1. Numerical Studies on Braess-like Paradoxes for Non-Cooperative Load Balancing in Distributed Computer Systems By Said Fathy El-Zoghdy, Hisao Kameda, and Jie Li

2. Contents Introduction System model 3. Numerical experiments
4. Conclusion

3. Introduction 1- Overall optimum (System optimum or Social optimum)
We can choose between several distinct objectives for performance
1- Overall optimum (System optimum or Social optimum)
2- Individual optimum (Wardrop equilibrium)
3- Class optimum (Nash equilibrium)
We examine a Braess-like paradox; i.e.,
increased capacity of a part of a system may sometimes lead to the degradation in the benefits of all users in class optima in distributed computer systems.

4. 2. System Model
A distributed computer system that consists of m nodes (hosts) numbered 1, 2,…, m and a communication means that connects them.
Where:
is the external job arrival
rate to node i
is the job processing rate
at node i
is the job flow rate
form node i to node j
(A distributed Computer System)

5. 2.1 System model and assumptions
Each node consists of a single exponential server with service rate
2. We classify jobs arriving at node i into class i, i=1,2,…,m
Jobs arrive to node i according to a Time-invariant Poisson process, with the average external job arrival rate , out of which the rate of jobs are processed at node i, and the rate of jobs are forwarded through the communication means to the other node j to be processed there and the results of those jobs are returned back through the communication means to node i. Then it follows that 　　　Denote by the vector and by The set of that satisfy constraints by .
Each node has one decision maker, also numbered i, i=1,2,…,m
The load at node i is and is denoted by , and the expected processing (including queueing) time of a job that is processed at node i, is

6. 6. The expected communication (including queueing) time of forwarding a job arriving at node i to node j is denoted by
Thus the expected response time of a job that arrives at node i is
Then the overall response time of a job that arrives at the system is

7. As the communication means we consider the following two cases (A) and (B).
It consists of m(m-1) communication lines. The line ij is used for forwarding of jobs that arrive at node i to node j. The expected time length for forwarding and sending back a job is
(B) It consists of a single or multiple communication line that has no queueing delay. Thus the expected communication time of a job arriving at node i and being processed at node is expressed as

8. We have three optima, the overall, the individual, and the class
The overall optimum is given by such as satisfies the following,
, with respect to
The individual optimum is given by such as satisfies the following
for all i ,
, such that
The class optimum is given by such as satisfies the following
for all i,

9. With respect to the system parameter setting, we can define the three symmetries
Overall symmetry if the following condition holds
for all i
Individual symmetry if the following condition holds
Complete symmetry if the overall and individual symmetry conditions hold
or equivalently if and are constant, for all i. .

10. Where Braess-like paradox occurs if the following holds:
for some t such that
Where
is the mean response time for class i jobs, computed at the Nash Equilibrium, when the communication time is t.
, is a value of the mean communication time such that the communication means is not used any more at equilibria if the mean communication time is larger than
And the worst-case ratio of performance degradation in the paradox is defined by

11. 3. Numerical experiments
Complete symmetry for Type B
1.01
1.001
1.0001
(%)

12. Complete symmetry for Type A
1.01
1.001
1.0001
(%)

13. ( Type A)
Overall symmetry

14. ( Type A)
Individual symmetry

15. (Type A)
No symmetry

16. ( Type B)
Complete symmetry

17. ( Type B)
Individual symmetry

18. (Type B)
No symmetry

19. Plus asymmetric worst-cases for m=2

20. Plus asymmetric worst-cases for m=4

21. 4. Concluding Remarks
We have examined a number of numerical examples for the Braess-like paradox wherein adding a communication capacity to the system leads to the performance degradation for all users in the class optimum for load balancing.
It has already been shown that, in complete symmetry, for the communication means of type B, the ratio of performance degradation in the paradox can increase without bound as the arrival rate approaches the processing rate.
On the other hand, in the numerical examples, for the communication means of type A, the worst-case ratio of the paradox is bounded .

22. In the numerical examples, as the system parameter setting departs from complete symmetry, the ratio of performance degradation decreases. It decreases slowly (resp. more slowly) when the system parameter setting departs from complete symmetry while keeping the individual (resp. overall) symmetry property.
It is observed that for all these cases, the worst-case ratio of performance degradation is asymptotically reached in complete symmetry as the arrival rate approaches the processing rate.