Little’s Law and modification

1.

Given that power is expensive, it is common practice to leave servers on only
when they are being used and to turn them off whenever they are not in use.
Assume that the following power-aware algorithm is used: When a job arrives,
it instantly turns on a fresh server (assume zero setup cost). When the job
completes service, it instantly turns off that server. Assume that there is always
a server available for every job (i.e., there is no queueing). Your goal is to derive
the time-average rate at which power is used in our system. Assume that when
a server is on, it consumes power at a rate of P = 240 watts. Assume λ = 10 jobs arrive per second and that the service requirement of jobs is Uniformly
distributed ranging from 1 second to 9 seconds.

2.

For the interactive system in Figure 6.14, suppose that we are given the following
information:
mean user think time = 5 seconds
expected service time at device i = .01 seconds
utilization of device i = .3
utilization of CPU = .5
expected number of visits to device i per visit to CPU = 10
expected number of jobs in the central subsystem (the cloud shape) = 20
expected total time in system (including think time) per job = 50 seconds.
How many jobs are there in the queue portion of the CPU on average, E
Ncpu
Q

?

3.

Proportional Power –

based on [69]
In the world of power distribution, one reasonable approximation is that the
power that is allocated to a machine is proportional to the speed at which that
machine can run. In this problem we assume that, if a machine is allocated
power w, then that machine processes jobs at speed w jobs/sec.
Consider a closed batch system with two servers and N users, where N is
assumed to be high. Assume that each job, with probability p, is routed to server
1 for processing and, with probability 1 − p, is routed to server 2. It may help
to look at Figure 7.6 here.
You are given a total power budget W, which you need to distribute between
the two machines. You can choose any way of dividing the power budget W
between the two machines, and you can also choose any value you want for p,
the routing probability.
(a) What choice for dividing W and for picking p will maximize the throughput
of your system?
(b) Suppose that N was small. Would your answer still be the same? If so,
explain why. If not, derive the optimal strateg