Credit Pre-reservation Mechanism for UMTS Prepaid Service
2.3 Numerical Examples
This section uses numerical examples to investigate the performance of the CPM. For the pre-sentation purpose, we assume that the packet termination probability is α = 0.01 and the session arrival rate (normalized by the message delivery rate) is γ = µ/20. For other α and γ/µ values, we observed similar results, which are not presented in this chapter. In Figures 2.2 and 2.3, the packet arrivals in a session have a Poisson distribution and the RU operation delay has an Erlang distribution. These Expontial-like assumptions are relaxed in Figures 2.4 and 2.5 by
considering the Pareto and the Gamma distributions. The effects of the input parameters λ/µ, C, θ and δ are described as follows.
Effects on Pr. Figure 2.2 (a) shows how Pr is affected by θ and λ/µ. From (2.12) and (2.13), we have
θ→0limPr = 1 and lim
θ→∞Pr = 0 (2.15)
Therefore, it is obvious that Pr is a decreasing function of θ.
When λ/µ is very small or vary large, we have
λ/µ→0lim Pr= (1 − α)θ−1 and lim
λ/µ→∞Pr = 0 (2.16)
When λ/µ increases, more packets arrive during an RU operation. When more than θ+δ packets arrive during one RU operation, an extra RU operation will be immediately executed. Consequently, Princreases. Therefore Pris an increasing function of λ/µ.
Effects of λ/µ. Figure 2.2 (b) shows that B is increasing functions of λ/µ. From (2.14), we have
λ/µ→0lim B = 0 and lim
λ/µ→∞B = (1 − α)δ
α (2.17)
When λ/µ increases, it is likely that more than δ packets will arrive during the interval of the RU operation. Note that W correlates positively with B. Therefore both B and W increase as λ/µ increases.
When λ/µ → ∞, we found that the B value in (2.17) is higher than the simulation result (not shown in this Figure), which is explained as follows: When λ/µ → ∞, all packets for a session will arrive before the end of the first RU operation, and therefore the B value in (2.17) is determined by α. In simulation, the session is always in the “low-credit” status
(in the LC period), and every time an RU operation is performed, the number of buffered packets at the end of the operation is reduced. Therefore the expected value B is smaller than that shown in (2.17). To avoid buffer overflow, the B value in (2.17) should be considered in the system setup.
Figure 2.3 shows that Pnc increases as λ/µ decreases. Since α is fixed, when λ/µ de-creases, the session holding times become longer, and it is likely that more new sessions will arrive during the holding time of an existing session. Therefore, when λ/µ decreases, more sessions will exist at the same time. Suppose that the credit in the OCS suffices to support these sessions if they are sequentially delivered. It is clearly that the OCS may not be able to support these sessions if they are delivered simultaneously. In this case, a newly incoming session is rejected because the credit in the OCS is depleted (while there are unused credit units held in the multiple in-progress sessions). Therefore, Pnc increases as λ/µ decreases.
In Figure 2.3, Xs is a decreasing function of Pnc because the number of RU operations performed in a force-terminated session is less than that in a complete session. Therefore, Xsincreases as λ/µ increases.
Effects of C. Figure 2.3 shows that the output measures (Pncand Xs) are only affected by the
“end effect” of C. As C increases, it is more likely that the remaining credit units in the OCS suffice to support one RU operation and such end effect becomes insignificant.
Similar to the λ/µ impact, Pnc is a decreasing function of C, and Xs is an increasing function of C. Figure 2.3 indicates that when C is sufficiently large (e.g. C ≥ 600/α), the end effect of C can be ignored. Same phenomenon is observed for B and W , and the results are not shown.
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Figure 2.4: Effects of θ and the packet interarrival time distribution (α=0.01, γ/µ=1/20, δ=0.3θ, C = 600/α, and λ/µ=3)
Since δ = 0.3θ in Figure 2.4 (b), from (2.13) and (2.14), we have
θ→∞lim B = lim
δ→∞B = 0 and
θ→0limB = lim
δ→0B = λ (2µ + αλ)
(1 − α) (µ + αλ)2 (2.18)
The non-trivial result is that there is a threshold θ value (θ ≈ 100λ in Figure 2.4) such that beyond this threshold value, increasing θ does not improve the performance. On the other hand, Figure 2.4 (c) shows that Pnc linearly increases as θ increases. When θ increases, more credit units are reserved in an RU operation, and the credit in the OCS is consumed fast. Therefore, a newly incoming session has less chance to be served, and an in-progress session is likely to be force-terminated.
Effects of packet interarrival time distribution. Figure 2.4 considers the packet arrival times with the Exponential and the Pareto distributions with mean 1/λ. In the Pareto distribu-tion, the shape parameter b describes the “heaviness” of the tail of the distribution. It has been shown that the Pareto distribution with 1 ≤ b ≤ 2 can approximate the packet traffic very well [26, 51].
Figure 2.4 shows that B, W and Pnc are decreasing functions of b. When b decreases, the tail of the distribution becomes longer, and more long packet interarrival times are observed. Since the mean value 1/λ is fixed for the Pareto distribution in Figure 2.4, more long packet interarrival times also imply more short packet interarrival times. The number of short interarrival times must be larger than that of long interarrival times because the minimum of the interarrival time is fixed but the maximum of the interarrival time is infinite. Thus, it is likely that more packets will arrive during an RU operation, and B and W increase. With a small b, it is likely that the last session for a user accommodated by
to arrive, and are rejected by the OCS, which contributes to Pnc. Therefore Pnc increases as b decreases.
Effects of δ. Similar to the effect of θ, Figure 2.5 shows that B and W are decreasing functions of δ (see (2.18)), and Pnc is a linearly increasing function of δ. A non-trivial observation is that when δ ≥ 0.6θ, B and W approach to zero. It implies that selecting δ value larger than 0.6θ will not improve the performance. Figure 2.5 (c) shows that Xsis an increasing function of δ. When the amount of the credit in a session is less than δ, a CCR message is sent to the OCS. Therefore, for a fixed θ, when δ is increased, Xsincreases.
Effects of RU operation delay distribution. Figure 2.5 considers the Erlang with b = 2 (which is a Gamma distribution with variance V = 18/λ2) and Gamma distributed RU operation delays with variances V = 0.01/λ2, 1/λ2, and 100/λ2, respectively.
The figure indicates that Pnc and Xs are not significantly affected by the RU operation delay distribution. On the other hand, B and W increase as V increases. As V increases, more long and short RU operation delays are observed. In long RU operation delays, it is likely that more than δ packets arrive. Therefore the packets are more likely to be buffered and delayed processed.
2.4 Conclusion
In this chapter, we investigated the prepaid services for the UMTS network where multiple prepaid and postpaid sessions are simultaneously supported for a user. We described the pre-paid network architecture based on UMTS, and proposed the credit pre-reservation mechanism (CPM) that reserves extra credit earlier before the credit at the GGSN is actually depleted.
An analytic model was developed to compute the average number B of packets buffered
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Figure 2.5: Effects of δ and the RU operation delay distribution (α=0.01, γ/µ=1/20, θ=100λ, C = 600/α, λ/µ=3, and the packet arrival times have the Pareto distribution with the mean 1/λ and b = 2)
during a reserve units (RU) operation and the probability Prthat more than one RU operation is executed during a low credit (LC) period. Simulation experiments are conducted to investigate the performance of CPM. We have the following observations:
• B and W increase as λ/µ increases. The probability Pncthat a session is not completely served decreases as λ/µ increases. The average number Xs of RU operations performed in a session is an increasing function of λ/µ.
• B, W, Xsand Pnc are only affected by the end effect of C. When C is sufficiently large (e.g., C ≥ 600/α, where α is the probability that an arrival packet is the last one of the session), the end effect can be ignored.
• B, W and Xsdecrease but Pncincreases as θ increases. There is a threshold θ value (e.g., θ ≈ 100λ) such that beyond this threshold value, increasing θ does not improve the CPM performance.
• B and W decrease but Xsincreases as δ increases. When δ is large (e.g., δ ≥ 0.6θ), both B and W approach zero.
• Pr increases when λ/µ increases or θ decreases.
• B, W and Pncincrease as the tail of the packet arrival time distribution becomes longer.
• B and W increase as the variance of the RU operation delay increases.
Our study provides guidelines to select the CPM parameters. Specifically, it is appropriate to select C ≥ 600/α, θ ≈ 100λ, and δ ≈ 0.6θ.