3) Consider the hierarchical network of data centers networks and suppose that the data center needs to support e--mail and video distribution among other applications. Suppose four racks of servers are reserved for e--mail and four racks are reserved for video. For each of the application, all four racks must lie below a single tier--2 switch since the tier--2to tier--1 links do not have sufficient bandwidth to support the intra--application traffic. For the e--mail application, suppose that for 99.9% of the time only three racks are used, and that the video application has identical usage patterns.
a. For what fraction of time does the e--mail application need to use
a fourth rack?
b. Assuming e--mail usage and video usage are independent, for what fraction of time do (equivalently, what is the probability that) both applications need their fourth rack?
c. Suppose that it is acceptable for an application to have a shortage of servers for 0.001 percent of the time or less (causing rare periods of performance degradation for users). Discuss how the topology in Figure 5.31 can be used so that only seven racks are collectively assigned to the two applications (assuming that the topology can support all the traffic).