Numerical Examples

This section uses the simulation experiments to investigate the performance of the PAS.

The input parameter threshold X^T and the output measure E[t^d] are normalized by the mean of the call holding time E[tc] = 1/µ. For the purposes of demonstration, we assume that the prepaid message service is charged for Tm = 5 CUs and the expected inter-call arrival time 1/λc = 50 TUs.

Effects of the expected call holding time E[t^c]. Fig. 2.6 plots P^{U F T} and E[t^d] against the threshold XT and E[tc], where X = 50E[tc] and λm = 5λc. This figure shows

In the remainder of this section, the expected inter-message arrival time 1/λm and the initial prepaid credit X are normalized by E[tc] = 4Tm.

Effects of the variance Vc of the call holding time tc. Fig. 2.7 plots PU F T and E[td] against XT and the variance Vc of the call holding time tc with Gamma distribution, where X = 25E[tc] and 1/λm = 0.5E[tc]. This figure shows that both PU F T and E[td] increase as Vc increases. This phenomenon is explained as follows: As the variance Vc of tc increases, more long and short tc periods are observed. The pre-paid message (i.e., the random observer) is more likely to fall in the long t^c periods than the short tc periods, and larger residual call holding times td are expected for larger variance Vc. Therefore, the performance of both PU F T and E[td] degrade as Vc increases.

Effects of the expected inter-message arrival time 1/λm . Fig. 2.8 plots PU F T and E[td] against XT and λm, where X = 25E[tc]. Fig. 2.8 (a) shows that PU F T increases as λ^m increases. When λ^m increases, more message deliveries are likely to occur during an in-progress call. Therefore the UFT probability PU F T increases. For X^T = 2E[t^c], when 1/λ^mdecreases from 50E[t^c] to 5E[t^c] and from 5E[t^c] to 0.5E[t^c], PU F T increases by 9.13 and 9.01 times, respectively. This effect becomes insignificant

phenomenon can be explained as follows: Since the prepaid messages are random observation points of the prepaid call holding interval, the delivery delays are not significantly affected by the message arrival rate λm.

Effects of the initial prepaid credit X. Fig. 2.9 plots PU F T and E[td] against XT and X, where 1/λm = 0.5E[tc]. This figure shows that both PU F T and E[td] decrease as X increases. When the initial prepaid credit X increases, there are more CUs left for a prepaid call, and it is more likely that there are enough CUs for both the remaining call and the message (i.e., the amount of the prepaid credits left is larger than XT+ Tm ≥ tc+ Tm). Therefore, the performance of both PU F T and E[td] improve as X increases. For X^T = 3E[t^c], when X increases from 20E[t^c] to 50E[t^c], PU F T decreases by 58.64% and E[td] decreases by 36.71%. When X increases from 50E[t^c] to 100E[t^c], P^{U F T} decreases by 49.96% and E[t^d] decreases by 50.87%. When X > 150E[tc] (i.e., the initial prepaid credit can support more than 150 voice calls), increasing X only has insignificant impacts on PU F T and E[td].

Effects of the threshold XT. From Figs. 2.6-2.9, when XT is small, increasing XT re-duces PU F T significantly. When XT ≥ 5E[tc], effects of XT on PU F T becomes insignificant. On the other hand, increasing XT always increases E[td]. Therefore

in these scenarios, it is appropriate to choose XT = 5E[tc] in the PAS.

2.6 Summary

This chapter proposed a SIP-based prepaid application server to handle both the prepaid IMS-to-PSTN calls and messaging services in UMTS. When both voice and messaging are simultaneously offered, a strategy is required to determine if a prepaid message can be sent out during an in-progress call without force-terminating this call. To avoid unnecessary force-termination, a threshold amount XT of prepaid credit is set to protect the in-progress IMS-to-PSTN call. This paper provided guidelines to select an appropriate XT. The output measures are the UFT (unnecessary force-termination) probability PU F T and the expected unnecessary delay E[td]. We investigated how these two output measures are affected by input parameters including the expected call holding time E[tc], the variance V^c of the call holding time, the message arrivals rate λ^m, X^T and the initial prepaid credit X. We make the following observations:

• When the proportion of X and E[tc] is fixed, PU F T and E[td] are not affected by the change of E[tc].

• The performance of both PU F T and E[td] degrade as Vc increases.

• PU F T increases as λm increases. This effect becomes insignificant when XT is large (e.g., XT ≥ 6E[tc] in our examples). On the other hand, E[td] is insignificantly affected by λ^m.

• Both PU F T and E[td] decrease as X increases.

• When XT is small, increasing XT reduces PU F T significantly. When XT is large (e.g. XT > 5E[tc] in our examples), increasing XT only has insignificant impact on PU F T. On the other hand, increasing XT always increases E[td].

Based on the above discussion, the network operator can select the appropriate XT values for various traffic conditions based on our study.

2.7 Notation

The notation used in this chapter is listed below.

• 1/λc: the expected interval between when the previous prepaid call completes and when the next prepaid call arrives

• λm: the arrival rate of the instant messages

• Nc: the number of prepaid call completed

• Nm: the number of prepaid message arrived

• N(t): the number of prepaid messages occurring in period t

• PU F T: the UFT (unnecessary force-termination) probability of an in-progress call in the prepaid application server

• τi: the interval between when the i-th prepaid call completes and when the i + 1-st prepaid call starts

• tc: the prepaid call holding time

• td: the extra delay between when an instant message service request arrives and when the instant message is delivered

• Tm: the charge for a prepaid instant message delivery

• Vc: the variance of the call holding time t^c

• X: the amount of the initial prepaid credit

• Xc: the credit units spend for Nc calls

• XT: the credit threshold that avoids force-termination of prepaid calls

• x^∗: the remaining credit in a user’s prepaid account

Chapter 3

Modeling Online Credit Reservation Procedure

Prepaid telecommunications service requires a user to make an advanced payment before enjoying the service. Usage of prepaid service does not require deposit and monthly bill. Instead the usage fee is directly deducted from the user’s prepaid account. Four billing technologies have been used in mobile prepaid service: hot billing approach [24], service node approach [23, 25], Intelligent Network (IN) approach [35] and handset-based approach [36]. By creating hybrid online/offline billing models, 3GPP Releases 5 and 6 propose the IP-based Online Charging System (OCS) [10] to allow both prepaid and postpaid subscribers to be charged in real-time.

As described in Section 1.3, online charging provides real-time charging information to the billing system. This chapter investigates credit reservation for the OCS. We assume that a mobile user may access n types of session-based IMS services. After an online service user has purchased some credit units, she is allowed to enjoy multiple sessions simultaneously. Each service type has its own traffic characteristics and communications parameters. For example, the average call holding time for a VoIP call session is 1-3 minutes, and the average session holding time for the interactive mobile gaming sessions may range from 10 to 30 minutes [18, 26]. Note that the service sessions can be charged according to time (duration) or transmitted packet volumes. For a session of type i, each time the OCS grants θi credit units to the session. When these credit units are consumed, the OCS grants extra credit units to the session through the Diameter credit reservation procedure. When the balance of the user account at the OCS is below a recharge threshold Cmin, the OCS does not allow the user to initiate new sessions, and reminds the user to refill the prepaid account by sending a recharge message. This chapter shows how to

select an appropriate recharge threshold Cmin. Note that the usage of Cmin is different from that of XT in Chapter 2. The notation used in this chapter is listed in Section 3.8.

3.1 Recharge Threshold-based Credit Reservation