Design of IMS Prepaid Application Server for SIP-based Services
Step 5. The IP-Message Gateway sends the SIP 200 OK message to UE1 through the PAS and the CSCF
3.4 Approximate Analytic Model for Multiple-type ServicesServices
tr,1=Cmin+˜θfr,1(tr,1)f˜θ(˜θ)dtr,1d˜θ
= θ1
˜θ=0
∞
tr,1=Cmin+˜θµ1e−µ1tr,1
&
µ1eµ1˜θ eµ1θ1 − 1
'
dtr,1d˜θ
=µ1θ1e−µ1Cmin
eµ1θ1 − 1 (3.11)
3.3.2 Derivation for the Unused Credit Units
It is clear that E[Cd] = E[limt→∞Cr(t)] assuming that the prepaid account is not recharged at the end of the RTCR execution. In Fig. 3.2, the critical RU operation occurs at t7
when Cx(t7) = Cmin+ ˜θ. If Cmin+ ˜θ > tr,1, then Cd= Cmin+ ˜θ − tr,1. Otherwise, Cd= 0.
Therefore, E[Cd] is expressed as
E[Cd]=E[max{Cmin+ ˜θ − tr,1, 0}] = E[Cmin+ ˜θ − tr,1; Cmin+ ˜θ > tr,1] (3.12) From (3.5) and (3.10), Eq. (3.12) is derived as
E[Cd]=
θ1
˜θ=0
Cmin+˜θ
tr,1=0 (Cmin+ ˜θ − tr,1)fr,1(tr,1)f˜θ(˜θ)dtr,1d˜θ
=Cmin +θ1(eµ1θ1 + e−µ1Cmin) eµ1θ1 − 1 − 2
µ1 (3.13)
3.4 Approximate Analytic Model for Multiple-type Services
In this subsection, we propose an approximate analytic model for multiple-type services (i.e., n ≥ 2). This model is accurate when θi value is small. We present the derivations of Pf and E[Cd] for n = 2. The derivations for n > 2 can be directly extended and will be briefly described at the end of this subsection.
3.4.1 Derivation for the Force-termination Probability
We first derive the force-termination probability Pf for n = 2. When an RU operation is performed, one of the following three cases occurs.
Case A. There is one active type-1 service session, and the residual session holding time is tr,1.
Case B. There is one active type-2 service session, and the residual session holding time is tr,2.
Case C. Both type-1 and type-2 service sessions are active, and the residual session
holding times are tr,1 and tr,2, respectively.
When the critical RU operation occurs, let PA, PB and PC be the probabilities of Cases A, B and C, respectively. For a sufficiently small θi value, Pf can be computed as
Pf = PAPr[tr,1 > Cmin] + PBPr[tr,2 > Cmin] + PCPr[tr,1+ tr,2 > Cmin] (3.14) In (3.14) we assume that Cx(t7) = Cmin, where t7is the time that the critical RU operation occurs. Note that Cmin ≤ Cx(t7) < Cmin+
iθi. Therefore, (3.14) incurs error when θi
is large. Substituting (3.5) into (3.14) to yield Pf=PA
∞
tr,1=Cmin
fr,1(tr,1)dtr,1+ PB ∞
tr,2=Cmin
fr,2(tr,2)dtr,2
+PC
∞
t=Cmin
t
tr,1=0fr,2(t − tr,1)fr,1(tr,1)dtr,1dt
=PAe−µ1Cmin+ PBe−µ2Cmin + PC
µ1e−µ2Cmin− µ2e−µ1Cmin µ1− µ2
(3.15) In (3.15), the probabilities PA, PB and PC are derived as follows: We first compute the probability pi that a type-i service session is active at a random observation point. In Fig.3.1, a random observation point occurs at t2 in the renewal period [t0, t5]. From the alternating renewal theory [48], pi is expressed as
pi = E[th,i]
E[th,i] + E[ta,i] = λi µi+ λi
(3.16) At t2, there is only one active type-1 service session with probability p1(1− p2), there is only one active type-2 service session with probability p2(1− p1), and both type-1 and type-2 service sessions are active with probability p1p2. The critical RU operation is issued by a type-1 and a type-2 service sessions in Cases A and B, respectively. In Case C, the critical RU operation may be issued by a type-1 or a type-2 services. Since the sending of recharge message can be modeled as a random observer for sufficiently small θi value in a service session, from (3.16), the ratio PA: PB : PC can be computed as
PA : PB : PC ≈ p1(1− p2) : p2(1− p1) : 2p1p2 = λ1µ2 : µ1λ2 : 2λ1λ2 (3.17)
Since PA+ PB+ PC = 1 and from (3.17), we have
PA≈ λ1µ2+µλ11λµ22+2λ1λ2, PB≈ λ1µ2+µµ11λλ22+2λ1λ2, PC ≈ λ1µ2+µ2λ11λλ22+2λ1λ2 (3.18) Substitute (3.18) into (3.15) to yield
Pf=
λ1µ2e−µ1Cmin+ µ1λ2e−µ2Cmin + 2λ1λ2
µ1e−µ2Cmin− µ2e−µ1Cmin µ1− µ2
×
1
λ1µ2+ µ1λ2+ 2λ1λ2
(3.19) For n > 2, Eq. (3.14) can be extended by including all active session combinations (there are 2n− 1 combinations). Then Pf can be computed following the same derivations for (3.15)-(3.19).
3.4.2 Derivation for the Unused Credit Units
For n = 2, E[Cd] is derived as follows: For sufficiently small θi values, Cx(t∗) ≈ Cmin. Therefore, the Cd values are max{Cmin − tr,1, 0}, max{Cmin − tr,2, 0} and max{Cmin − tr,1− tr,2, 0} in Cases A, B and C, respectively. We have
E[Cd]=PAE[max{Cmin− tr,1, 0}] + PBE[max{Cmin− tr,2, 0}]
+PCE[max{Cmin− tr,1− tr,2, 0}]
=PAE[Cmin− tr,1; Cmin > tr,1] + PBE[Cmin− tr,2; Cmin > tr,2]
+PCE[Cmin− tr,1− tr,2; Cmin> tr,1+ tr,2] (3.20) Substitute (3.5) and (3.18) into (3.20) to yield
E[Cd]=PA
Cmin
tr,1=0(Cmin− tr,1)fr,1(tr,1)dtr,1+ PB
Cmin
tr,2=0(Cmin− tr,2)fr,2(tr,2)dtr,2
+PC
Cmin
t=0 (Cmin− t) t
tr,1=0fr,2(t − tr,1)fr,1(tr,1)dtr,1dt
=Cmin−
1
λ1µ2+ µ1λ2+ 2λ1λ2
λ1µ2(1− e−µ1Cmin)
µ1 +µ1λ2(1− e−µ2Cmin) µ2
+2λ1λ2(
µ12(1− e−µ2Cmin)− µ22(1− e−µ1Cmin)) µ1µ2(µ1− µ2)
(3.21)
For n > 2, E[Cd] can be computed through the same derivations for (3.20)-(3.21) by considering 2n− 1 active session combinations.
Table 3.1: Comparison of the Analytic and Simulation Results (n = 1)
Cmin (Unit: c) 1 2 3 4 5
Simulation 21.4081% 7.8867% 2.9089% 1.0616% 0.3926%
Analytic 21.4097% 7.8762% 2.8975% 1.0659% 0.3921%
Error 0.01% −0.13% −0.39% 0.41% −0.11%
(a) Pf (θ1 = 1/µ1)
Cmin (Unit: c) 1 2 3 4 5
Simulation 0.8004 1.6481 2.5971 3.5880 4.5820 Analytic 0.7961 1.6607 2.6110 3.5926 4.5859 Error −0.55% 0.76% 0.53% 0.13% 0.08%
(b) E[Cd] (unit: c) (θ1 = 1/µ1)
3.5 Simulation Validation
The analytic model developed in this section is validated against the simulation exper-iments. The discrepancies between analytic analysis (specifically, Eqs. (3.11), (3.13), (3.19) and (3.21)) and simulation are within 2% in Tables 3.1 and 3.2. The simulation model follows the discrete event approach described in Chapter 2, and the details are omitted. The input parameter θi is normalized by the mean 1/µi of the service session holding time. The input parameter Cmin and output measure E[Cd] are normalized by the expected credit units c consumed in a session, where c is derived as follows:
Consider n = 2 and let n1 and n2 be the numbers of session completions for type-1 and type-2 services in an observation period, respectively. In (3.16), pi represents the fraction of time that the type-i service is active in an observed period, and the expected service session holding time for a type-i service is 1/µi, the ratio n1 : n2 can be computed as
n1 : n2 = p1µ1 : p2µ2 = λ1µ1
µ1+ λ1
: λ2µ2
µ2+ λ2
(3.22) From (3.22), c can be computed as
c=n1/µ1+ n2/µ2
n1+ n2
= λ1(µ2+ λ2) + λ2(µ1+ λ1)
λ1µ1(µ2+ λ2) + λ2µ2(µ1+ λ1) (3.23)
Table 3.2: Comparison of the Analytic and Simulation Results (n = 2)
Cmin (Unit: c) 1 2 3 4 5
Simulation 57.3683% 31.1314% 16.3689% 8.5267% 4.4320%
Analytic 57.5872% 31.2126% 16.4590% 8.5647% 4.4274%
Error 0.38% 0.26% 0.55% 0.44% −0.10%
(a) Pf (θi = 0.01/µi, λ1 = µ1 and µ2= λ2 = 2µ1)
Cmin (Unit: c) 1 2 3 4 5
Simulation 0.2281 0.7970 1.5672 2.4478 3.3837 Analytic 0.2257 0.7937 1.5628 2.4419 3.3792 Error −1.06% −0.41% −0.28% −0.24% −0.13%
(b) E[Cd] (unit: c) (θi = 0.01/µi, λ1 = µ1 and µ2= λ2 = 2µ1)
3.6 Numerical Examples
This section uses numerical examples to investigate the performance of the RTCR mech-anism. For the examples in Figs. 3.4 and 3.5, n = 2, λ1 = µ1 and λ2 = µ2 = 2µ1. In Fig. 3.6, 1 ≤ n ≤ 4 and λi = µi = iµ1. Similar results are observed for other parameter values and are not presented. The effects of the input parameters are described below.
Effects of θi on E[Nr,i]. Fig. 3.3 plots the expected number E[Nr,i] of RU operations executed in a type-i service session against the granted credit θi. Note that E[Nr,i] is not affected by Cmin and n. This figure shows a trivial result that E[Nr,i] decreases as θi increases. A non-trivial observation is that when θi ≥ 2.5/µi, E[Nr,i] ≈ 1.
It implies that selecting θi value larger than 2.5/µi will not improve the E[Nr,i] performance.
Effects of Cmin. Fig. 3.4 plots the force-termination probability Pf and the expected credit E[Cd] against θi and Cmin, where n = 2, λ1 = µ1 and λ2 = µ2 = 2µ1. Fig. 3.4 (a) shows that Pf decreases as Cmin increases. When the critical RU operation occurs, more unused credit units are available in the prepaid account when Cmin
increases. Therefore, the possibility of force-termination reduces. For θi = 1/µi, when Cmin increases from 2c to 4c, Pf decreases from 18.78% to 4.99%. Fig. 3.4 (b) shows that E[Cd] increases as Cmin increases. It is apparent that when the critical RU operation occurs, the exact unused credit units Cx(t∗) for the user increases as
Cmin increases. That is, the amount of consumed credit reduces, and the expected credit E[Cd] increases. For θi = 1/µi, when Cmin increases from 2c to 4c, E[Cd] increases from 1.60c to 3.41c. In this scenario, we expect that 3.41 − 1.60 = 1.81 more sessions are complete when Cmin = 2c than when Cmin = 4c.
Effects of θi. Fig. 3.4 (a) shows that Pf is a decreasing function of θi. When θi in-creases, more credit units are granted to the AS. Therefore, the possibility of force-termination reduces. For Cmin = 6c, when θi increases from 1/µi to 2.5/µi, Pf
decreases from 1.32% to 0.37%. This effect becomes insignificant when θi is large (e.g., θi ≥ 5/µi). Fig. 3.4 (b) shows that E[Cd] is an increasing function of θi. For Cmin = 6c, when θi increases from 1/µi to 2.5/µi, the E[Cd] increases from 5.37c to 7.64c. Fig. 3.4 (b) also quantitatively indicates how the θi and Cmin values affect E[Cd]. When θi ≤ 1/µi, E[Cd] ≈ Cmin. On the other hand, E[Cd] >> Cmin as θi
increases. For example, when Cmin = 6c and θi = 10/µi, E[Cd] = 23.63c >> 6c.
Effects of Vh,i. Fig. 3.5 plots Pf and E[Cd] against Cmin and the variance Vh,i of the Gamma service session holding time th,i, where n = 2, θi = 2.5µi, λ1 = µ1 and λ2 = µ2 = 2µ1. Fig. 3.5 (a) shows that Pf increases as Vh,iincreases. This phenomenon is explained as follows: As Vh,iincreases, more long and short th,i periods are observed.
The recharge message is more likely to be sent in the long th,i periods than the short th,i periods, and larger residual service session holding time tr,i are expected.
Therefore, Pf increases as Vh,i increases. Fig. 3.5 (b) shows that E[Cd] decreases as Vh,i increases. As Vh,i increases, the recharge message is likely to be sent in the long th,i periods. Then the possibility that tr,i ≥ Cmin (i.e., Cd= 0) increases. Therefore E[Cd] decreases as Vh,i increases.
Effects of n. Fig. 3.6 plots Pf and E[Cd] against θi and the number n of the service session types. Fig. 3.6 (a) shows that Pf is an increasing function of n. As n increases, the number of in-progress service sessions increases when the recharge message is sent, and therefore Pf increases. In Fig. 3.6 (b), when θi is small (e.g.
θi ≤ 1/µi), E[Cd] decreases as n increases. On the other hand, when θi is large (e.g.
θi ≥ 2.5/µi), E[Cd] increases as n increases. When the recharge message is sent, the number of simultaneous in-progress service sessions increases as n increases. There are two conflicting effects as n increases. First, more credit units will be consumed by these sessions and E[Cd] decreases. Second, the “net” unused credit units which have granted to the AS increases, and E[Cd] increases. When θi is small, the first effect will dominate. On the other hand, the second effect will dominate when θi is large.
3.7 Summary
This chapter studied the Recharge Threshold-based Credit Reservation (RTCR) mecha-nism for UMTS Online Charging System (OCS). In RTCR, when the remaining amount of prepaid credit is below a threshold, the OCS reminds the user to recharge the prepaid account. It is essential to choose an appropriate recharge threshold to reduce the prob-ability that the in-progress service sessions are forced-terminated. An analytic model is developed to compute the expected number E[Nr,i] of the RU operations executed in a type-i service session, the force-termination probability Pf and the expected credit E[Cd] left in the user account. We make the following observations:
• E[Nr,i] decreases as the granted credit θi increases. When θi ≥ 2.5/µi, increasing θi
will not improve the E[Nr,i] performance.
• Pf decreases as the recharge threshold Cmin or θi increases. This effect becomes insignificant when θiis large (e.g., θi ≥ 5/µi). E[Cd] increases as Cminor θiincreases.
• Pf increases as the variance Vh,i of the service session holding time increases, and E[Cd] decreases as Vh,i increases.
• Pf increases as the number n of the service session types increases. When θi is small (e.g. θi ≤ 1/µi), E[Cd] decreases as n increases. On the other hand, when θi is large (e.g. θi ≥ 2.5/µi), E[Cd] increases as n increases.
Based on the above discussion, a mobile operator can select the appropriate Cmin and θi values for various traffic conditions.
3.8 Notation
The notation used in this chapter is listed below.
• C: the amount of the initial prepaid credit for a mobile user
• Cmin: the recharge threshold in RTCR
• Cr: the amount of the remaining prepaid credit in the OCS
• Cr(t): the remaining credit units in the OCS for the user at a particular time point t
• Cx(t): the exact unused credit units for the user at a particular time point t
• E[Cd]: the expected amount of unused credit units in the user account at the end of RTCR execution (before recharging)
• E[Nr,i]: the expected number of the RU operations executed during a type-i session
• λi: the rate of ta,i; i.e., λi = 1/E[ta,i]
• µi: the rate of th,i; i.e., µi = 1/E[th,i]
• n: the number of types of session-based IMS services
• PA: the probability that the critical RU operation occurs when there is one active type-1 service session
• PB: the probability that the critical RU operation occurs when there is one active type-2 service session
• PC: the probability that the critical RU operation occurs when both type-1 and type-2 service sessions are active
• Pf: the probability that an in-progress session is forced to terminate (for all service type-i)
• pi: the probability that a type-i service session is active at a random observation point
• t∗: the critical time when Cx(t∗) = Cmin+ θ1
• θi: the amount of credit units that the OCS grants in each RU operation for a type-i session
• ˜θ + Cmin: the remaining credit units left when the critical RU operation arrives
• ta,i: the inter-arrival time of the type-i service session
• th,i: the holding time of the type-i service session
• tr,i: the residual holding time of the type-i service session
• tu: the consumed credit units in the AS when a random observer arrives
• ty: the elapsed holding time of the service session when a random observer arrives