6.5 Realistic schemes
6.5.3 Scheme 3: A client consumed fixed bandwidth for complete
Table 6.23: Simulation Result: Realistic Scheme 3
Grouped Approximate Solu- tion (group size = 20)
Session Approximate Solu- tion
Simple Heuristic
Total Income 2,242,191 1,962,629 1,662,017
Percentage change in total income as compared to total income in the simple
heuristic solution
+35% +18% -
Total delay 4,102,986 10,140,375 9,493,557
Percentage change in total delay as compared to total delay in the simple
heuristic solution
-57% +7% -
Total rejections 194,573 209,716 284,293
Percentage change in total rejection as compared to total rejection in the simple
heuristic solution
-32% -26% -
6.5.3 Scheme 3: A client consumed fixed bandwidth for com-
Conclusion and Future Work
The internet has revolutionized the working and living of people. Clients want to have a good, reliable, and predictable internet service. The number of people with internet access and the internet access speeds are different in different parts of the world. Many regions of the world face shortage of bandwidth. We have presented a scheme and solutions that contains assurance from service providers to clients regarding good internet service and if clients do not get assured service, service providers pay penalties to clients. As in such schemes with penalties, a side effect is to balance load among service providers.
7.1 Conclusion
Initially we explored the problem using game theory. We developed a game theoretic solution based on obtaining Nash equilibrium. This solution assumes that there are just two ISPs and each client can request just one unit of bandwidth. The algorithmic complexities of the solutions is shown in Table 7.1. We have compared the two solutions by running them and observed the expected income when both the service providers used the accurate solution and when both the service providers used the approximate solutions. There were six comparison with multiple variations of bandwidth. The result shows that when both the service providers used the approximate Nash equilibrium instead of the accurate Nash equilibrium solution, the performance decline is at most 15% when the congestion is high to almost no difference when the congestion is low.
The Nash equilibrium solutions have many limitations. These have high time and space complexities. In addition to that, if the same method to find the Nash equilibrium is extended for more than two service providers, the complexities increase
Table 7.1: Complexities of our Nash Equilibrium solutions
Size of solution Time complexity Space complexity Accurate Nash
Equilibrium
O(mmax2) Best case: O(
mmax7.12×mmaxT×mmax)
Worst case: O(
mmax7.12×mmax2×T×mmax)
O(mmax6)
Approximate Nash Equilib- rium
O(mmax) Best case: O((mmax)3+T)
Worst case:
O((mmax)3+2×T)
O(mmax)2
exponentially. We do not have a proof that a Nash equilibrium will always exist.
Therefore, we have also presented non-Nash equilibrium solutions. We presented a model in which an arriving client can request for more than one unit bandwidth and there is no restriction on the number of service providers.
We presented an accurate solution for the non-game theoretic model. The method to find the accurate solution is to start with a solution and then improve it; and this improvement continues till no more improvement is possible. The accurate solutions has significantly lower time and space complexities than the accurate Nash equilibrium solution. However, as the size of service providers increase, there was a need for solutions with lower complexities. Therefore, we presented approximate solutions: the session approximate solution, the grouped approximate solution and the session grouped approximate solution. The complexities of the accurate solution and the approximate solutions are given in Table 7.2. The accurate solution can run for very small sized service providers (when mmax' 10 and e=2) and for medium sized service providers only when the maximum bandwidth that can be requested by a client is 1 (when mmax'100 and e=1). The session approximate solution can run for medium sized service providers (when mmax'100 and e=2) but cannot run for large sized service providers (whenmmax>1000ande≥2). The grouped approximate solution and the session grouped approximate solution can run for large sized service providers.
The session approximate solution, the grouped approximate solution and the ses- sion grouped approximate solution are approximate and we have evaluated these solutions by simulating them. As already mentioned, we compared these solutions with a simple heuristic. The accurate solution and the session approximate solution were compared with the simple heuristic solution for small and medium ISPs. The decline in income when the session approximate solution is used instead of the ac- curate solution is approximately 10% and both performed significantly better than
TH-1482_08610101
Table 7.2: Complexities of our Non Game-theoretic solutions
Size of solution Time complexity Space complexity Accurate Solu-
tion
O(mmaxe+1×Be) Best case:
Whene6= 1: O((mmax× B)2.4×e×T)
When e=1:
O((mmax)2.4×T)
O((mmax×B)2×e)
Session Approxi- mate Solution
O(mmax) Finding solution for the first time
O(mmaxe+1×Be+mmaxT+1) Updating solution if just the mean arrival rates change
O(mmaxT+1)
O(mmax)
Grouped Ap- proximate
Replace mmax with group sizemmax and B with group sizeB in the ac- curate solution complexities
Session Grouped Approximate So- lution
Replacemmax with group sizemmax andB with group sizeB in the ses- sion approximate solution complexities
the simple heuristic. We also did simulations for medium and large sized service providers and found that there is an improvement in income and quality of ser- vice in our accurate and approximate solutions (the session approximate solution, the grouped approximate solution and the session grouped approximate solution) as compared to the simple heuristic in almost all the simulations. We also did simula- tions for real scenarios in which there are multihomed clients and service providers observe arrival rate of clients instead of being told arrival rates when a given price is charged. The simulations show that our accurate and approximate solutions perform better as compared to the simple heuristic solution in most of our simulations.