Repacking on Demand for the Hierarchical WiMAX Networks
4.4 Results and Discussions
We compare NR, AR, PRoD, RoD-R and RoD-L in terms of the output measures de-scribed above. In our numerical examples, the radio channel number is c = 10 for every microcell and the expected microcell residence time for slow MSs is 5 times the value for fast MSs (i.e., 1/ηs= 5/ηf). In most cases, we have the input parameters with the default setup: the call holding times are exponentially distributed with mean 1/µ = 1 minute (i.e., Vc= 1/µ2), the microcell residence times for slow MSs are exponentially distributed with mean 1/ηs = 10/µ (i.e., Vm,s = 100/µ2 and Vm,f = Vm,s/25 = 4/µ2), the number of macrocell channel is C = 8, the call arrival rate to a microcell is λ = 7/µ, and the proportion of fast calls is β = 9%. The effects of the input parameters are described as follows.
Effect of the Macrocell Channel Number C. Figures 4.9 (a) - (d) plot Pf f, Pb, Pf and Pnc as functions of C, where the values for the input parameters except C follow the default values listed above. These figures show an intuitive result that for all approaches, Pf f, Pb, Pf and Pnc decrease as C increases. We also observe that the Pf f, Pb, Pf and Pnc are more sensitive to the change of C for small C values than for large C values. Since the macrocell channels are the bottleneck resources when C is small, increasing C significantly reduces Pf f, Pb, Pf and Pnc. Figures 4.10 (a) - (d) plot Hmm, HmM, HR and H as functions of C, where the values for the input parameters except C follow the default values listed above. Figure 4.10 (a) shows that for all approaches, Hmm is a decreasing function of C. This phenomenon is due to the fact that when C increases, more calls (especially fast calls) will occupy macrocell channels, and the number of microcell to microcell handoffs will decrease.
0.0
0.000
Figure 4.10 (b) shows that for AR, HmM is an increasing function of C. Increasing C results in more idle macrocell channels, and more calls (especially slow calls) will be handed off from microcells to macrocells. For NR, PRoD and RoD, HmM
increases and then decreases as C increases. When C is small, macrocell channels are the bottleneck resources and increasing C results in more calls handed off from microcells to macrocells (especially fast calls that overflow to microcells). When C is large (C > 5 in Figure 4.10 (b)), macrocell channels are no longer the bottleneck resources and less fast calls overflow to microcells. In this case, increasing C results in decreasing of HmM. The performance figures for HM m and HM M are similar to that for HmM, and the details are omitted. Figure 4.10 (c) shows that for AR, HR increases as C increases. Increasing C results in more slow calls that would overflow to macrocells, and thus more calls are repacked. On the other hand, for PRoD and RoD, HR increases and then decreases as C increases. This non-trivial phenomenon is explained as follows. When C is small (C < 5 for RoD and C < 12 for PRoD in Figure 4.10 (c)), increasing C results in more M-to-m repacking candidates, and more on-demand repackings are exercised. When C is large, macrocell channels are no longer the bottleneck resources. Increasing C results in less blockings as well as force-terminations, and less on-demand repackings are needed. Therefore HR decreases as C increases in this case. Figure 4.10 (d) shows the net effects of repackings and all types of handoffs. In this figure, as C increases, H decreases for NR and increases for AR. For PRoD and RoD, H increases and then decreases as C increases.
Comparison of NR, AR, PRoD, RoD-R and RoD-L. Figures 4.9 (a) - (d) com-pare NR, AR, PRoD, RoD-R and RoD-L on Pf f, Pb, Pf and Pnc, respectively.
These figures show that RoD-R and RoD-L have smaller Pf f, Pb, Pf and Pnc values than NR, AR and PRoD. Furthermore, AR has higher Pnc than NR when C is small, and the opposite result is observed when C is large (C > 12.5 in Figure 4.9
(d)). This non-trivial phenomenon is explained as follows. Consider the case when C is much less than the number of fast calls. In AR, m-to-M repacking is always exercised, and therefore macrocells have less idle channels in AR than in NR. In this case, the Pnc value is higher for AR than for NR. On the other hand, when C is large, the effect of M-to-m repacking for slow calls becomes more significant. Thus macrocells have more idle channels in AR than in NR. Specifically, if the HWN is engineered at Pnc = 2% (see the horizontal dashed line in Figure 4.9 (d)), C = 5.5 for RoD, C = 13 for PRoD, C = 14.5 for AR, and C = 15 for NR. Thus RoD can save at least 7 macrocell channels over other approaches.
Figure 4.10 (c) shows that HR,RoD−R > HR,RoD−L > HR,AR > HR,P RoD > HR,N R = 0 when C is small (C < 12). When C is large (C > 12), HR,AR > HR,RoD−R >
HR,RoD−L > HR,P RoD > HR,N R = 0. Similar phenomena for H are observed in Figure 4.10 (d). Although NR, AR and PRoD have smaller HR and H values than RoD, the increase of HR and H does not results in the increase of Pf f, Pb, Pf and Pnc in RoD (see Figures 4.9 (a) - (d)).
Effect of the Proportion β of Fast MSs. Figure 4.11 plots Pnc as a function of β, where the values for the input parameters except β follow the default values listed above. In the figure, the call arrival rates for fast and slow MSs are βλ and (1 − β)λ, respectively. This figure shows that for all approaches, Pnc increases as β increases. Increasing of fast calls results in the increase of handoffs and hence force-terminations. This figure also shows that RoD approaches (i.e., RoD-R and RoD-L) are less sensitive to β than other approaches. Furthermore, when β is large (β > 8%
for AR and β > 18% for PRoD), AR and PRoD have higher Pnc values than NR.
The reason is that for AR, when β is large, the effect of m-to-M repackings is more significant than that of M-to-m repackings. Thus, less idle channels are available in macrocells for AR than for NR, and more calls are blocked or forced to terminate in
0.0
Figure 4.11: Effect of the Proportion β of Fast MSs on Incomplete Probability Pnc AR. For PRoD, when β is large, more fast calls repack slow calls, which occupy more macrocell channels. Therefore, more slow calls are blocked or forced to terminate.
Effect of the MS Mobility (i.e., Mean Microcell Residence Times). Figure 4.12 plots Pnc as a function of the microcell mobility rates (i.e., ηs for slow MSs and ηf = 5ηs for fast MSs), where the values for the input parameters except 1/ηs and 1/ηf follow the default values listed above. This figure shows that Pnc increases as ηs increases. This figure also shows that to keep the same Pnc performance, (e.g., Pnc = 7%), RoD can support much faster MSs (at least 23 times the ηs value) than other approaches.
Effect of the Arrival Rate λ. Figure 4.13 plots Pncas a function of λ, where the values for the input parameters except λ follow the default values listed above. This figure shows that Pnc increases as λ increases. It also shows that to keep the same Pnc performance (e.g., Pnc = 2%), RoD can support more call arrivals (at least 18%) than other approaches.
Effect of the Variance Vc for the Call Holding Times. Figure 4.14 plots Pnc and H as functions of Vc, where the values for the input parameters except Vc follow
0
Figure 4.13: Effect of the Arrival Rate λ on Incomplete Probability Pnc
0
Figure 4.14: Effects of the Variance Vc for the Call Holding Times
the default values listed above. Figure 4.14 shows that Pnc and H decrease as Vc increases. Note that, for the call holding time distributions with the same mean value 1/µ, the standard deviation σ = √
Vc. By the Chebyshev’s Inequality, the probability that the call holding times are out of range [1/µ − 5√3Vc, 1/µ + 5√3Vc] is smaller than 36% for all Vc values. For example, if Vc= 100/µ2, then 5√3Vc = 50/3µ and the probability that the call holding time exceeds 53/3µ is smaller than 36%.
As Vcincreases, more long and short call holding times are observed. More short call holding times implies that more calls are completed before next new call attempts arrive or next handoff attempts are exercised. Thus the numbers of blocked calls, force-terminated calls and handoffs decrease.
Effect of the Variances for the Microcell Residence Times. Figure 4.15 plots Pnc and H as functions of variances (i.e., Vm,s for slow MSs and Vm,f = Vm,s25 for fast MSs) for the microcell residence times, where the values for the input parameters except Vm,s and Vm,f follow the default values listed above. Figure 4.15 shows that
0
Pnc and H decrease as Vm,s increases. From the residual life theorem [42], the mean value of the first microcell residence time increases as Vm,s increases, which implies that more calls will complete in the first microcell before they are handed off to the next cells. Therefore, both Pnc and H drop as Vm,s increases.
4.5 Summary
By considering the moving speeds of MSs, this study proposed the repacking on demand (RoD) approach for channel assignment in the WiMAX systems with the hierarchical cell structure (i.e., the Hierarchical WiMAX Network; HWN). We developed simulation models to investigate the RoD performance on the blocking probability Pb, the force-termination probability Pf, the incomplete probability Pnc and the expected number of handoffs H during a call (for both slow and fast calls). We compared RoD with other repacking channel assignment approaches including No Repacking (NR), Always Repacking (AR) and Partial RoD (PRoD). Our study indicated that
• If the requests for the macrocell channels can not be satisfied (e.g., when the number of the macrocell channel is small, the call arrival rate and MS mobility are high, and so on), macrocell channels are the bottleneck resources. In this case, RoD significantly reduces Pb, Pf, and Pnc as compared with other approaches.
• The Pb, Pf, and Pnc performance for RoD is not sensitive to the proportion of fast call arrivals as compared with other approaches. That is, the increase of fast calls does not affect RoD as much as other approaches.
• With the same Pnc performance, RoD can support much faster MSs and/or more call arrivals than other approaches.
• In RoD, Random RoD (RoD-R) and Load Balancing RoD (RoD-L) have the same Pb, Pf, and Pnc performance. Note that in repacking, macrocell is a resource pool used to adjust traffic load of each microcell. That is, repacking already conducts the load balancing function to effectively balance the system workload, and the load-balancing improvement by RoD-L becomes insignificant. Therefore, the per-formance resulted from random selection for repacking candidates (i.e., RoD-R) is similar to that for RoD-L. This result is very important for network operators because RoD-R is much easier to implement than RoD-L.