3.5.1 Dual Priority MINs vs.Single Priority Ones
In this chapter we address the performance evaluation of the 2-class priority scheme for MINs, aiming to get insight on the e ects of each factor on the overall performance of this MIN class. In this section, we present our ndings and compare di erent con gurations of 2-class priority MINS; we also compare the performance metrics of 2-class priority MINs against the single-priority MIN class.
Figure 3.4: T htotal of nite-bu ered, 10-stage, dual- vs. single-priority MINs
Figure 3.4 illustrates the gains on total normalized throughput of a MIN using a 2- class priority scheme versus a single priority one. In the diagram, curve 2P[10]B[b]H[20] depicts the total normalized throughput of a 2-class priority, 10-stage MIN, under various bu er-length setups (b = 1; 2; 4), when the ratio of high priority packets is 20%. Similarly, curve 1P[10]B[b] shows the corresponding normalized throughput of a single priority, 10- stage MIN, under the same bu er-length setups (b = 1; 2; 4). In this gure, all curves represent the performance factor of normalized throughput at di erent o ered loads ( = 0:1; 0:1 · · · 1).
We can notice here that the gains on total normalized throughput of a 2-class priority scheme for a 10-stage MIN versus a single priority one are 23%, 12.6%, and 7.4%, when the bu er-lengths are 1, 2, and 4 respectively, under a high-priority appearance of 20%, and full load trac conditions. The throughput gains can be mainly attributed to the exploitation of the extra bu er spaces available in the SEs of the 2-class priority MINs: recall that SEs in single-bu ered MIN supporting one priority class have a single bu er available per incoming link; in single-bu er MINs supporting two priorities, however, SEs have one bu er for high-priority packets and one bu er for low-priority packets per input link. The normalized throughput of single-bu er MINs supporting two priorities appears though inferior to that of double-bu ered single-priority MINs in gure 3.4, because the extra bu er available in dual-priority MINs is exploited only for high-priority packets
(20% of the total trac), and thus remains unexploited when no high-priority packets are available. Contrary to that, double-bu ered single-priority MINs can exploit the extra bu er space for any packet, with no restriction whatsoever. In gure 3.4 we can nally notice that the input load at which the dual-priority MIN's performance starts to have an edge over its single-priority counterpart is smaller for single-bu ered MINs ( = 0:3) and smaller for double-bu ered ( = 0:5) and quad-bu ered MINs ( = 0:6). These loads correspond to the points where the probability that single-priority MIN bu ers are full (leading thus to packet blockings) exceeds a certain threshold, having therefore observable e ects.
Figure 3.5: RT h(h) of nite-bu ered, 10- stage, dual-priority MIN
Figure 3.6: RT h(l) of nite-bu ered, 10- stage, dual-priority MIN
Figure 3.5, depicts the metric of relative normalized throughput for high priority pack- ets in a MIN using the 2-class priority scheme and the (overall) relative normalized throughput for single-priority MINs. All measurements apply to a 10-stage MIN, and when packets of two priorities are considered, they account to 20% of the overall trac; measurements have been collected for bu er lengths b = 1; 2; 4.
It is worth noting that the relative normalized throughput of high priority packets is improved dramatically for all con guration setups, approaching the optimal value (T hmax = 1), especially when b >= 2, under full load trac conditions. Practically,
for b >= 2 and under the examined conditions, blockings events for high-priority packets were very rare.
Figure 3.6 illustrates the relative normalized throughput for low priority packets in a MIN using the 2-class priority scheme and the (overall) relative normalized throughput for single-priority MINs. Considering the performance curves for MIN pairs (dual-priority and single-priority) with equal bu er sizes (b = 1; 2; 4), we can identify three segments:
• An initial segment where the performance of single-priority MINs is identical to that of its dual-priority counterpart. This segment corresponds to the load range that
the available bu er space in the single-priority MIN is adequate, and blockings are mostly due to packets in the same SE contending for the same output link, rather than due to bu er unavailability at the next MIN stage.
• A middle segment, where the normalized throughput of the low-priority packets in the two-priority MIN is superior to the (overall) normalized throughput in a single- priority MIN. The beginning of this segment corresponds to the load point where blockings due to bu er unavailability begin to play a part in the MIN performance. In this segment, the gains obtained from the exploitation of the extra bu er space in the dual priority MINs is higher than the penalization incurred for low-priority packets, due to the fact that they yield to high-priority ones.
• An ending segment, where the normalized throughput of the low-priority packets in the two-priority MIN is inferior to the (overall) normalized throughput in a single- priority MIN. This corresponds to the load range where the yielding of low-priority packets incurs higher penalty than the gains obtained due to the availability of the extra bu er space. Especially at loads close to 1, bu er space for low-priority packets is already saturated and low-priority packets are further delayed because high-priority packets are preferred for transmission, when present.
In all cases, the maximum deterioration recorded is 15.6% for b = 1, 13% for b = 2 and 8.48% for b = 4. This deterioration can be considered as tolerable, especially considering the gains achieved for high-priority packets.
Figure 3.7: D(h) of nite-bu ered, 10-stage, dual-priority MIN
Figure 3.8: D(l) of nite-bu ered, 10-stage, dual-priority MIN
Figure 3.7 represents the corresponding decrements on normalized delays for high priority packets of 2-class priority scheme vs. single priority one for a 10-stage MIN,
under a rate appearance of 20% for high priority o ered loads. It noteworthy, that the improvement of high priority packet delays is considerable for all above bu er-length con gurations of MIN. It follows that normalized delay is reduced dramatically to D(h) = 1:07 · · · 1:09 approaching the optimal value Dmin = 1. It also follows that the minimization
of normalized delays for high-priority packets in a 2-class priority scheme is stronger at larger bu er-length con gurations, where the packet delays have greater values in the corresponding single priority MINs.
Figure 3.8 illustrates the normalized delay for low priority packets in a MIN using the 2-class priority scheme vs. single priority one. Similarly to the case of normalized throughput for low-priority packets, when examining the performance curves for MIN pairs (dual-priority and single-priority) with equal bu er sizes (b = 1; 2; 4), we can identify three segments respectively.
Figure 3.9: RT h(h) of nite-bu ered, k- stage, dual-priority MIN
Figure 3.10: RT h(l) of nite-bu ered, k- stage, dual-priority MIN
Figures 3.9 and 3.10 depict the relative normalized throughput for high and low priority packets respectively, in a k-stage MIN, where k = 6; 8; 10, using a 2-class priority scheme, under a packet appearance of 30% for high priority o ered load, and full trac conditions versus the bu er-length of MIN. A high-priority packet ratio of 30% was used in these diagrams, to make the e ects of the introduction of priority handling more discernible, especially for low priority packets (results for high-priority packets for the 20% ratio case are similar, showing only a slight improvement for b = 1). We noticed again that the relative normalized throughput of high priority packets is improved dramatically for all network size setups, approaching the optimal value (T hmax = 1), especially when
b >= 2. On the other hand, the loss of normalized throughput for corresponding low priority packets ranged from 9% to 24.6%, which is tolerable for all network size and
bu er-length con gurations.
We can also notice that the relative normalized throughput appears to drop as the number of MIN stages increases (for low-priority packets and for single-priority MINs): this happens because although the overall number of packets traversing the network in the unit of time increases along with the number of stages, this increment is less than the theoretical growth of the MIN routing capacity, which the de nition of the relative normalized throughput takes into account (recall that the normalized throughput metric divides the number of packets traversing the network in the unit of time by the network size, to express the extent to which the MIN's routing capacity is exploited). An equivalent reading of this phenomenon is that less packets per input source reach their destination per unit of time, when the MIN size increases. This performance degradation is due to the fact that each extra MIN stage introduces an additional point that blockings may occur, mainly due to contentions for the same output link. This is especially true under the full load condition considered in Figure 3.10, while for lighter MIN loads, the drop is less observable. MIN designers should take into account this fact when they need to upsize their network installations, and take additional actions if they want to maintain the throughput per input source; two prominent approaches are the super-linear increase of the network size (leaving some inputs unconnected) and the addition of extra bu er space in the SEs.
Figure 3.11: Upf(h) of nite-bu ered, 10- stage, dual-priority MIN
Figure 3.12: Upf(l) of nite-bu ered, 10- stage, dual-priority MIN
Finally, gures 3.11 and 3.12 illustrate the relation of the combined performance in- dicator Upf of a 2-class, 10-stage MIN to the o ered load , for high and low priority packets respectively, under di erent bu er size con gurations (b = 1; 2; 4), when the ratio of high priority o ered load is 20%. Recall from section 3.3, the combined performance
indicator Upf depicts the overall performance of a MIN, considering the weights of each individual performance factor (throughput and packet delay) are of equal importance. In gure 3.11 we notice that the value of the universal performance factor decreases (thus MIN performance is improved) when the bu er size increases, except for the case of single- priority MINs with b = 4 and operating under medium and high loads ( >= 0:6), in which case the universal performance factor deteriorates. This holds because the delay in these cases increases rapidly, while gains in the throughput are very small.
In gure 3.12 we can observe the behaviour of the universal performance factor for low priority packets in dual-priority MINs, and the (overall) universal performance factor for single-priority MINs when considering di erent o ered loads. Consistently with the respective ndings for normalized throughput and delay, three segments are identi ed when examining the performance curves for MIN pairs (dual-priority and single-priority) with equal bu er sizes (b = 1; 2; 4): an initial segment with identical performance among pairs, a middle segment where the dual-priority MIN outperforms the single-priority one and a nal segment where the dual-priority MIN lags behind the single-priority one. This is to be expected since the universal performance factor combines the individual metrics of normalized throughput and delay, and since these metrics exhibit a common behaviour, this behaviour is also exhibited in the combined metric.
3.5.2 Dual Priority MINs with Asymmetric-sized Bu er Queues
Figure 3.13: T htotal of asymmetric-sized, 10-stage, dual-priority MIN
In this subsection we introduce a variation of double-bu ered SEs that uses asymmetric bu er sizes in order to o er di erent quality-of-service parameters to packets that have di erent priorities, while providing in parallel optimal overall network performance. We
note here that the particular bu er sizes have been chosen since they have been reported at previous subsection to provide optimal overall network performance: indeed, it is observed that for smaller bu er sizes (1) the network throughput drops due to high blocking probabilities, whereas for higher bu er sizes (4 and 8) packet delay increases signi cantly (and the SE hardware cost also raises).
In gure 3.13, curves 1P[10]B[b] depict the normalized throughput of a 10-stage MIN, under a single priority mechanism, when the bu er-length is b = 2; 4. Similarly, curves 2P[10]B[bl,bh]H[20] show the total normalized throughput of a 10-stage MIN, under a
2-class priority mechanism, when the bu er-length of low and high priority packets is bl= 2; 3 and bh = 2; 1 respectively, and the probability of high priority packet appearance
is 20%. According to this gure the gain for total normalized throughput of a double- bu ered MIN, employing a 2-class priority mechanism (curve 2P[10]B[2,2]H[20]) vs. the corresponding single priority one (curve 1P[10]B[2]) is 12.6%, under full trac load. Con- sidering that the rate of high priority packets is relatively low, and con guring thus a asymmetric bu er-sized system (curve 2P[10]B[3,1]H[20]) the total normalized through- put is further improved 14.1%, approaching that of a single priority mechanism, when bu er-length is b = 4, where all bu ers serve all packets.
Figure 3.14: RT h(h) of asymmetric-sized, 10-stage, dual-priority MIN
Figure 3.15: RT h(l) of asymmetric-sized, 10- stage, dual-priority MIN
Figures 3.14 and 3.15 depict the relative normalized throughput of high and low priority packets respectively. According to gure 3.14 both curves employing the 2-class priority mechanism approach the optimal value T hmax = 1 of this performance factor. It is
obvious that, when the setup of bu er-length for high priority packets is bh = 2 (curve
2P[10]B[2,2]H[20]), the relative normalized throughput appears further improved, but the gains are marginal. Figure 3.15 presents the case of low-priority packet throughput; in
this gure we can observe that the relative normalized throughput of low priority packets is considerably better, when the setup of bu er-length for high priority packets is bh = 1
(curve 2P[10]B[3,1]H[20]), as compared to the case of having equal-size bu ers (curve 2P[10]B[2,2]H[20]), for high and low priority packets. The performance di erence between the two setups is approximately 20% for medium and high network loads ( >= 0:5). We can also observe that the asymmetric-sized bu er setup o ers superior service to the low-priority packets as compared to the single-priority scheme, mainly owing to the one additional bu er position available in the asymmetric setup to packets of this class. The performance improvement appears for medium and high network loads ( >= 0:5) and ranges from 8% to 21%.
Figure 3.16: D(h) of asymmetric-sized, 10- stage, dual-priority MIN
Figure 3.17: D(l) of asymmetric-sized, 10- stage, dual-priority MIN
Figures 3.16 and 3.17 present the ndings for the normalized delay performance metric. In gure 3.16 we observe that both 2-priority schemes (i.e. the equal-sized bu er and the asymmetric-sized bu er scheme) have a clear edge over the single-priority mechanism, which ranges from 18% at 30% load to over 96% at full load. The di erence however between the performance of the equal-sized bu er scheme and the asymmetric-sized bu er scheme is very small, less than 4% in all cases. Conversely, when low priority packets are considered ( gure 3.17), the equal-sized bu er scheme is found to have delays close to the single-priority scheme, with the worst case being a deterioration of 6.7% at o ered load = 1. In the asymmetric-sized bu er setup however, the deterioration is considerable, especially at high loads (13% at = 0:6 rising up to 24.4% as compared to the equal-sized bu er setup at = 1).
Figures 3.18 and 3.19 depict the behavior of the universal performance factor metric for high- and low-priority packets, respectively, in correlation to the o ered load. We can
Figure 3.18: Upf(h) of asymmetric-sized, 10- stage, dual-priority MIN
Figure 3.19: Upf(l) of asymmetric-sized, 10- stage, dual-priority MIN
observe in gure 3.18 that when the load of the network is relatively low ( <= 0:4), all con gurations have identical performance; however, when the network load increases, the overall performance of the single-priority con guration quickly deteriorates, as compared to the setups supporting two priorities. The asymmetric-sized bu er con guration shows almost identical performance to the equal-size bu er con guration in this case, and both these performances are close to the optimal one.
Regarding low-priority packets, again the overall performance of all con gurations is identical for light network loads ( <= 0:4). Beyond this point, the single-priority setup exhibits the most stable behaviour, with the value of the universal performance factor for low-priority packets Upf(l) being close to 1.5; the single-priority setup has a clear advantage over the dual-priority schemes for o ered loads >= 0:7.
The con gurations supporting two priorities exhibit a wider performance uctuation, with the asymmetric-sized bu er con guration having a performance edge for network loads between 0.5 and 0.6, while for network loads >= 0:7 the performance advantage moves to the equal-size bu er con guration side, not exceeding though 5.5% in any case.
3.6 Conclusions for Dual Priority MINs
In this chapter we have addressed the performance evaluation of dual priority MINs. We have presented an analytical model for their operation, employing a scheme that takes into account both the previous and the last state of the switching elements, providing thus better accuracy than schemes considering only the last state. We have also evaluated the
performance of 2-class priority MINS under varying o ered loads and bu er sizes, consid- ering the high-priority and low-priority packet classes, as well as cumulative performance for the MIN, and compared these metrics against the corresponding performance gures of single-priority MINs. In this chapter, we have taken into account the two most im- portant network performance metrics namely throughput and packet delay. The diagrams and discussions given may be used by network designer to tune parameters for their in- stallations so as to obtain optimal performance for the communication requirements of their environments.
Moreover we have introduced an asymmetric bu er size con guration for MINs sup- porting two packet priority classes and compared its performance against both the single- priority scheme and the equal-sized bu er con guration of two packet priority classes MINs under di erent trac loads. The asymmetric-sized bu er con guration has been found to better exploit network resources and capacity, since available bu ers are more t- tingly allocated to the priority class needing them. More speci cally, when comparing the asymmetric bu er size con guration against its equal-sized bu er counterpart, we found that the former provides better overall throughput and signi cantly better low-priority packet throughput; for high-priority packets on the other hand, the performance of the two schemes is almost identical, with the equal-sized bu er scheme having a small edge.