0.1. PROBLEMS
0.1
1
Problems
Exercise 0.1 Let xr be the rate allocated to user r in a network where users routes are xed.
Link l is called a bottleneck link for user r if l r, and
yl = cl
and
xs xr
s such that l s,
i.e., link l is fully utilized and user
0.1. PROBLEMS
0.1
1
Problems
Exercise 0.1 Consider the primal congestion controller with r (x) = 1 x and r. Using the
Lyapunov function
(xr xr )2 ,
r
to prove that the controller is globally, asymptotically stable, where x is the global maximizer of
W (x)
8.1. PROBLEMS
91
3.
(n1 , , nK )
n =
cfw_ni :
=
=
i ni =n
K
e n
n!
e n
n!
i=1
n
pi
i
.
Exercise 8.10 Let X be a CTMC over a state-space S and suppose that X is time-reversible in
steady-state. Now consider a CTMC Y which is a restriction of X to the state
Chapter 5
Scheduling in Wireless Networks
5.1
Problems
Exercise 5.1 Consider a cellular wireless network consisting of a base station and two receivers,
mobile 1 and mobile 2. The network can be in two channel states: c1 = (1, 2) and c2 = (3, 1), equally
Chapter 4
Scheduling in Packet Switches
4.1
Problems
Exercise 4.1 In this exercise, we use an example to illustrate the throughput loss induced by HOL.
Consider a 22 switch, and assume input queues are innitely backlogged. The destinations (output
ports)
Chapter 8
Queuing Theory in Continuous Time
8.1
Problems
Exercise 8.1 Prove Result 7.2.1 (the sum of two independent Poisson process is a Poisson process)
and Result 7.2.2 (K random processes generated from a Poisson process are K independent Poisson
proc