Performance of hot billing mobile prepaid service

Performance of hot billing mobile prepaid service

Ming-Feng Chang, Yi-Bing Lin

_{, Wei-Zu Yang}

Dept. of Computer Science and Information Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan, ROC Received 19 September 2000; accepted 10 January 2001

Responsible Editor: I.F. Akyildiz

Abstract

Prepaid service has become an important mobile application with rapid growth for subscription rate in the recent years. Implementation of prepaid service may generate large network trac that signi®cantly a€ects the performance of a mobile network. This paper studies the hot billing solution for prepaid service. We investigate how the amount of prepaid credit and the frequency of call detail record (CDR) transmissions a€ect network signaling and potential bad debt that a service provider may bear. Our study suggests that a prepaid service provider should encourage customers to buy large prepaid credits by giving them discounts. Furthermore, based on call trac, an optimal CDRtransmission frequency can be found by using our modeling technique. Ó 2001 Elsevier Science B.V. All rights reserved.

Keywords: Bad debt; Call detail record; Hot billing; Prepaid service center; Recharge

1. Introduction

Prepaid telecommunication services were o€ered in Europe and Asia in 1982 and became popular in the US in 1992 [1]. During the past few years, the mobile prepaid service has been growing exponentially all over the world. In the US, more than thirty prepaid solution vendors, such as Corsair Communications, Boston Communications Group and Vicorp, are competing for carrier business [13]. Today, more than 100 million prepaid cards have been issued [2], and revenues of more than US $650 million had been generated from the prepaid service in the US by the year 2000. In 1997, there were about 60 million GSM subscribers across the world and 8% of them subscribed to prepaid service. It is predicted that in 2001, the number of GSM subscribers will increase to 140 million and 25% of the customers will subscribe to the prepaid service [15]. Asian countries such as Philippine, Australia, Hong Kong, Singapore and Taiwan have already shown successful examples for prepaid services. For example, Islacom in Philippine launched prepaid service in November 1997 and has a comparable number of prepaid and postpaid customers now [14]. In Australia, Telstra started prepaid service with an initial capacity of 100,000 customers and has exceeded the capacity

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E-mail addresses: [email protected] (M.-F. Chang), [email protected] (Y.-B. Lin), [email protected] (W.-Z. Yang).

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in early 1999. In Taiwan, FarEastone reported that more than 47% of the customers subscribed to prepaid service in March 1999.

Several factors have contributed to the rapid growth of mobile prepaid service [2]. Firstly, the growth rate of cellular subscribers appears to decrease while the competition among carriers keeps in¯aming. The service providers are searching for new ways of increasing revenues, reducing expenses and improving the customer satisfaction. Secondly, as the customer base grows to cover customers with poor credit, providing new services that can minimize or avoid fraud usage is becoming more and more critical today. Mobile prepaid service o€ers a desirable solution that satis®es the aforementioned requirements.

In the GSM prepaid service, a customer subscribes to the GSM service with a prepaid credit. This credit is either coded into the subscriber identity module (SIM) card or kept in the network [1]. In many areas, the initialization of prepaid services must be completed within a certain number of days after subscription. In Taiwan, prepaid service is available immediately after purchasing the service. Whenever the customer originates a prepaid call, the corresponding payment is decremented from the prepaid credit. Status report of the credit balance can be obtained from the SIM card or the network.

If the balance is depleted, the customer cannot originate calls, but may be allowed to receive phone calls for a period (e.g., 6 months). To recover the prepaid service, the balance needs to be recharged by pur-chasing a top-up card. The top-up card is like a lottery scratch card. When the seal is scratched o€, a secret code appears. The customer dials a toll-free number and follows the instructions of an interactive voice response (IVR) to input the Mobile Station ISDN Number (MSISDN, i.e., the GSM phone number) and the secret code. The system will verify and refresh the account if it is a valid code. On the other hand, if the prepaid balance is not depleted at the end of a valid period, the balance is automatically reset to zero. After a certain amount of time, the unused prepaid credit may be considered abandoned and becomes the government's property.

Four solutions have been proposed to implement the prepaid services: hot billing approach, service node approach, intelligent network approach and based approach [4]. The hot billing and the handset-based approaches provide solutions without major changes to the network infrastructure. Intelligent net-work solution o€ers real-time rating and real-time call control, but is not widely deployed today. The service node approach, which utilizes extra voice circuits and switching resources for prepaid calls, provides a variant to the intelligent network solution.

This paper studies the hot billing approach. The other three approaches are out of the scope of this paper. Details of these approaches can be found in [13]. We ®rst describe the hot billing approach. Then, we inves-tigate the performance of this approach by both analytical and simulation models. We assume that the reader is familiar with the GSM terms such as SIM, mobile switching center (MSC), home location register (HLR), authentication center (AuC), MSISDN and international mobile subscriber identity (IMSI). Details of these terms can be found in [6,9,11]. For the reader's bene®t, Appendix A lists the notations used in this paper. 2. Hot billing solutions

Hot billing uses the call detail records (CDRs) produced by the wireless switch (i.e., MSC) to process the prepaid usage. The information in a CDRincludes the type of service, date/time of usage, user identi®-cation and loidenti®-cation information [8]. When calls are completed, the CDRs are generated and transported from the MSC to the prepaid service center. The balance of the customer's account is decremented ac-cording to the CDRs. As a customer uses up the prepaid credit, the HLR and the AuC are noti®ed to prevent further service access and the prepaid service center instructs the network to route the next prepaid call attempt to an IVRto play an announcement indicating that the balance has been depleted. The IVR can also communicate with the customer to replenish the prepaid credit by using a top-up card, a credit/ debit card or credit transfer from the bank account.

The architecture of the hot billing approach is depicted in Fig. 1. The CDRfor a prepaid call is created in the MSC based on the destination of the call and the connection time for the call. This call record can be sent from the MSC to the prepaid service center by using protocols such as common management infor-mation service element (CMISE) [5]. The same protocol can be used for communication between the prepaid service center and the HLR. The HLR communicates with the MSC by invoking GSM MAP service primitives [6]. The IVRgenerates automatic messages that allow the customer accounts to be queried and reloaded. The voice trunks between the IVRand the MSC are set up by SS7 ISUP (ISDN User Part) messages [3].

Fig. 2 illustrates the initialization of prepaid service. This procedure is described in the following steps: Step 1. The customer subscribes to the prepaid service. The prepaid service center creates a subscriber data record including IMSI, MSISDN, prepaid credit and other authentication-related information. Step 2. The prepaid service center sends the customer data to the HLRand activates the prepaid service. The prepaid call origination procedure is illustrated in Fig. 3 with the following steps:

Step 1. When a customer originates a prepaid call, the IMSI is sent to the MSC. Step 2. The MSC instructs the HLRto check if it is a valid service.

Fig. 1. Hot billing architecture for prepaid service.

Fig. 2. Initialization of prepaid service.

Step 3. If the veri®cation is successful, the HLRdownloads the customer data and a prepaid tag to the MSC.

Step 4. When the call terminates, a billing record is created and sent to the prepaid service center. Step 5. The prepaid service center decrements the prepaid credit based on the received billing record. Step 6. If the balance is negative, the prepaid service center instructs the HLRto suspend the prepaid service or to delete the customer record.

A customer can query his/her current credit through the following steps (see Fig. 4): Step 1. The customer makes a service query call that is typically free of charge.

Step 2. The MSC sends the request together with the MSISDN of the customer to the IVRand sets up a voice path to the IVR.

Steps 3 and 4. The IVRqueries the prepaid service center for the balance information. Steps 5 and 6. The IVRplays an announcement to answer the customer.

When the prepaid credit has been decremented below a threshold, the prepaid service center automat-ically calls the customer and plays a warning message that reminds the customer to recharge the prepaid credit. The customer may recharge the prepaid credit using the top-up card described in Section 1. This recharging procedure is similar to the credit query procedure illustrated in Fig. 4.

If the prepaid credit depletes during a phone call, the credit becomes negative at the end of the phone call. The negative credit is potential bad debt. If the customer does not recharge the credit, this negative credit becomes a real bad debt of the service provider. To avoid bad debt, some approaches (such as service node) decrement the prepaid credit by seconds during a phone call. However, in the hot billing approach, sending these ``real time'' CDRs to the prepaid service center and processing these CDRs at the center may incur heavy overheads to the network. Thus, the CDRs are delivered and processed on a per-call basis and in some cases, on a multiple-call basis. In the hot billing approach, it is important to select the CDRsending frequency such that the sum of the CDRsending/processing cost and the bad debt is minimized. This paper utilizes analytical and simulation models to investigate the performance of hot billing. Our study provides the guidelines to select the CDRsending frequency and the amount for the initial prepaid credit.

3. The analytical model

In this section, we propose an analytical model to study the performance of the hot billing approach. In our model, a CDRis sent from the MSC to the prepaid service center for every m complete prepaid call, where m P 1. The prepaid service center decrements the customer's credit according to the CDRs received. When a customer's credit becomes negative, the customer is not allowed to make a phone call. We will estimate the number of CDRs transmitted and the amount of potential bad debt.

Let B be the prepaid credit and K be the total number of CDRs sent to the prepaid service center when the prepaid service ends. Let xi(i ˆ 1; 2; . . .) be the charge indicated in the ith CDR. If a CDR is sent per m

call completion, then xirepresents the net charge for the ‰m…i 1† ‡ 1Šst call to the (m  i)th call. We

as-sume that xiis a random variable with the density function f …xi† and the expected value E‰xiŠ ˆ m=c (i.e., the

expected charge for a call is 1=c). Let BL be the amount of potential bad debt. That is, BL ˆPKiˆ1xi B.

The amount BL becomes a real bad debt if the customer does not recharge the prepaid credit. Fig. 5

illustrates the charges for the prepaid calls. The horizontal line is the ``prepaid credit line'' that illustrates the decrement of the prepaid credit due to the CDRtransmission events (the vertical arrows).

The theory of a renewal process [12] can be applied to evaluate the hot billing approach. A renewal process is a counting process for which the interarrival times of events are independent and identically distributed. The transmission of CDRs in the hot billing approach is a renewal process, since the inter-arrival times between two CDRtransmissions are independent and identically distributed. Hence, the number of CDRtransmissions, the expected value and the second moment of potential bad debt can be estimated by the approximate solutions of a renewal process. For the large prepaid credit (i.e., B is su-ciently large), E‰KŠ can be approximated as

E‰KŠ  B E‰xiŠ‡ E‰x2 iŠ 2…E‰xiŠ†2 : …1†

The mean of BL can be approximated as

E‰BLŠ  E‰x 2 iŠ

2E‰xiŠ: …2†

The second moment of BL can be approximated as

E‰B2 LŠ  E‰x

3 iŠ

3E‰xiŠ: …3†

The approximate solutions suggest that E‰BLŠ and E‰B2LŠ be independent of the prepaid credit as long as

the credit is large enough. Later in this paper, we will derive an exact solution for the expected potential bad debt. The solution will show that the prepaid credit may a€ect the expected value and variance of the potential bad debt when the prepaid credit is small.

In PCS services, the call holding times are usually assumed to be exponentially or Erlang distributed [7,9,10]. Since a CDRis sent per m call completions and the charge of a call is proportional to its call holding time, we assume that xi has an Erlang distribution with mean E‰xiŠ ˆ m=c and variance

Var‰xiŠ ˆ m=…lc2† (i.e., the shape parameter and scale parameter of xiare lm and lc, respectively). From (1),

E‰KŠ is approximated as E‰KŠ 2lcB ‡ lm ‡ 1_2lm : From (2), E‰BLŠ is approximated as

E‰BLŠ lm ‡ 1_2lc : …4† From (3), E‰B2 LŠ can be approximated as E‰B2 LŠ  lm…lm ‡ 1†…lm ‡ 2† …lc†3 " #, 3m c   ˆ…lm ‡ 1†…lm ‡ 2† 3l2_c2 : …5†

From (4) and (5), the variance of the potential bad debt Var‰BLŠ can be approximated as

Var‰BLŠ …lm ‡ 1†…lm ‡ 2†_3l2c2 …lm ‡ 1† 2

4l2_c2 ˆ

…lm ‡ 1†…lm ‡ 5† 12l2_c2 :

In the following subsections, we consider two cases for prepaid credit B. In case I, a customer is given a small prepaid credit and the customer does not recharge after the credit depletes. In case II, B is a random variable with an arbitrary distribution. Case II represents a customer who may recharge the prepaid card when the credit runs out.

3.1. Case I: small prepaid credit

Small prepaid credit may be provided to promote the prepaid service. In this case, the prepaid credit is a constant. Let yn be the accumulated charge of the calls for the ®rst n CDRs. That is, ynˆPniˆ1xi. Let

Fn…y† ˆ Prfyn< yg be the distribution function for yn. Let N…y† ˆ maxfn j yn< yg. Then, N…y† represents

the number of CDRs transmitted in …0; yŠ and is a renewal process. It is apparent that K ˆ N…B† ‡ 1. Let PrfN…B† ˆ ng be the probability that n CDRs have been sent to the prepaid service center just before the credit runs out. Then, the expected value E‰KŠ is derived as

E‰KŠ ˆX1 nˆ0 n PrfN…B† ˆ ng ‡ 1 ˆX1 nˆ0 n F‰ n…B† Fn‡1…B†Š ‡ 1 ˆ X1 nˆ1 Fn…B† ‡ 1: …6†

We assume that the call charges are Erlang distributed with mean E‰xiŠ ˆ m=c and variance

Var‰xiŠ ˆ m=…lc2†. From (6), E‰KŠ can be expressed as

E‰KŠ ˆX1 nˆ1 1 " X lmn 1 jˆ0 …lcB†je lcB j! # ‡ 1 ˆ e lcB X1 nˆ1 X1 jˆlmn …lcB†j j! " # ‡ 1 ˆ e z_G m…z† ‡ 1; …7† where z ˆ lcB and Gm…z† ˆP1nˆ1 P₁

jˆlmnzj=j!. From (7), a di€erential equation can be obtained as follows.

Gm…z† ‡dG_dzm…z†‡d 2_G m…z† dz2 ‡


‡ dlm 1_G m…z† dzlm 1 ˆ zez: …8†

Using the Laplace transform and following the process described in Appendix B, E‰KŠ can be expressed as E‰KŠ ˆ 1 2 4 8 < : X lm 1 iˆ1 …xi " 8 < : 1†2 Y 1 6 j 6 lm 1;j6ˆi …xi _xj_† # 1 _Y 1 6 j 6 lm 1;j6ˆi …2 xj_† 9 = ; 3 5 , Y 1 6 j 6 lm 1 …2 xj_† 9 = ; ‡_{1 x†…1 x}elcB<sub>2†


…1 x</sub>1 _{lm 1†}‡Xlm 1 iˆ1 …xi " 8 < : 1†2 Y 1 6 j 6 lm 1;j6ˆi …xi _xj_† # ₁ elcB…xi _1† 9 = ; ‡ 1; …9†

where x is the principle …l  m†th root of 1 (i.e., x ˆ cos…2p†=…lm† ‡ i sin…2p†=…lm†). Based on Wald's equation [12], the expected value E‰BLŠ can be expressed as

E‰BLŠ ˆ E XK iˆ1 xi B ! ˆ E‰KŠE‰xiŠ B: …10†

From (C.17) in Appendix C, E‰B2

LŠ is expressed as E‰B2 LŠ ˆ X1 nˆ0 X lm 1 jˆ0 ‰…j ‡ 2†…j ‡ 1†Š …lc†_{lmn ‡ j ‡ 2†!}lmn‡jBlmn‡j‡2e lcB " # 2lmX1 nˆ0 Xlm jˆ0 …j ‡ 1† …lc†lmn‡j 1_{lmn ‡ j ‡ 1†!}Blmn‡j‡1e lcB " # ‡X1 nˆ0 X lm‡1 jˆ0 lm…lm ‡ 1† …lmn ‡ j†!   …lc†lmn‡j 2Blmn‡j_e lcB h i : …11†

From (10) and (11), Var‰BLŠ can be obtained.

3.2. Case II: recharged credit

In the recharged credit case, a customer may recharge his/her prepaid card before the credit runs out. Assume that B has a density function h…b† with the Laplace transform h_{s†. As described in the previous}

section, we assume that the charge xi …i ˆ 1; 2; . . .† indicated in the ith CDRhas an Erlang distribution with

mean E‰xiŠ ˆ m=c and variance Var‰xiŠ ˆ m=…lc2†. The probability that n CDRs have been sent to the

prepaid service center before total credit runs out is expressed as Prfyn< b < yn‡ xn‡1g ˆ Z ₁ bˆ0 Z _b ynˆ0 Z ₁ xn‡1ˆb yn h…b† …lc†lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " # dxn‡1dyndb ˆlm 1X jˆ0 Z ₁ bˆ0h…b†b lmn‡j …lc†lmn‡j …lmn 1†!j!e lcb " # Xj iˆ0 … " ( 1†i‡j j_i   1 lmn ‡ j i # db ) : …12†

From (C.5) in Appendix B, we have Xj iˆ0 … 1†i‡j j_i   1 lmn ‡ j i   ˆ…lmn 1†!j!_{lmn ‡ j†!} : …13† Substituting (13) into (12), we have

Prfyn< b < yn‡ xn‡1g ˆ X lm 1 jˆ0 …lc†lmn‡j …lmn ‡ j†! " _{# Z} 1 bˆ0h…b†b lmn‡j_e lcb_db   ˆlm 1X jˆ0 … lc†lmn‡j …lmn ‡ j†! " # d…lmn‡j†_h_s† ds…lmn‡j† " _#   sˆlc 8 < : 9 = ;: …14†

If B has a Gamma distribution with the shape parameter a and the scale parameter b (i.e., the mean of the distribution is E‰BŠ ˆ a=b and the variance is Var‰BŠ ˆ a=b2_{), then the Laplace transform of B is}

h_{s† ˆ} b

s ‡ b  _a

The …lmn ‡ j†th order derivative for h_{s† is} h…lmn‡j†_{s† ˆ} … 1†lmn‡j…a ‡ lmn ‡ j 1†! …a 1†! " # ba …s ‡ b†a‡lmn‡j " # : …15†

Substituting (15) into (14), we have Prfyn< b < yn‡ xn‡1g ˆ X lm 1 jˆ0 a ‡ lmn ‡ j 1 lmn ‡ j   …lc†lmn‡j_ba …lc ‡ b†lmn‡j‡a " # ( ) : From (6), the expected value E‰KŠ can be expressed as

E‰KŠ ˆX1 nˆ0 …n ‡ 1† lm 1X jˆ0 a ‡ lmn ‡ j 1 lmn ‡ j   …lc†lmn‡jba …lc ‡ b†lmn‡j‡a " # ( ) : …16†

From (10), E‰BLŠ can be expressed as

E‰BLŠ ˆ E XK iˆ1 xi " # E‰BŠ ˆ m c   X1 nˆ0 …n ( ‡ 1†lm 1X jˆ0 a ‡ lmn ‡ j 1 lmn ‡ j   …lc†lmn‡j_ba …lc ‡ b†lmn‡j‡a " #) a b: …17†

From Appendix D, E‰B2

LŠ can be expressed as E‰B2 LŠ ˆ X1 nˆ0 X lm 1 jˆ0 …j ( ‡ 1†…j ‡ 2† …lc†lmn‡jba …lc ‡ b†a‡lmn‡j‡2 " # C…a ‡ lmn ‡ j ‡ 2† C…a†…lmn ‡ j ‡ 2†!  ) 2lmX1 nˆ0 Xlm jˆ0 …j ( ‡ 1† …lc†lmn‡j 1ba …lc ‡ b†a‡lmn‡j‡1 " # C…a ‡ lmn ‡ j ‡ 1† C…a†…lmn ‡ j ‡ 1†!  ) ‡ lm…lm‰ ‡ 1†ŠX1 nˆ0 X lm‡1 jˆ0 ba_lc†lmn‡j 2 …b ‡ lc†a‡lmn‡j " # C…a ‡ lmn ‡ j† C…a†…lmn ‡ j†!   ( ) : …18†

The variance Var‰BLŠ of the potential bad debt can be derived from (17) and (18). Di€erent distributions

(e.g., shift-Gamma) for the recharged credit can be derived with the similar approach. However, it will not be included in this paper.

4. Numerical examples

This section investigates the performance of the hot billing approach based on the analytical model developed in the previous section. Computer simulations have been conducted to validate the analytical results. Each simulation experiment was repeated 500,000 times to ensure stable results.

Tables 1±3 compare the results of simulation, exact solution and approximation for the large, small prepaid credit and recharged credit cases. To re¯ect the situation of prepaid service in Taiwan, the expected charge of a call is assumed to be NT$36 dollars. The expected prepaid credit B considered in large and recharged credit cases is NT$100, NT$300, NT$400 and NT$500 dollars, respectively. For the small pre-paid credit case, B ˆ NT$6, NT$12 and NT$18. The tables indicate that the exact solutions are consistent

with the simulation results in all cases, while the approximate solution is good only in large prepaid credit case. Note that the discrepancy between the exact and approximate solution can be over 20% when the prepaid credit is small.

4.1. E€ects of the variance Var‰xiŠ of CDR charges when B is large

Figs. 6 and 7 depict the e€ects of the variance Var‰xiŠ of CDRcharges on Var‰BLŠ and E‰BLŠ=E‰BŠ for

both large credit and recharged credit cases. In these two ®gures, a CDRrecord is sent to the prepaid service center for every call completion (i.e., m ˆ 1) and the charge for a call has an Erlang distribution with shape parameter l and mean E‰xiŠ ˆ 1=c ˆ NT$36. In the recharged credit case, we consider Gamma

prepaid credit distribution with the variances Var‰BŠ ˆ 104_{, 9  10}4_{, 16  10}4 _{and 25  10}4 _{(i.e., the scale}

parameter b is 1=100, 1=300, 1=400 and 1=500), respectively. Both ®gures show that E‰BLŠ and Var‰BLŠ

increase as Var‰xiŠ increases. Table 1

Comparison of simulation, exact solution and approximation for large prepaid credit (m ˆ 1, E‰xiŠ ˆ NT$36 and Var‰xiŠ ˆ 432)

B Simulation Exact solution Approximation

E‰KŠ 100 3.444 3.444 3.444 300 9.000 9.000 9.000 400 11.774 11.778 11.778 500 14.551 14.556 14.556 E‰BLŠ 100 24.028 24.000 24.000 300 23.955 24.000 24.000 400 23.949 24.000 24.000 500 24.005 24.000 24.000 Var‰BLŠ 100 385.321 384.000 384.000 300 382.482 384.000 384.000 400 382.443 384.000 384.000 500 386.892 384.000 384.000 Table 2

Comparison of simulation, exact solution and approximation for small prepaid credit (m ˆ 1, E‰xiŠ ˆ NT$36 and Var‰xiŠ ˆ 432)

B Simulation Exact solution Approximation

E‰KŠ 6 1.014 1.014 0.500 12 1.080 1.081 0.667 18 1.194 1.196 0.833 E‰BLŠ 6 30.536 30.519 24.000 12 26.897 26.912 24.000 18 24.993 25.043 24.000 Var‰BLŠ 6 423.431 423.833 384.000 12 403.510 403.274 384.000 18 389.654 388.344 384.000

The phenomenon that E‰BLŠ increases as Var‰xiŠ increases is explained as follows. In the prepaid credit

line of Fig. 5, the value 0 can be treated as an observer of the periods xi. From an argument similar to the

one for excess life theorem [12], longer xi's are more likely to be observed by point ``0'' in the credit line.

When Var‰xiŠ is large, there are more large xi's and small xi's. Thus, the value 0 is more likely to observe

longer xias Var‰xiŠ increases. We conclude that if the call pattern of a prepaid customer is very irregular, it is

more likely that the service provider will lose revenue due to bad debt.

Experience indicates that prepaid customers tend to purchase small prepaid credits to make the credit control easier and more economical. On the other hand, Figs. 6(b) and 7(b) indicate that large prepaid credit helps the service provider to reduce the potential bad debt per prepaid dollar (i.e., E‰BLŠ=E‰BŠ). It also

reduces the cost for distributing the prepaid cards and allows collecting the capitals quickly. Thus, based on the above analysis, a prepaid service provider should encourage the customers to buy large prepaid credits by giving them discounts.

Table 3

Comparison of simulation, exact solution and approximation for recharged credit (m ˆ 1, a ˆ 1=2, E‰xiŠ ˆ NT$36 and Var‰xiŠ ˆ 432)

B Simulation Exact solution Approximation

E‰KŠ 100 3.519 3.512 3.444 300 9.041 9.040 9.000 400 11.766 11.812 11.778 500 14.640 14.586 14.556 E‰BLŠ 100 26.456 26.449 24.000 300 25.440 25.430 24.000 400 25.264 25.240 24.000 500 25.157 25.110 24.000 Var‰BLŠ 100 410.231 412.088 384.000 300 401.863 401.800 384.000 400 399.029 399.666 384.000 500 398.816 398.165 384.000

Fig. 6. E€ect of the variance of call charges for large credit case (m ˆ 1, E‰xiŠ ˆ NT$36): (a) variance of the potential bad debt;

4.2. E€ects of the variance Var‰xiŠ of CDR charges when B is small

Fig. 8 shows the e€ects of the variance Var‰xiŠ of CDRcharges on E‰BLŠ when B is small. The mean

charge of a CDR E‰xiŠ ˆ NT$36 and the variance Var‰xiŠ is varied as 648.0, 324.0 and 216.0. We assume

that the CDRis sent whenever a call completes (i.e., m ˆ 1). The small prepaid credit considered in our case ranges from NT$6 to NT$72. The results show that when B is small, most of the customers can only originate one call and the bad debt E‰BLŠ  E‰xiŠ B. As the prepaid credit increases, the potential bad

decreases and converges to the approximate value derived from the renewal theory when B is larger than E‰xiŠ. It is interesting to note that when Var‰xiŠ is small (i.e., the call pattern is regular), there exists an

optimal prepaid credit such that the expected bad debt is minimal. 4.3. E€ects of the prepaid credit variance

Fig. 9 shows the e€ects of the prepaid credit variance Var‰BŠ on E‰BLŠ and Var‰BLŠ. We consider the case

where every CDRis sent per single call completion (i.e., m ˆ 1). We only present the results for the cases where the charge xi for a CDRhas an Erlang distribution with mean E‰xiŠ ˆ NT$36 and the variance

Var‰xiŠ ˆ 648 (i.e., l ˆ 2). Note that similar conclusions can be drawn for xi with various variances. We

Fig. 8. E€ect of the variance of call charges for small prepaid credit case (m ˆ 1; E‰xiŠ ˆ NT$36).

Fig. 7. E€ect of the variance of call charges for the recharged credit case (m ˆ 1, E‰xiŠ ˆ NT$36): (a) variance of the potential bad debt;

assume that the prepaid credit B consists of two parts: the initial credit and the net recharged credit. At the beginning, the customer purchases an initial credit BI and then recharges the credit several times. The net

recharged credit is the sum of all recharged credits. We consider two cases where E‰BŠ ˆ NT$300. In the ®rst case, the prepaid credit B has a shifted Gamma distribution. In this case, the net recharged credit is generated from a Gamma random number generator and the initial credit BIis varied as NT$0, NT$10,

NT$20, NT$36 and NT$50. In the second case, the net recharged credit has a geometric distribution. We assume that every time the customer recharges the prepaid credit with probability p, and the amount of a single recharged credit is Br. Thus,

E‰BŠ ˆ BI‡ _{1 p}p   Br and Var‰BŠ ˆ p …1 p†2 " # B2 r:

We illustrate two sets of input parameters for the geometric prepaid credit distribution: (BIˆ NT$50,

Brˆ NT$50, p ˆ 5=6) and (BIˆ NT$50, Brˆ NT$125, p ˆ 2=3). Fig. 9 shows that for BIP NT$36, Var‰BŠ

has insigni®cant e€ect on E‰BLŠ and Var‰BLŠ in all cases. The ®gure also indicates that Var‰BŠ has e€ect on

BL when BI< NT$36. This phenomenon is referred to as the ``small BIe€ect''. We conclude that for every

prepaid credit value, the customer's recharging behavior does not a€ect the amount of the potential bad debt if the initial credit is larger than the cost for a single call. For prepaid service planning, a service provider can ignore the recharging behavior of customers.

4.4. E€ects of multiple CDR transmissions

Fig. 10 shows the e€ect of multiple CDRtransmissions on E‰KŠ and E‰BLŠ in the recharged credit cases

where the charge for one call is exponentially distributed with mean NT$36 and the expected value for the prepaid credits E‰BŠ is varied as NT$100, NT$300, NT$400 and NT$500. Similar results are observed for large credit case and are not presented here. In the recharged credit case, the prepaid credits have Gamma density functions with variances Var‰BŠ ˆ 104_{, 9  10}4_{, 16  10}4_{and 25  10}4_{(i.e., b ˆ 1=100, 1=300, 1=400}

and 1=500), respectively. The ®gure shows the intuitive results that E‰KŠ decreases as m increases, while E‰BLŠ increases as m increases. Fig. 10(b) shows that E‰BLŠ increases as E‰BŠ decreases. The e€ect becomes

signi®cant when m is large. This phenomenon is similar to the small BIe€ect observed in Fig. 9. When m

increases, the ratio of prepaid credit to the charge indicated in a CDR(E‰BŠ=E‰xiŠ) becomes smaller, which

ampli®es the small BI e€ect.

Two costs are associated with the prepaid service: the CDRtransmission cost and the bad debt. Fig. 10 indicates that the potential bad debt can be reduced by increasing the CDRtransmission frequency. In other words, CDRtransmission frequency and bad debt are con¯icting factors. Consider a cost function

C ˆ E‰KŠ ‡ /E‰BLŠ where / is a weighted cost that normalizes the bad debt with respect to the CDR

transmission cost. The cost C provides the net e€ect of CDRtransmissions and bad debt. The / value is determined by two factors: the signaling cost Cs for a CDRtransmission and the probability Pd that the

potential bad debt becomes real bad debt. That is, / ˆ Pd=Cs. According to OFTEL (cost analysis

docu-ments of OFTEL can be found in http://www.oftel.gov.uk/numbers/number.htm), the signaling cost is Csˆ 0:05 pence  NT$0:025. If 0:1% < Pd< 1%, then / ranges from 0.04 to 0.4. Fig. 11 plots C as a

function of / and m with l ˆ 1 and E‰xiŠ ˆ NT$36 in large credit case. In this ®gure, the bullets in the

curves represent the cost for the optimal m values. Consider the case when B ˆ NT$500. For / ˆ 0:25, the potential bad debt costs are high and m ˆ 2 should be selected. For / ˆ 0:05, the CDRtransmission costs are high and m ˆ 4 should be selected. If / > 1, m ˆ 1 in all cases studied in this paper. Also, for the same / value, the optimal m values increase as B increases. Therefore, if the prepaid credit is large (e.g., B P NT$500), a CDRshould be transmitted after multiple call completions. Although the above result is intuitive, our analysis quantitatively computes the prepaid service overheads to select the optimal m values for speci®c network setup.

Fig. 11. The cost for large credit case (l ˆ 1; E‰xiŠ ˆ NT$36). (a) B ˆ NT$100; (b) B ˆ NT$500.

Fig. 10. E€ect of m for the recharged credit case (l ˆ 1, E‰xiŠ ˆ NT$36): (a) number of CDRtransmissions; (b) expected potential bad

5. Conclusions

This paper studied the hot billing solution for prepaid service. We described the system architecture and the procedures for prepaid service initialization, call origination and credit recharging. An analytical model was proposed to analyze the performance in the large, small prepaid credit and the recharged credit cases. The analytical results were validated by simulation experiments. We observed the following results: · If the call pattern of a prepaid customer is very irregular, it is more likely that the service provider will

lose revenue due to bad debt.

· Large prepaid credit (i.e., the customer either purchases large initial credit or recharges the prepaid credit several times) helps the service provider to reduce the expected and the variance of the potential bad debt. Thus, the service provider may encourage the customers to buy large initial prepaid credit by giving them discounts.

· When the prepaid credit is small, the expected potential bad debt is a€ected by the amount of prepaid credit. If the call pattern is regular, there exists an optimal prepaid credit such that the expected bad debt is minimal.

· If the initial credit is larger than the cost for a single call, the customer's recharging behavior does not a€ect the amount of the potential bad debt.

· A cost function was used to determine the minimal cost for prepaid service. The minimal cost can be achieved by properly setting the CDRtransmission frequency to balance the network trac with the bad debt overhead. This optimal CDRtransmission frequency can be determined by using our modeling technique.

Acknowledgements

Lin's work was sponsored in part by the MOE Program of Excellence Research under contract 89-E-FA04-4, CCL/ITRI, FarEastone, National Science Council under contract NSC 89-2213-E-009-203, the Lee and MTI Center for Networking Research, NCTU. Chang's work was sponsored in part by MOE Program of Excellence Research under contract 89-E-FA04-4, and National Science Council under con-tract NSC 89-2213-E-009-201.

Appendix A. Notations

The notations used in this paper are listed as below.

a the shape parameter of the prepaid credit in the recharged credit case B the amount of the prepaid credit

BI the initial credit of a prepaid credit

BL the amount of potential bad debt

Br the amount of a single recharged credit when the prepaid credit is geometric distributed

b the scale parameter of the prepaid credit in the recharged credit case C the net cost of the CDRtransmissions and the bad debt

Cs the signaling cost for a CDRtransmission

Fn…
† the distribution function of yn

f …
† the density function of xi

E‰xiŠ ˆ 1=c the expected charge of a call

h_{s† the Laplace transform of the prepaid credit distribution}

K the total number of CDRs sent to the prepaid center when the prepaid service ends m the number of complete prepaid calls accumulated in a CDR

lm the shape parameter of xi

N…y† the largest n such that yn< y

Pd the probability that the potential bad debt becomes real bad debt

p the probability that a prepaid customer will recharge his/her credit

/ the weighted cost that normalizes the bad debt with respect to the CDRtransmissions xi the charge indicated in the ith CDR

yn the accumulated charge of the calls in the ®rst n CDRs

Appendix B. Deriving E‰KŠ for the small prepaid credit case

This appendix derives E‰KŠ for the small prepaid credit case. Let G

m…s† be the Laplace transform of

Gm…z†. Then, (8) is re-written as G m…s† ‡ sGm…s† Gm…0† ‡ s2Gm…s† sGm…0† dG_dzm…z†   zˆ0‡


‡ s lm 1_G m…s† slm 2Gm…0†


dlm 2_{dzlm 2}Gm…z† zˆ0ˆ 1 …s 1†2: …B:1† Note that Gm…0† ˆ dG_dzm…z†   zˆ0 ˆ


ˆ dlm 2_{dzlm 2}Gm…z† zˆ0 ˆ 0: Thus, (B.1) can be re-written as

G m…s† ˆ _{s 1†}2_{1 ‡ s ‡ s}1<sub>2‡


‡ slm 1†</sub> …B:2† ˆ A1 s 1‡ A2s ‡ A3 …s 1†2 ‡ C1 s x‡ C2 s x2‡


‡ Clm 1 s xlm 1 …B:3† ˆ A_{s 1}1‡ A2‡A2‡ A3 …s 1†2‡ C1 s x‡ C2 s x2‡


‡ Clm 1 s xlm 1;

where x is the principle …l  m†th root of 1 (i.e., x ˆ cos…2p†=…lm† ‡ i sin…2p†=…lm†). Hence, Gm…z† can be

expressed as

Gm…z† ˆ …A1‡ A2†ez‡ …A2‡ A3†zez‡ C1exz‡ C2e…x2z†‡


‡ Clm 1e…xlm 1z†: …B:4†

From (B.2) and (B.3), we have

1 ˆ A1…s 1†…s x†…s x2†


…s xlm 1† ‡ …A2s ‡ A3†…s x†…s x2†


…s xlm 1†

‡ C1…s 1†2…s x2†…s x3†


…s xlm 1† ‡ C2…s 1†2…s x†…s x3†


…s xlm 1†

‡


‡ Clm 1…s 1†2…s x†…s x2†


…s xlm 2†: …B:5†

By letting s ˆ xi _{i ˆ 1; 2; . . . ; lm 1† in (B.5), we have}

Ciˆ …xi " 1†2 Y 1 6 j 6 lm 1;j6ˆi …xi _xj_† # 1 : …B:6†

With s ˆ 1 in (B.5), we have A2‡ A3ˆ_{1 x†…1 x2}1<sub>†


…1 xlm 1†</sub>: …B:7† With s ˆ 2 in (B.5), we have A1‡ 2A2‡ A3ˆ 1 Plm 1_iˆ1 CiQ1 6 j 6 lm 1;j6ˆi…2 xj† h i Q 1 6 j 6 lm 1…2 xj† : …B:8†

From (B.7) and (B.8), we have A1‡ A2ˆ 1 Plm 1_iˆ1 CiQ1 6 j 6 lm 1;j6ˆi…2 xj† h i Q 1 6 j 6 lm 1…2 xj† 1 …1 x†…1 x2<sub>†


…1 x</sub>lm 1_†: …B:9†

From (7), (B.4), (B.6), (B.7) and (B.9), E‰KŠ is expressed as (9) in Section 3.1. Appendix C. Deriving E‰B2

LŠ for the small prepaid credit case

This appendix derives E‰B2

LŠ for the small prepaid credit case. From the assumptions in Section 3.1,

E‰B2 LŠ ˆ X1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn …xn‡1 ( ‡ yn B†2 …lc† lmn …lmn 1†!ynlmn 1e lcyn " #  _{lm 1†!}…lc†lm xlm 1 n‡1 e lcxn‡1 " #) dxn‡1dyn ˆX1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn …x2 n‡1 ( ‡ y2 n‡ B2 2Bxn‡1 2Byn‡ 2xn‡1yn†  …lc†lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyn ˆ D1‡ D2‡ D3‡ D4‡ D5‡ D6; …C:1†

where the ®rst item of the right-hand side in (C.1) is D1ˆ X1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn x2 n‡1 …lc† lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " # ( ) dxn‡1dyn ˆX1 nˆ0 X lm‡1 jˆ0 …lc†lmn‡j 2 …lmn 1†!j! " # lm…lm  ( ‡ 1†Blmn‡j_e lcB Xj iˆ0 … " 1†i‡j j i   1 lmn ‡ j i  #) : …C:2† The third item of the right-hand side in (C.2) can be re-written as

Xj iˆ0 … 1†i‡j j_i   1 lmn ‡ j i   ˆ _lmn1  _{ X}j iˆ0 … 1†i j_i   lmn lmn ‡ i   : …C:3†

For the derivation purpose, we de®ne X…h; j† as X…h; j† ˆXj iˆ0 … 1†i j i   h h ‡ i   ˆXj iˆ0 … 1†i j 1 i    ‡ j 1_{i 1}   h h ‡ i   ˆ X…h; j 1† _{h ‡ 1}h   X…h ‡ 1; j 1† ˆ h ‡ j_j   1 : …C:4†

From (C.4) and let h ˆ lmn, (C.3) is re-written as Xj iˆ0 … 1†i‡j j_i   1 lmn ‡ j i   ˆ…lmn 1†!j!_{lmn ‡ j†!} : …C:5† Substituting (C.5) into (C.2), we have

D1ˆ X1 nˆ0 X lm‡1 jˆ0 lm…lm ‡ 1† …lmn ‡ j†!   …lc†lmn‡j 2Blmn‡j_e lcB_{: …C:6†}

The second item of the right-hand side in (C.1) is D2ˆ X1 nˆ0 Z B ynˆ0 Z 1 xn‡1ˆB yn y2 n …lc† lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " # ( ) dxn‡1dyn ˆX1 nˆ0 X lm 1 jˆ0 …lc†lmn‡j …lmn 1†!j!Blmn‡j‡2e lcB " # 1 lmn ‡ 2   X…lmn  ‡ 2; j†  : …C:7† From (C.4), (C.7) is re-written as D2ˆ X1 nˆ0 X lm 1 jˆ0 lmn…lmn ‡ 1† …lmn ‡ j ‡ 2†!   …lc†lmn‡jBlmn‡j‡2_e lcB_{: …C:8†}

The third item of the right-hand side in (C.1) is D3ˆ X1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn B2 …lc†lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " # ( ) dxn‡1dyn ˆ B2X1 nˆ0 X lm 1 jˆ0 …lc†lmn‡j …lmn 1†!j!Blmn‡je lcB " # 1 lmn   X…lmn; j†   : …C:9† From (C.4), (C.9) is re-written as D3ˆ X1 nˆ0 X lm 1 jˆ0 1 …lmn ‡ j†!   …lc†lmn‡j_Blmn‡j‡2_e lcB_{: …C:10†}

The fourth item of the right-hand side in (C.1) is D4ˆ 2B X1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn …lc†lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlmn‡1e lcxn‡1 " # ( ) dxn‡1dyn ˆ 2lmX1 nˆ0 Xlm jˆ0 …lc†lmn‡j 1 …lmn 1†!j!Blmn‡j‡1e lcB " # 1 lmn   X…lmn; j†   : …C:11†

From (C.4), (C.11) is re-written as D4ˆ 2lm X1 nˆ0 Xlm jˆ0 1 …lmn ‡ j†!   …lc†lmn‡j 1Blmn‡j‡1_e lcB_{: …C:12†}

The ®fth item of the right-hand side in (C.1) is D5ˆ 2B X1 nˆ0 Z B ynˆ0 Z 1 xn‡1ˆB yn …lc†lmn …lmn 1†!ynlmne lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " # ( ) dxn‡1dyn ˆ 2X1 nˆ0 X lm 1 jˆ0 …lc†lmn‡j …lmn 1†!j!Blmn‡j‡2e lcB " # 1 lmn ‡ 1   X…lmn  ‡ 1; j†  : …C:13† From (C.4), (C.13) is re-written as D5ˆ 2 X1 nˆ0 X lm 1 jˆ0 lmn …lmn ‡ j ‡ 1†!   …lc†lmn‡j_Blmn‡j‡2_e lcB_{: …C:14†}

The sixth item of the right-hand side in (C.1) is D6ˆ 2 X1 nˆ0 Z _B ynˆ0 Z ₁ xn‡1ˆB yn xn‡1yn … † …lc†lmn …lmn 1†!ylmn 1n e lcyn " # …lc†lm …lm 1†! xlm 1n‡1 e lcxn‡1 " # ( ) dxn‡1dyn ˆ 2lmX1 nˆ0 Xlm jˆ0 …lc†lmn‡j 1 …lmn 1†!j!Blmn‡j‡1e lcB " # 1 lmn ‡ 1   X…lmn  ‡ 1; j†  : …C:15† From (C.4), (C.15) is re-written as D6ˆ 2lm X1 nˆ0 Xlm jˆ0 lmn …lmn ‡ j ‡ 1†!   …lc†lmn‡j 1Blmn‡j‡1_e lcB_{: …C:16†}

From (C.1), (C.6), (C.8), (C.10), (C.12), (C.14) and (C.16), E‰B2

LŠ is expressed as E‰B2 LŠ ˆ X1 nˆ0 X lm 1 jˆ0 ‰…j ‡ 2†…j ‡ 1†Š …lc†_{lmn ‡ j ‡ 2†!}lmn‡jBlmn‡j‡2e lcB " # 2lmX1 nˆ0 Xlm jˆ0 …j ‡ 1† …lc†lmn‡j 1_{lmn ‡ j ‡ 1†!}Blmn‡j‡1e lcB " # ‡X1 nˆ0 X lm‡1 jˆ0 lm…lm ‡ 1† …lmn ‡ j†!   …lc†lmn‡j 2_Blmn‡j_e lcB h i : …C:17†

Appendix D. Deriving E‰B2

LŠ for the recharged credit case

This appendix derives E‰B2

LŠ for the recharged credit case. From the assumptions in Section 3.2,

E‰B2 LŠ ˆ X1 nˆ0 Z ₁ bˆ0 Z _b ynˆ0 Z ₁ xn‡1ˆb yn …yn ( ‡ xn‡1 b†2 b a C…a†ba 1e bb    _{lmn 1†!}…lc†lmn ylmn 1 n e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyndb ˆ E1‡ E2‡ E3‡ E4‡ E5‡ E6; …D:1†

where the ®rst item of the right-hand side in (D.1) is E1ˆ X1 nˆ0 Z ₁ bˆ0 Z _b ynˆ0 Z ₁ xn‡1ˆb yn y2 n b a C…a†ba 1e bb   (  …lc†lmn …lmn 1†!ynlmn 1e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyndb ˆX1 nˆ0 X lm 1 jˆ0 ba_lc†lmn‡j C…a†…lmn 1†!j! " # C…a ‡ lmn ‡ j ‡ 2† …b ‡ lc†a‡lmn‡j‡2 " # 1 lmn ‡ 2   X…lmn  ( ‡ 2; j† ) : …D:2† From (C.4), (D.2) is expressed as E1ˆ X1 nˆ0 lmn…lmn ‡ 1†Xlm 1 jˆ0 ba_lc†lmn‡j …b ‡ lc†a‡lmn‡j‡2 " # C…a ‡ lmn ‡ j ‡ 2† C…a†…lmn ‡ j ‡ 2†!   ( ) : …D:3†

The second item of the right-hand side in (D.1) is E2ˆ X1 nˆ0 Z ₁ bˆ0 Z _b ynˆ0 Z ₁ xn‡1ˆb yn x2 n‡1 b a C…a†ba 1e bb   (  _{lmn 1†!}…lc†lmn ylmn 1 n e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyndb ˆ lm…lm ‡ 1† …lc†2 " # X1 nˆ0 X lm‡1 jˆ0 ba_lc†lmn‡j …lc ‡ b†a‡lmn‡j " # C…a ‡ lmn ‡ j† C…a†…lmn 1†!j!   1 lmn   X…lmn; j†   ( ) : …D:4† From (C.4), (D.4) is re-written as E2ˆ lm…lm ‡ 1† X1 nˆ0 X lm‡1 jˆ0 ba_lc†lmn‡j 2 …b ‡ lc†a‡lmn‡j " # C…a ‡ lmn ‡ j† C…a†…lmn ‡ j†!   ( ) : …D:5†

The third item of the right-hand side in (D.1) is E3ˆ X1 nˆ0 Z 1 bˆ0 Z b ynˆ0 Z 1 xn‡1ˆb yn b2 ba C…a†ba 1e bb   (  _{lmn 1†!}…lc†lmn ylmn 1 n e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyndb ˆX1 nˆ0 X lm 1 jˆ0 ba_lc†lmn‡j C…a†…lmn 1†!j! " # C…a ‡ lmn ‡ j ‡ 2† …b ‡ lc†a‡lmn‡j‡2 " # 1 lmn   X…lmn; j†   ( ) : …D:6† From (C.4), (D.6) is re-written as E3ˆ X1 nˆ0 X lm 1 jˆ0 ba_lc†lmn‡j …b ‡ lc†a‡lmn‡j‡2 " # C…a ‡ lmn ‡ j ‡ 2† C…a†…lmn ‡ j†!   ( ) : …D:7†

The fourth item of the right-hand side in (D.1) is E4ˆ 2 X1 nˆ0 Z 1 bˆ0 Z b ynˆ0 Z 1 xn‡1ˆb yn ba C…a†bae bb   (  _{lmn 1†!}…lc†lmn ylmn n e lcyn " # …lc†lm …lm 1†!xlm 1n‡1 e lcxn‡1 " #) dxn‡1dyndb ˆ 2X1 nˆ0 X lm 1 jˆ0 ba_lc†lmn‡j …lc ‡ b†a‡lmn‡j‡2 " # C…a ‡ lmn ‡ j ‡ 2† C…a†…lmn 1†!j!   1 lmn ‡ 1   X…lmn  ( ‡ 1; j† ) : …D:8† From (C.4), (D.8) is E4ˆ 2lm X1 nˆ0 n lm 1X jˆ0 ba_lc†lmn‡j …b ‡ lc†a‡lmn‡j‡2 " # C…a ‡ lmn ‡ j ‡ 2† …lmn ‡ j ‡ 1†!C…a†   ( ) : …D:9†

The ®fth item of the right-hand side in (D.1) is E5ˆ 2 X1 nˆ0 Z ₁ bˆ0 Z _b ynˆ0 Z ₁ xn‡1ˆb yn ba C…a†bae bb   (  _{lmn 1†!}…lc†lmn ylmn 1 n e lcyn " # …lc†lm …lm 1†!xlmn‡1e lcxn‡1 " #) dxn‡1dyndb ˆ 2lmX1 nˆ0 Xlm jˆ0 ba_lc†lmn‡j 1 …b ‡ lc†a‡lmn‡j‡1 " # C…a ‡ lmn ‡ j ‡ 1† C…a†…lmn 1†!j!   ₁ lmn   X…lmn; j†   ( ) : …D:10† From (C.4), (D.10) is re-written as E5ˆ 2lm X1 nˆ0 Xlm jˆ0 ba_lc†lmn‡j 1 …b ‡ lc†a‡lmn‡j‡1 " # C…a ‡ lmn ‡ j ‡ 1† C…a†…lmn ‡ j†!   ( ) : …D:11†

The sixth item of the right-hand side in (D.1) is E6ˆ 2 X1 nˆ0 Z 1 bˆ0 Z b ynˆ0 Z 1 xn‡1ˆb yn ba C…a†ba 1e bb   (  _{lmn 1†!}…lc†lmn ylmn n e lcyn " # …lc†lm …lm 1†!xlmn‡1e lcxn‡1 " #) dxn‡1dyndb ˆ 2lmX1 nˆ0 Xlm jˆ0 ba_lc†lmn‡j 1 …b ‡ lc†a‡lmn‡j‡1 " # C…a ‡ lmn ‡ j ‡ 1† C…a†…lmn 1†!j!   1 lmn ‡ 1   X…lmn  ( ‡ 1; j† ) : …D:12† From (C.4), (D.12) is re-written as E6ˆ 2…lm†2 X1 nˆ0 n Xlm jˆ0 ba_lc†lmn‡j 1 …b ‡ lc†a‡lmn‡j‡1 " # C…a ‡ lmn ‡ j ‡ 1† C…a†…lmn ‡ j ‡ 1†!   ( ) : …D:13†

From (D.1), (D.3), (D.5), (D.7), (D.9), (D.11) and (D.13), E‰B2

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Ming-Feng Chang received his B.S. and M.S. degrees in electrical engineering from the National Taiwan University in 1982 and 1984, respectively, and his Ph.D. degree in computer science from the University of Illinois at Urbana-Champaign in 1991. He is currently an Associate Professor in the Department of Computer Science and Information Engineering, Chiao-Tung University, Taiwan, Republic of China. His research interests include Internet communication, mobile computing and VLSI system design.

Yi-Bing Lin received his BSEE degree from the National Cheng Kung University in 1983, and his Ph.D. degree in Computer Science from the University of Washington 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 in the Department of Computer Science and Information Engineering (CSIE), National Chiao Tung University (NCTU). In 1996, he was appointed as the Deputy Director of Micro-electronics and Information Systems Research Center, NCTU. During 1997±1999, he was elected as Chair-man 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 asso-ciate editor of IEEE Network, an editor of IEEE J-SAC: 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 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 Eighth Workshop on Distributed and Parallel Simulation, General Chair for the Ninth Workshop on Dis-tributed and Parallel Simulation. Program Chair for the Second 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 special issue on Active, Programmable, and Mobile Code Networking. Lin is the author of the book Wireless and Mobile Network Architecture (co-author with Imrich Chlamtac; published by Wiley). Lin received 1998 and 2000 Outstanding Research Awards from National Science Council, ROC, and 1998 Outstanding Youth Electrical Engineer Award from CIEE, ROC. Lin is an Adjunct Research Fellow of Academia Sinica. Lin's email address is [email protected].

Wei-Zu Yang received his M.S. degree from the Department of Computer Science and Information Engi-neering, National Chiao Tung University in 1992. He is currently a Ph.D. candidate in the Chiao Tung University. His research interests include performance modeling of PCS and ATM networks.