Chapter 3 Localized State-Dependent Routing
3.6 Boosting with Dynamic Weights Adjusted by the Delay Time of State
After source S receives the RESV message of RSVP, source S records the moment of message feedback from path i, i.e. t0[i]. The delay time,dt , of a new call attempt at time t is [i]
Different from the previous two methods which choose one of these two routing approaches to select one best path, we non-linearly combine MDP Routing and Proportional Routing by the exponentially decayed weights. We introduce this method in detail as follow:
For path i, i=1,2,Κ ,K, we calculate the path-combine-proportion pcp , using the [i]
where γ is a configurable parameter, and
fp is the flow-proportion of path i using VCR algorithm in proportional routing,
And then we normalize the path-combine-proportion pcp for path i, [i] i=1,2,Κ,K
When the average load of source is heavy, the state-information is fresh because the update interval (delay-time informationdt ) of the source is short. When the state information [i] is fresh, we have better to choose the MDP Routing scheme to find the best path. On the other hand, if the state information is delayed and outdated, we need to use the Proportional Routing scheme. Therefore, we have to multiply (1−dc[i]) by e −γ×dt [ i] and multiply fp by [i]
]
1 − e
−γ×dt[i in order to cooperate these two schemes adaptively and choose one path proportionally. So we use the different weight with exponentially decayed for MDP routing and proportional routing. And by sending RESV message which additionally contains the state information back to source, source obtains the local network information feedback from the routers, and therefore we do not increase the communication overhead.In the summary, we illustrate our methods by the following block diagram.
Fig. 7 Boosting process
Chapter 4
Simulation Results
In the following we illustrate a simple network topology how our scheme works better than adaptive proportional routing when the load of source is varying with time.
Consider the fish topology shown in Fig. 8.
Fig. 8 Fish topology
The nodes 1, 2, 3, and 4 are the source nodes and node 12 is the destination node. Each of node 1 and node 2 has two min-hop paths (1Æ5Æ6Æ12, 1Æ5Æ7Æ12 and 2Æ5Æ6Æ12, 2Æ5Æ7Æ12) and two alternative paths (1Æ5Æ8Æ9Æ12, 1Æ5Æ10Æ11Æ12 and 2Æ5Æ8Æ9Æ12, 2Æ5Æ10Æ11Æ12) to the destination node 12. Other two sources, nodes 3 and node 4, have just one min-hop path (3Æ8Æ9Æ12 and 4Æ10Æ11Æ12) to the destination node 12.The alternative path of source node1 and node 2 share the bottleneck link 9 Æ 12 and 11Æ 12 with the min-hop paths of source node 3 and node4. We assume that the capacities c1, c2, c3, and c4 of the bottleneck links are all set to 80 units of bandwidth and others are infinite bandwidth.
The follow are our simulator design for this fish topology. At first, the Object Model Diagram in UML is illustrated in Fig. 9 below. There are four Source objects constructed by system in our simulation model. Source S1 and source S2 have four feasible paths and source S3
and source S4 have only one feasible path.
Fig. 9 Object Model Diagram in UML
4.1 Comparison of Proportional Routing and MDP Routing
Before displaying the simulation performance of our proposed routing, we first compare these two routing approaches “State-Dependent Separable Routing formulated by MDP
“(called MDP routing for shorted) and “Adaptive Proportional Routing” (called Proportional routing for shorted). We observe the blocking probability of source S1 by increasing the update interval of S1 when the average load of all sources is set to 40.
0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5
update interval
blocking probabil ity
MDP_Routing
Proportional Routing
Fig. 10 Comparison of Proportional Routing and MDP Routing
In Fig. 10, we find that the performance of MDP Routing degrades rapidly as the update interval increases. As for the scheme of Proportional Routing, the blocking probability of S1 is keeping at about 0.3 and is even better than the scheme of MDP Routing when its update interval is longer than one minute. It is because that the proportional routing adopts the average available bandwidth information instead of stale state information to make path selection.
From the observation in Fig. 10, we proposed three boosting methods, which are mentioned at Chapter 3 in detail. Before we illustrate how these three routing methods adaptively adjust the load proportion of feasible paths as the load increases, we firstly introduce some parameters in our simulator.
4.2 Simulator Parameters Setting
We assume that the average offered load of S1, S2, S3, and S4 are 40, 40, 5, and 5 respectively in the beginning. And we set the “Update Proportional Interval” as 10 minutes,
“Update Lambda Interval” as 1 minute and “Update Weight Interval” as 1 minute. For source S1, we set the initial flow proportion of path 1, 2, 3, and 4 to 0.4999, 0.4999, 0.0001, and 0.0001 respectively. So does source S2. And we set the constant parameter γ in Eq.(3.2) to 0.01.
Consider the scenario: when the number of calls generated by source S1 is equal to 1000, we increase the average load of S3 and S4 to 20. And when the number of calls generated by source S1 is equal to 3000, we increase the average load of S3 and S4 to 40. By increasing the average load of S3 and S4, we study how source S1 and S2 adjust their flow proportion on the feasible paths in order to decrease the overall blocking probability.
The following table is the setting of parameters in our simulator.
Parameter Initial value
Path1_Flow-proportion of S1 and S2 0.499 Path2_Flow-proportion of S1 and S2 0.499 Path3_Flow-proportion of S1 and S2 0.001 Path4_Flow-proportion of S1 and S2 0.001 Average Load of S1 and S2 40 calls/min
Average Load of S3 and S4 5 calls/min Link Capacity (c1 , c2 , c3, c4) 80 calls
psi (ψ ) 0.95
Update Proportional Interval 10 minutes
Update Lambda Interval 1 min
Update Weight Interval 1 min
Total Arrival of Source1 5000 calls
Gamma(γ ) 0.01
Table .1 Setting of parameters in our simulator
In the following sections, we will introduce our simulator for these three methods and their simulation results.
4.3 Simulator Design and Results for BFW
In this method, we assign fix weight δ to Proportional Routing and fix weight 1−δ to MDP Routing. We define this method as “Boosting with Fixed Weight” (BFW for shorted).
4.3.1 System Operations
Fig. 11 System diagram of BFW in UML
In Fig.11, there are four sub-states. In System_Timer sub-state, timer is running and the parameter systemTime plus one per second. And in Update_Path_Proportion sub-state, system calculates the total arrival of sources during the “Update Proportion Interval”, and triggers the event evUpdate1 of sources to calculate the new path flow proportion ( fp ) by VCR [⋅] algorithm. And in Update_Path_Arrival_Rate sub-state, system triggers the event evUpdate2 of sources to calculate the path arrival rate (λi) during the “Update Lambda Interval”. Finally, system can change the arrival rate of source S3 and S4 in the Change_S3S4_Arrival_Rate sub-state.
4.3.2 Source Operations
Fig. 12 Source diagram of BFW in UML
In Fig. 12, there are three sub-states. In the gen_call sub-state, source generates calls by Poisson arrival process. Each call performs callArrival() to find a path by Path-Selection-Process based on the value of δ , path flow-proportion ( fp ), and path [⋅] delta-cost (dc ). And then source uses RSVP to setup this call along the selected path. When [⋅]
the number of call generated by source equals to Total_Arrival, source stops generating new call. In Update_Path_Proportion sub-state, the event evUpdate1 is triggered from system and source updates the path flow-proportion ( fp[⋅] ) by VCR algorithm. At last, in Update_Path_Arrival_Rate sub-state, the event evUpdate2 is triggered from system and source updates the path arrival rate (λi) in order to calculate the path-delta-cost, (dc ) in Eq. (2.21). [⋅]
4.3.3 Path Operations
Fig. 13 Path diagram of BFW in UML
In Fig. 13, it demonstrates that the selected path accepts this call which consumes one unit of bandwidth (count plus one) and triggers one server object. At the same time, the selected path has to inform source, which generates this call, the latest state information by RESV message.
4.3.4 Server Operations
In Fig. 14, it describes that call holding time is exponentially distributed with meanµ. After the serviceTime, this call stops serving and the server object will be terminated.
Fig. 14 Server diagram of BFW in UML
In summary, our simulator model could be illustrated in the sequence diagram below.
Fig. 15 Sequence diagram for BFW in UML
4.3.5 Simulation Result
Comparsion of BFW with different weight
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 10 20 30 40 50 60 70 80 90 100 110 120
time
blocking probability
MDP Routing without delay MDP Routing with delay Proportional Routing BFW(0.25)
BFW(0.5) BFW(0.75)
Fig. 16 Comparison of BFW with different weight (δ )
In Fig.16, the scheme of MDP Routing without delay is nearly optimal because sources obtain instantaneous network state information. We can find that when the average load of S3
and S4 are low, the performance of Proportional Routing is worst. As the average load of S3
and S4 are increasing, the performance of MDP Routing with delayed information gets worse.
When the average loads of S3 and S4 increase to 40, the performance of MDP Routing with delayed information gets worse than Proportional Routing. Consider our proposed method BFW with different value of δ which is the weight of using Proportional Routing to select one path. No matter what the value δis, the performance of BFW is better than Proportional Routing and is better than MDP Routing with delayed information when the load is heavy.
4.4 Simulator Design and Results for BDW-BP
In this method, we dynamically boost the weight adjusted by the observed blocking probability during a fixed time interval. We define this method as “Boosting with Dynamic Weights Adjusted by Blocking Probability” (BDW-BP for shorted). The server operation in UML is the same as previous method.
4.4.1 System Operations
In order to update the new weights of the new Update Weight Interval, we have to calculate the blocking probability of calls using Proportional Routing and calls using MDP Routing in the Update_Blocking_Probability sub-state in Fig. 17.
Fig. 17 System diagram of BDW-BP in UML
4.4.2 Source Operations
Fig. 18 Source Diagram of BDW-BP in UML
Different from the source operations in BFW, when the event evUpdate3 is triggered, source has to calculate the blocking probability of calls using Proportional Routing and calls using MDP Routing in order to find out the new weights using Eq. (3.1a) and Eq. (3.1b).
In “ n-th ” Update Weight Interval:
New weight for using Proportional Routing:
(3.1a)
New weight for using MDP Routing:
(3.1b)
And then source chooses one path proportionally based on the new weightδ(n), path flow-proportion (fp ), and path delta-cost ([⋅] dc ). [⋅]
4.4.3 Path Operations
Fig. 19 Path Diagram of BDW-BP in UML
Different from path operations in BFW, path has to check the blocked call which is using Proportional Routing or using MDP Routing, and then informs source that generated this blocked call.
In summary, our simulator model could be illustrated in the sequence diagram below.
Fig. 20 Sequence diagram for BDW-BP in UML
4.4.4 Simulation Result
Comparison of BDW-BP
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 10 20 30 40 50 60 70 80 90 100 110 120
time
blocking probability
MDP Routing without delay MDP Routing with delay
Proportional Routing BDW-BP
Fig. 21 Comparison of BDW-BP
The performances of “MDP Routing without delay”, “MDP Routing with delay” and
“Proportional Routing” are the same as the results in Fig. 16. Consider the performance of BDW-BP, we find that the performance is comparable with the MDP Routing with delayed information, but is worse than the performance of Proportional Routing when the load is heavy.
4.5 Simulator Design and Results for BDW-DT
In this method, we dynamically boost the weight adjusted by the delay time of state information. We define this method as “Boosting with Dynamic Weight Adjusted by the Delay Time of State Information” (BDW-DT for shorted). The system operations and server operations in UML are the same as BFW.
4.5.1 Source Operations
The source diagram in UML is the same as the source operations in BFW. For each new call arrival, source has to calculate the delay-time (dt ) for all feasible paths and [⋅] proportionally selects one path using Eq. (3.2).
For path i,i=1,2,Κ,K, we calculate the path-combine-proportion pcp [i]
]) [ 1
( ]
[ )
1 ( ]
[i e [ ] fp i e [ ] dc i
pcp = − −γ⋅dt i ⋅ + −γ⋅dt i ⋅ − (3.2)
where γ is a configurable parameter, and ]
[i
dt is the delay-time of this call for path i, ]
[i
dc is the delta-cost of this call for path i using Eq .(2.21) in MDP routing ]
[i
fp is the flow-proportion of path i using VCR algorithm in proportional routing,
4.5.2 Path Operations
Fig. 22 Path diagram of BDW-DT in UML
No matter the call is accepted or blocked, path not only feedbacks the state information, but also asks source to record the ack_time[] for this path in order to calculate the delay-time of state information.
4.5.3 Simulation Result
Comparison of BDW-DT with different gamma
MDP Routing without delay MDP Routing with delay
Proportional Routing BDW-DT(1)
BDW-DT(0.1) BDW-DT(0.01)
Fig. 23 Comparison of BDW-DT with different gamma (γ )
In Fig. 23, the performances of “MDP Routing without delay”, “MDP Routing with delay” and “Proportional Routing” are the same as the results in Fig. 16. By changing the value of γ in Eq. (3.2), we find that the performances of “BDW-DT (0.1)” and “BDW-DT (0.01)”
are better than “BDW-DT (1)”, and are even better than “MDP Routing with delay” when the load is heavy. Consider Eq.(3.2).
When the load is heavy, the link state information is fresh enough. So we increase the weight
] [i
e
−γ⋅dt on “1−dc[i]” and decrease the weight( 1 − e
−γ⋅dt[i])
on fp[i]. On the other hands, the link state information may be delayed, i.e. dt[i] is long. So the biggere
−γ⋅dt ][i compensates the uncertainty of “1−dc[i]”. No matter what the value of dt it is, we [i] adaptively proportion flows in any situation using Eq. (3.2).flow proportion of BDW-DT(0.01)
0 0.2 0.4 0.6 0.8 1 1.2
0 10 20 30 40 50 60 70 80 90 100 110 120
time
fp_minhop fp_alternative
Fig. 24 Flow proportion of BDW-DT (gamma=0.01)
In Fig. 24, it illustrates that the flow proportion of min-hop paths and alternative paths. In the beginning, the flow proportion of min-hop paths is 0.998 and the flow proportion of alternative path is 0.002. With the increasing load of S3 and S4, the flow proportional of alternative paths increases because the blocking probability of min-hop paths is higher than alternative paths. And when the blocking probability of alternative path is increasing with the increasing load of S3 and S4, the system starts decreasing the proportion of alternative in order to equalize the overall blocking rate.
Comparison of BDW-DT with different gamma
0 0.05 0.1 0.15 0.2 0.25 0.3
0 10 20 30 40 50 60 70 80 90 100 110 120
time
blocking probability
BDW-DT(1) BDW-DT(0.1) BDW-DT(0.2) BDW-DT(0.3)
Fig. 25 Comparison of BDW-DT with different gamma (γ )
In Fig .25, we compare four scenarios where gamma (γ ) in Eq. (3.2) is equal to 1, 0.1, 0.2, and 0.3, respectively. From the simulation results, we find that the performance is not necessarily getting better if we decreaseγ . We consider that the performance is not only related toγ but also to the load of the network.
Finally, we compare our proposed three methods with MDP Routing and Proportional Routing in Fig. 26. We find that these three methods indeed boost MDP Routing and Proportional Routing in some network situation.
Comparison of Different Methods
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0 10 20 30 40 50 60 70 80 90 100 110 120
time
blocking probability
MDP Routing without delay MDP Routing with delay Proportional Routing BFW(delta= 0.5)
BDW-BP BDW-DT(gamma = 0.1)
Fig. 26 Comparison of all methods
Chapter 5 Conclusion
In this thesis, we first compare the state-dependent separable routing and adaptive proportional routing with different update interval and find that the latter works significantly worse than the former if the network state information obtained in the source is delayed not more than one minute. So we propose three possible boosting methods to compensate the shortcoming of proportional routing and state-dependent routing. As shown in Fig. 26, when the load is light, the performance of BDW-BP is better than other boosting methods. But as the load increases, we find that the performance of BFW is getting better and even better than MDP Routing with delayed state information when the load is heavy. Therefore, these three boosting methods are load-dependent and none of them can outperform others in any network situations.
Our future work will be conducting the simulation on the more complicated network model where the call arrival rates for most of the source-destination pairs are less frequent. We believe in such a model the network state tends to be more obsolete whereby configuring the weight on the state-dependent separable routing will become a more important issue.
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