an Erlang distribution with the shape parameter and the scale pa-rameter (and thus the variance of the distribution is ), then the probability of -call out-standing billing retrieval for Erlang [with parameters ] VS residence time is derived in Appendix I as [see (21)] given in the equation at the bottom of the page. Specifically from the derivations in Appendix I [see (22), (24), and (26)], we have

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It is observed that when is sufficiently small, i.e., the roamer stays in that visited area for sufficiently long time, the number of outstanding calls are equally probable, i.e., uniformly distributed. Indeed, we can easily show that

We conjecture that this observation is valid for any nonlattice distribution

When the VS residence time is exponentially distributed, from Appendix I we obtain

B. Checkpointing Model Based on Real-Time Interval In the previous section, we applied a threshold to the number of calls in the update of billing status. This approach may have a disadvantage when the call holding times are long. For example, when a fraudulent user, using another customer’s identification, makes a call to a 900 number in the United States, the call holding time tends to be quite long. During a typical billing period, the number of such calls may not be high, therefore, the previous strategy may not be appropriate. One solution to overcome this is to use real-time interval (threshold in time) for updating of billing record. In this section, we analyze this strategy.

Assume now that the billing information is sent back to the HS from the VS for every time units. As in the previous section, we assume that the VS residence time has a nonlattice density function with Laplace transform

and with expected residence time Let be its distribution function. Let be the retrieval point. Let denote the outstanding time, i.e., the time between the retrieval point and the last update instant. Since the customer billing record is forwarded to the HS every time units, is distributed in the

interval Let have density function .

Then, from Appendix III, we have

for for

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Fig. 2. Time diagram for checkpointing model based on real-time interval.

Suppose that has only isolated singular points. Let denote the set of poles of Let denote the residue operator at the pole ([6]). From Appendix III, can be expressed as follows:

(2)

Let be the number of outstanding calls as defined in the last section and the total calling time of outstanding calls, let denote the average call holding time of the outstanding calls. We can obtain (Appendix III)

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When the VS residence time is Erlang distributed, simple analytic results can be obtained. For simplicity, let

Let , and then we obtain (Appendix III)

(5) (6) (7)

In particular, when , i.e., the VS residence time is exponentially distributed with parameter , we have

(8) When , we have (Appendix III) (9), given at the bottom of the page. It is not difficult to show that when is sufficiently small, the density function for either or approaches , which implies that the outstanding time is uniformly distributed when the roamer stays in the visited area for sufficiently long. We conjecture that this is true for any nonlattice distributed VS residence time.

C. Checkpointing Model Based on Number of Calls and Real-Time Interval

Another strategy is to use both the number of calls and real-time interval for updating billing record. The thresholds used in this case and may be less restricted (larger) than those in previous strategies. In this strategy, the billing record update works as follows. Choose threshold for the number of calls and threshold for the real-time interval. The billing record at the VS keeps the records of both time, say, and the number of calls, say, When a customer enters a visited area, the billing record for it starts, and setting and , the clock for starts. If there are calls from or to the customer, is incremented by one. Whenever either or reaches the threshold, the billing record is forwarded to the HS at the same time and are reset to zeros and starts over again if the customer is still in the VS area. The methods used in previous two sections can be modified to analyze the performance of this model.

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D. How to Choose the Thresholds

In the above strategies, we need to determine how often the customer’s billing record should be updated, i.e., how to choose the thresholds, since if the billing record is forwarded from VS to the HS one by one in real time, the signaling channel may be congested.

To this end we assume that the call arrivals of a customer have Poisson distribution with parameter and let

denote the interarrival times whose expected value is We consider the first strategy. In this strategy, the number of outstanding calls is related to the call arrival process. Suppose that the billing system may tolerate the outstandingness of time for a period of , which is determined by the PCS network design objective and the availability of the bandwidth for signaling channel. Then the total time that call arrives is given by

and we need , which is equivalent to

So the threshold can be chosen as the integral part of the positive number

In the second strategy, the billing system will be tolerant of a fixed number of outstanding calls. This is interpreted as the probability of more than, say, calls in the interval of length is less than a preassigned number, say, Suppose that there are calls in the updating interval with length , then the above criterion can be translated into the following:

i.e.,

(10)

Notice that since for

any , so is a strictly decreasing function of Also,

and , and then for any , we can

always find solution for (10). For very small , (10) is always true, but the small value for may not be desirable due to the channel traffic intensity requirement. The best value would be the largest that satisfies (10). It is obvious that there is a unique solution for this value, which is denoted by We can choose in the second strategy. The thresholds in the third strategy can use either , a conservative choice, or from the previous two strategies based on the value

III. ILLUSTRATIVE EXAMPLES

We first consider the model based on the number of out-standing calls. Based on (1), Fig. 3 illustrates the effect of the Erlang shape parameter on the -call outstanding probability In the figures, the mean VS residence

time is can be several hours, days, or

months. For simplicity, the call arrival rate is normalized by in our numerical examples. Each figure plots the

Fig. 3. Effects ofm on Pr[k]:

curves for and (i.e., and ) and

and That is, we consider the

Erlang VS residence times with the same mean , but different variances , , and , respectively. The figures indicate that the shape parameter (or the variance) of the Erlang VS residence times do not have significant effect on the -call outstanding probability. This result is true for different values and for large values. Figs. 4 and 5 plot

for different values, where and and ,

respectively. These figures indicate that

for (11)

Let be the call to mobility ratio. When ,

may significantly larger than On the other

hand, if , then This phenomenon is

explained as follows. In the period , the last checkpointing is performed when the roamer moves out of the VS, and no more than call records are sent back to the HS. Thus, if a billing retrieval falls in this last checkpointing period, it is likely to have a small -call outstanding observation. For the other “normal” -checkpointing intervals, tends to be uniformly distributed (because the billing retrieval is a random observer). When , the end effect of the last checkpointing interval becomes insignificant and

On the other hand, when is not much larger than , the end effect results in large for a small Note that the situations of small are often observed in the existing system [1], and the end effect cannot be ignored. Suppose that checkpointing is performed for every phone calls (i.e., the checkpointing interval is ). Define as the probability that the retrieval is less than -call outstanding and as the number of checkpointing operations performed during In

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Fig. 4. Effects of on Pr[k] (n = 5):

cellular network engineering, maximizing and minimizing are two conflict goals.

It is easy to derive that

for From (27) in Appendix II

form Figs. 6 and 7 plot and for and

, respectively. From these two figures, several engineering questions can be answered. For example, when , if is changed from 12 to 8, then is increased by 16% and is increased by 130%. With appropriate weighting factors specific to the network under study, one can use the above data to determine whether it is beneficial to change from 12 to 8. Note that is more sensitive to the change of when is large than when is small. On the other hand, is more sensitive to the change of when is small than when is large. When , if is changed from 12 to 8, then is increased by 44%, and is increased by 57%. Thus, it is more cost effective to decrease when is large than when is small.

Next, we consider the billing model based on the real-time interval. Fig. 8 illustrates the density function for different

Fig. 5. Effects of on Pr[k] (n = 10):

the average VS residence time for the case when the VS residence time is exponentially distributed. As in the previous case, the density function is a decreasing function of time elapsed from the last updating. When the roamer stays sufficiently long at that visited area, the outstanding time is uniformly distributed. When the roamer stays sufficiently short at the visited area, the outstanding time is approximately exponentially distributed.

Fig. 9 shows the expected number of outstanding calls for different call arrival rate As expected, the expected number of outstanding calls is decreasing as is increasing, which implies that the shorter the roamer stays, the fewer the outstanding calls. The expected number of outstanding calls is also decreasing as the call arrival rate is decreasing, i.e., the fewer the arrival calls, the fewer the outstanding calls. These observations are consistent with our intuition.

Fig. 10 shows the density function of outstanding time is not much affected by the variance of the VS residence time (which is uniquely determined by ).

IV. CONCLUSIONS

This paper has proposed a number of strategies for ex-pediting billing record updates in the PCS networks. These strategies have been designed as a compromise between the requirements of traffic for network signaling and the timeliness of the roamer’s billing record (i.e., the accuracy of the billing information) at the HS. Performance analysis was performed for all proposed strategies and was shown to be useful in

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Fig. 6. The effect of n on 2;n (m = 1):

optimizing the strategies performance by proper setting of timing thresholds to balance control traffic overhead and the usefulness of the billing process. We therefore believe that the results presented here can be used in the future design of billing systems for PCS networks.

APPENDIX I

Consider Fig. 1. Assume that has a general den-sity function with the Laplace transform

and the mean Let the

calls to the roamer be a Poisson process with arrival rate and the billing retrieval point be a random observer. Let and be the density function and the Laplace transform of the interval in Fig. 1. If is nonlattice, then from the residual life theorem [10] we have

and (12)

The billing retrieval is -call outstanding if there are call arrivals during the period Since the call arrivals during the period are a Poisson process, the -call outstanding probability is expressed as

(13) From (12), (13) is rewritten as (14) where (15) and (16)

For two functions and , we have

and letting and , (16) is rewritten as

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Fig. 7. The effect of n on E[Nn] (m = 1):

From (14), (15), and (17), we have

From this, we can develop an algorithm to compute the probability Since is an analytic function at ,

hence

hence from (18), we obtain

(18)

Thus, we first expand the at as a Taylor series,

using this to express as and then using

the sequence and (18) to compute ,

hence, can be computed by This

algorithm is numerically easy to implement.

For special VS residence time distributions, an analytical expression may be obtained. For example, if the VS residence times have an Erlang distribution with the shape parameter and the scale parameter , then

and

(19)

Denote as the probability when the Erlang

VS residence distribution has the shape parameter and the scale parameter Substitute (19) in (18) and after careful rearrangement using the combinatorial identity techniques

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Fig. 8. The density function of outstanding time for different  (m = 1):

[13], we have

(20) Equation (20) can be presented in a recursive format as (21), given at the bottom of the page. For , we have

(22) For , we have (23) Since and for for (21)

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Fig. 9. The expected number of outstanding calls for different arrival rate (m = 1): Equation (23) is rewritten as (24) For (25) Since (25) is rewritten as (26)

When the VS residence time is exponentially distributed, i.e., , we have

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Fig. 10. The density function for different m:

and

APPENDIX II

Let be the probability that there are checkpointing operations during Then

If is exponentially distributed, then we have

and

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APPENDIX III

As before, let be the VS residence time and be the billing information retrieval time instant. Let and be the density function and distribution function of , respectively, and denotes its Laplace transform. Let and be the density function and Laplace transform of the interval Let denote the updating interval, i.e., the billing information will be forwarded from the VS to HS every time units whenever the roamer is still in that visited area. Using the residual life theorem ([10]), we have

and (28)

Let denote the time during which the outstanding calls may happen as shown in Fig. 2, i.e., the time between the last updating instant, say, and the checking point , so

where here means the residue modulo Let and denote the density function and distribution function

(12)

so the density function of is given by

(29) Suppose that the Laplace transform of the VS residence time has only isolated singular points in the open left-half complex plane, let be the set of poles of Let be such real number such that all poles have real parts smaller than It is obvious that and have the same poles. From the inverse Laplace transform formula and the Residue Theorem ([6]), we have (the contour used in the Residue Theorem is chosen to be the domain in the left-half plane to the left of vertical line

(30) where denotes the residue at the pole

Assume that the call arrivals for the roamer is Poisson with arrival rate Let denote the number of outstanding calls (the number of calls arriving after the last billing update instant and before the retrieval point, the billing records of these

(31) Following a similar procedure as in the derivation of (30), the expected number of outstanding calls is given by

(32)

Let be the call holding times of the roamer and assume that this random sequence is independently iden-tically distributed with expected value . Then the total outstanding calling time (the summation of total calling times for all outstanding calls) is given by

(13)

From Wald’s equation [14], we obtain the expected total outstanding time is given by

(34) For simplicity, let

(35) If the VS residence times have an Erlang distribution with the shape parameter and the scale parameter , then

This function has only one pole , i.e., From (30), (32), and (34), we obtain

(36) (37) (38) where denotes the th order derivative of function at point

In particular, when , i.e., the VS residence times are exponentially distributed, and we have

When , the outstanding time distribution is given by

The expected number of outstanding calls and the total out-standing calling time are given by the equations at the bottom of the previous page, respectively.

REFERENCES

[1] E. Bond, “IS-124 Standard for cellular roaming moving forward with implementation,” Billing World, pp. 30–37, Mar./Apr. 1996.

[2] Y. Fang, I. Chlamtac, and Y. B. Lin, “Modeling PCS networks un-der general call holding times and cell residence time distributions,”

IEEE/ACM Trans. Networking, vol. 5, pp. 893–906, Dec. 1997.

[3] , “Call performance for a PCS network,” IEEE J. Select. Areas

Commun., vol. 15, pp. 1568–1581, Oct. 1997.

[4] V. K. Garg and J. E. Wilkes, Wireless and Personal Communications

Systems. Englewood Cliffs, NJ: Prentice-Hall, 1996.

[5] F. P. Kelly, Reversibility and Stochastic Networks. New York: Wiley, 1979.

[6] W. R. LePage, Complex Variables and the Laplace Transform for

Engineers. New York: Dover, 1980.

[7] Y. B. Lin and S. K. DeVries, “PCS network signaling using SS7,” IEEE

Personal Communications Mag., pp. 44–55, June 1995.

[8] Y. B. Lin and I. Chlamtac, “Heterogeneous personal communications services,” IEEE Communications Mag., vol. 34, pp. 106–113, Sept. 1996.

[9] Y. B. Lin, “Signaling system number 7,” to be published.

[10] I. Mitrani, Modeling of Computer and Communication Systems. Cam-bridge, U.K.: Cambridge Univ. Press, 1987.

[11] S. H. Moran, “Cellular companies to expense up to $1.2 billion on billing and customer care in 1996,” Billing World, pp. 12–16, Mar./Apr. 1996. [12] A. R. Noerpel, L. F. Chang, and Y. B. Lin, “Polling deregistration for unlicensed PCS,” IEEE J. Select. Areas Commun., vol. 14, no. 4, pp. 728–734, 1996.

[13] J. Riordan, Combinatorial Identities. New York: Wiley, 1968. [14] S. M. Ross, Introduction to Probability Models, 5th ed. New York:

Academic, 1993.

Yuguang Fang (S’96–M’98) received the B.S. and M.S. degrees in mathematics from Qufu Normal University, Qufu, China, in 1984 and 1987, re-spectively, the Ph.D. degree in systems and control engineering from Case Western Reserve University, Cleveland, OH, in 1994, and the Ph.D. degree in electrical and computer engineering from Boston University, Boston, MA, in 1997.

From 1987 to 1988, he held research and teaching positions in both the Department of Mathematics and the Institute of Automation at Qufu Normal University. From September 1989 to December 1993, he was a Teaching and a Research Assistant in the Department of Systems, Control and Industrial Engineering, Case Western Reserve University, where he held a research associate position from January 1994 to May 1994. He held a post-doctoral position in the Department of Electrical Engineering, Boston University, from June 1994 to August 1995. From September 1995 to May 1997, he was a Research Assistant in the Department of Electrical and Computer Engineering, Boston University. From June 1997 to July 1998, he was a Visiting Assistant Professor in the Department of Electrical Engineering, University of Texas at Dallas, Richardson. Since July 1998, he has been an Assistant Professor in the Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newark. He has published more than 30 papers in professional journals and refereed conferences. He is the Area Editor for ACM Mobile Computing and Communications Review. His research interests include wireless networks and mobile communications, personal communication services, stochastic and adaptive systems, hybrid systems in integrated communications and controls, robust stability and controls, nonlinear dynamical systems, and neural networks.

Dr. Fang is the Program Vice Chair for the Seventh International Conference on Computer Communications and Networking(IC3N098) and a Member of the Technical Program Committee for the 1998 International Conference on Computer Communications (INFOCOM’98).

Imrich Chlamtac (M’86–SM’86–F’93) received the B.Sc. and M.Sc. degrees in mathematics with highest distinction and the Ph.D. degree in computer science in 1979, all from the University of Minnesota, Minneapolis.

He currently holds the Distinguished Chair in Telecommunications at the University of Texas at Dallas, Richardson, and is the President of Boston Communications Networks. He is the author of over 200 papers in refereed journals and conferences, multiple books, and book chapters. He is the Founding Editor in Chief of ACM-URSI-Baltzer Wireless Networks (WINET) and ACM-Baltzer Mobile Networking and Nomadic Applications (MONET).

Dr. Chlamtac served on the editorial board of the IEEE TRANSACTIONS ON

COMMUNICATIONSand other leading journals. He served as the General Chair and Program Chair of several ACM and IEEE conferences and workshops, was a Fulbright Scholar, and was an IEEE, Northern Telecom, and BNR Distinguished Lecturer. He is the founder of ACM/IEEE MobiCom and of the ACM Sigmobile of which he is the current Chairman. He is an ACM Fellow and an Honorary Member of the Senate of the Technical University of Budapest.

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Hsinchu, Taiwan, In 1996, he was appointed as Deputy Director of the Microelectronics and Information Systems Research Center, NCTU. Since 1997, he has been elected as Chairman of CSIE and NCTU. His current research interests include design and analysis of personal communications services network, mobile computing, distributed simulation, and performance modeling. He is 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 SIMULATION magazine,

an Area Editor of ACM Mobile Computing and Communication Review, an Editor of the Journal of Information Science and Engineering, Guest Editor for the ACM/Baltzer MONET Special Issue on Personal Communications, and a Columnist of the ACM Simulation Digest. He is a Member of the editorial boards of the International Journal of Communications Systems, ACM/Baltzer

Wireless Networks, and Computer Simulation Modeling and Analysis.

Dr. Lin is an Associate Editor of the IEEE NETWORKand Guest Editor for the IEEE TRANSACTIONS ONCOMPUTERSSpecial Issue on Mobile Computing. He is the Program Chair for the 8th Workshop on Distributed and Parallel Simulation and 2nd International Mobile Computing Conference, General Chair for the 9th Workshop on Distributed and Parallel Simulation, and Publicity Chair of ACM Sigmobile.

數據

Fig. 1. The timing diagram for the checkpointing model based on the number of calls.
Fig. 2. Time diagram for checkpointing model based on real-time interval.
Fig. 3. Effects of m on Pr[k]:
Fig. 5. Effects of  on Pr[k] (n = 10):
+6

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