In this subsection, we describe the source routing mechanism for choosing a best one from feasible paths in the Node. Based on the Maximum Flow Algorithm, in the beginning for each link the value λ1 (we mentioned above) is used as the maximum amount of calls that can flow through it per unit time in the steady-state situation.
Then we can find the maximum flow, a maximum capacity flow augmenting path, and other feasible flow paths. If any link broadcasts its routing information and changes the maximum amount of calls that can flow through it per unit time now, the maximum flow and flow paths are immediately computed again.
Before the algorithm, we denote some parameters below for discussion later:
rs: the source traffic rate.
fbs: the flow rate that a call blocked in the source. That if it is not zero is meant to block a call according to fbs / rs at source nodes. Here we do not hope to block any calls at source nodes instead of delivering calls through the shortest path.
The reason we choose the shortest path under this situation is that choosing one from feasible flow paths results in occupying resources on more links.
fmax: the total maximum flow.
cmax: the maximum capacity of a maximum capacity flow augmenting path.
randNr: a random number which the value is between 0 and 1.
The routing algorithm we proposed is below:
if ( rs > fmax ) fbs = rs – fmax ; else fbs = 0 ;
if ( randNr ≧ fbs / rs ) {
if ( rs ≦ cmax ) choose the maximum capacity flow augmenting path;
else choose one from all feasible paths randomly ; }
else choose the shortest path ;
Now we illustrate the operation of our algorithm. First, if the source traffic rate is greater than the total maximum flow, it shows that the network is not capable of this source traffic and it is preferred to block the exceeded amount of source traffic in the source. Thus we take fbs / rs as the average blocking probability in the source on arriving a new call. Second, if the call is not blocked in the source, the maximum capacity flow augmenting path is checked if it will be used. That rs ≦ cmax means the maximum capacity flow augmenting path can still load the source traffic, or we have
to randomly choose one path from all feasible ones while a new call is originated.
This algorithm is represented as a flow chart as follows:
Generates calls
Check if there is any traffic blocked at source
?
( Yes ) fbs = rs - fmax
( No ) fbs = 0
Generates a random number randNr :[0,1]
Choose path ? ( If randomNr <fbs / rs )
choose shortest path
Choose flow augmenting path ?
( else ) ( If rs <= cmax) choose maximum capacity
flow augmenting path
( else ) Choose one path from all feasible flow paths
randomly
Figure 4.2. Flow chart of the Maxflow source routing algorithm
Chapter 5
Numerical Results
In this chapter, performance of Maxflow source routing with various call arrival rates is evaluated via simulations and compared with performance of shortest path routing. Also the communication overhead of Maxflow source routing is computed for scalable test.
Here we take a simple example of a small network. Figure 5.1 shows the network example for our simulation.
0 2
7 8
6
1 5 4 3
3 4 Node
5 Node Node
0
Node Node 1 Node 2
( )
Figure 5.1. An example of a simple network
There are six nodes and nine links in this network example that is not fully connected.
The Nodes are the notions of “computer” above, and the links are black lines with a number ID used for the ID of trunk group. Each node with any other node will form a origin-destination pair for transmitting calls. In our simulation, the capacity of each link is 30 trunks and the network only support one class of traffic, which bandwidth requirement is one trunk. Despite of Node 5, any other nodes are attempted to carry calls with some arrival rate to Node 5. Thus all OD pairs in this situation are as follows:
( ) ( ) ( ) ( ) ( )
origin, destination : 0,5 , 1, 5 , 2, 5 , 3, 5 , 4, 5
In other words, Node 5 is a destination node but not a source node. That the arrival rate of traffic is assumed the same at all source nodes in one experiment is designed to
verify experiments under different arrival rates. The rates we chosen after are 15, 17.5, 20, 22.5, 25, 27.5 and 30.
Table 5.2 shows the result under different arrival rates when shortest path routing is used at each source node to elect minimum-hop routes for calls. This table shows that in this situation there are three critical links, which are Link 1, 5, 8. Among these critical links, the blocking probability of calls on Link 5 is very higher than ones of calls on Link 1 and Link 8. The reason for the higher probability is that Node 1, 2, and Node 3 all attempt to carry their traffic streams into Link 5 since the shortest path routing is used. Link 1 and Link 8 are loaded traffic streams of Node 0 and Node 4 respectively. (The shortest path of the OD pair (2, 5) is two-link path including Link 2 and Link 5. Also the shortest path of the OD pair (3, 5) is two-link path including Link 3 and Link 5.) This phenomenon is to explain why the shortest path routing is not always useful. With shortest path routing algorithm it is possible for carrying too many calls on some critical link to increase the blocking probabilities of the link and overall network.
In our simulation we design two types of the congestion level. They are as follows in Table 5.1.
Cases
Table 5.1 Types of congestion level.
Table 5.3 shows one result for Case I under different arrival rates when Maxflow source routing is applied at each source node to elect a suitable path for calls under different network routing information. This Table shows that for overall network using Maxflow source routing has less blocking probability than using shortest path
routing under any different arrival rates. Although with Maxflow source routing Link 1 and Link 8 have higher blocking probability than ones with shortest path routing, using Maxflow source routing make Link 5 lower blocking than using shortest path routing. This fully explains that originally with shortest path routing a portion of calls from Link 5 changes to Link 1 or Link 8 since Maxflow source routing is used. It is desired for decreasing overall network blocking probability as low as possible.
It is obvious in our experiment that for the compare of blocking probability on Link 5 under traffic rate 20 and 22.5 or traffic rate 17.5 and 20. Especially the difference of the blocking probability on Link 5 for traffic rate 17.5 and 20 exceeds twenty percent. Besides, the parameter for advancing or reducing threshold is designed as 0.1, i.e., if in a length of time interval the blocking probability of the link is greater than 0.1, the threshold is advanced one until threshold is equal to 30, and threshold is reduced one if the probability is equal or lower than 0.1. This design is not clear under traffic rate above 20 because the rate is too high to overload some links still.
Table 5.4 and Table 5.5 show other results for Case II and Case III under different arrival rates when Maxflow source routing is applied at each source node to elect a suitable path for calls under different network routing information. Equally, the designs of congestion level for Case II and for Case III decrease overall network blocking probability compared with one of shortest path routing. The difference among link probabilities between Case I and Case II is that for Case II the blocking rate of Link 5 is more reduced and the blocking rates of Link 1 and Link 8 are higher.
The behavior of Case III is like the one of Case I because they are only different in the design of the low congestion level.
Figure 5.2 shows that the compare for overall network blocking probability under various traffic arrival rates between shortest path routing and Maxflow source routing (Case I, Case II, and Case III). It is clear that the network blocking probability of shortest path routing decreases as the arrival rate decreases almost with a fixed slope. For Case I, the network blocking probability of Maxflow source routing from traffic rate 30 to 22.5 is like the behavior of shortest path routing with almost the same slope but with a lower blocking rate. Then from traffic rate 20 to 15, the network blocking probability decreases rapidly at a very oblique angle. Our illustration is that for traffic rate above 20, it makes critical links congested frequently and thus source nodes have higher possibility to choose shortest paths. For traffic rate equal to or below 20, Maxflow source routing works well because source nodes have
less possibility to choose shortest paths. Particularly the differences of network blocking between Maxflow source routing and shortest path routing under traffic rates 17.5 and 15 are twenty percent and upward.
For Case II and Case III, the behaviors of overall network blocking probability are similar to the one of Case I. Some difference between them is the following. The blocking probabilities of Case II from traffic rate 25 to rate 30 are all a little higher than ones of Case I. The reason is that the high congestion level of Case II is higher than the one of Case I. It makes congested links accepting more traffic streams continuously.
We run our simulation under various traffic rates according to different multipliers and observe that blocking probabilities of links and network are independent of different multipliers over a long period of time. This observation shows that the long-term behaviors of network and links will converge toward the same result although the initial multiplier value is different.
Finally, Table 5.6 shows the computation of link update overheads for scalability test. Here we list average overhead value and maximum deviation overhead value according to three cases under various arrival rates. First, we can see that the average values for traffic rates 15, 17.5 and 20 are higher than others. Especially, the value for traffic rate 17.5 is almost equal to one, and this means that the link routing information will be updated one time per second averagely. Luckily it is acceptable since the link will equally broadcast its link information 0.33 times per second for that total overheads is distributed into three critical links. The reason of smaller value for traffic rate from 22.5 to 30 is that critical links under these traffic rates frequently attempt to notify source nodes that it is in the high congestion level. Thus in this situation the communication overhead will be lightened. Second, for Case III the value of link update under traffic rate 17.5 is highest. If the value 1.6905 is distributed into three critical links, each critical link floods its link routing information to overall network every two seconds almost. Thus it shows an acceptable situation. In the low-congested traffic condition, it remains a higher overhead value, and in the high-congested traffic condition, it decreases to a lower overhead value.
Arrival rate Blocking (%)
15 17.5 20 22.5 25 27.5 30
Link 0 0 0 0 0 0 0 0
Link 1 0 0.0008 0.0053 0.0143 0.0357 0.0710 0.0968
Link 2 0 0 0 0 0 0 0
Link 3 0 0 0 0 0 0 0
Link 4 0 0 0 0 0 0 0
Link 5 0.3526 0.4309 0.4995 0.5457 0.5835 0.6261 0.6508
Link 6 0 0 0 0 0 0 0
Link 7 0 0 0 0 0 0 0
Link 8 0 0.0018 0.0035 0.0192 0.0302 0.0818 0.0897
Network 0.2115 0.2591 0.3017 0.3341 0.3633 0.4063 0.4278
Table 5.2. Blocking probabilities of shortest path routing under various arrival rates.
Arrival rate Blocking (%)
15 17.5 20 22.5 25 27.5 30
Link 0 0 0 0.0798 0.0942 0.0972 0.0988 0.0999
Link 1 0.0073 0.0592 0.0956 0.1177 0.1873 0.2534 0.3178
Link 2 0 0 0.0261 0.0052 0.0082 0.0090 0.0132
Link 3 0.0007 0.0021 0 0 0 0 0
Link 4 0 0 0.0002 0 0 0 0.0003
Link 5 0.0063 0.0537 0.2685 0.4310 0.4547 0.4903 0.5324
Link 6 0.0038 0.0241 0.0131 0.0006 0.0015 0.0002 0.0002
Link 7 0 0 0 0 0 0 0
Link 8 0.0084 0.0524 0.0997 0.1040 0.1473 0.1672 0.2074
Network 0.0083 0.0581 0.1853 0.2730 0.3107 0.3491 0.3964
Table 5.3. Blocking probabilities of Maxflow source routing using Case I under various arrival rates.
Arrival rate Blocking (%)
15 17.5 20 22.5 25 27.5 30
Link 0 0 0.0046 0.0311 0.1528 0.1993 0.2174 0.2559
Link 1 0.0053 0.0696 0.0992 0.2912 0.3464 0.4103 0.4714
Link 2 0 0.0153 0.0544 0.0961 0.0968 0.1004 0.1019
Link 3 0.0016 0.0015 0 0 0.0043 0.0003 0.0029
Link 4 0 0 0 0 0 0 0
Link 5 0.0106 0.0784 0.1934 0.1149 0.1248 0.1614 0.1870
Link 6 0.0036 0.0739 0.0997 0.0956 0.0962 0.0989 0.1044
Link 7 0 0 0 0.0096 0.0038 0.0024 0.0029
Link 8 0.0153 0.0922 0.0975 0.2419 0.3098 0.3731 0.4323
Network 0.0120 0.0962 0.1642 0.2722 0.3180 0.3696 0.4201
Table 5.4. Blocking probabilities of Maxflow source routing using Case II under various arrival rates
Arrival rate Blocking (%)
15 17.5 20 22.5 25 27.5 30
Link 0 0 0 0.0867 0.0932 0.0980 0.0982 0.1003
Link 1 0.0054 0.0513 0.0990 0.1137 0.1967 0.2912 0.3358
Link 2 0 0.0002 0.0048 0.0081 0.0187 0.0131 0.0244
Link 3 0.0003 0.0009 0.0039 0.0091 0.0301 0.0327 0.0494
Link 4 0 0 0.0002 0 0 0.0007 0
Link 5 0.0109 0.0788 0.3136 0.4087 0.4466 0.5057 0.5306
Link 6 0 0 0 0 0 0 0
Link 7 0 0 0 0 0 0 0
Link 8 0.0036 0.0407 0.0971 0.1005 0.1231 0.1768 0.2143
Network 0.0063 0.0565 0.2017 0.2614 0.3086 0.3710 0.4046
Table 5.5. Blocking probabilities of Maxflow source routing using Case III under various arrival rates
0.0581
Maxflow Source Routing - Case I Maxflow Source Routing - Case II Maxflow Source Routing - Case III
Figure 5.2. Network Blocking Comparison for shortest path routing and Maxflow source routing
Arrival Rate Case I Case II Case III
Average (/sec) Deviation (/sec)
Table 5.6. Computation of Link Update Overheads
Chapter 6 Conclusion
State-dependent routing is preferred for the real-time control of network performance in the past. It needs to use the network state information in order to select a route for a call. In this work, we study various routing strategies including shortest-path routing, QoS routing, and other state-dependent routing strategies with different objectives and ideas. Based on the above learning and other background knowledge, we propose an aggregation scheme and a source-based routing algorithm to provide network scalability.
In our simulation we run the MDP function on links to aggregate routing information for providing network scalability and design the Maxflow source routing algorithm as the source routing algorithm used at source nodes. The MDP function is used for partitioning the congestion level sets of each link. The Maxflow source routing algorithm chooses the appropriate routes according to above-mentioned congestion information of each link. Based on these ideas, we can decrease the network probability that results from shortest path routing greatly and reduce the number which links flood their routing information into overall network as few as possible. As the simulation result shown, the Maxflow source routing is superior to shortest-path routing.
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