Problem Set 2
ECE 534
R. Srikant
Fall 2009
Due: Sept. 22
1. Consider the following congestion control algorithm:
wr
qr ,
xr
pl = hl (yl cl )+ ,
pl
xr = r
where qr = l:lr pl , yl = r:lr xr , and r and
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
Problem Set 1
ECE 534
R. Srikant
Fall 2009
Due: Sept. 8
1. Let xr be the rate allocated to user r in a network where each users route is xed. A link l is
called a bottleneck link for a user r if l r,
Problem Set 5
ECE 567
R. Srikant
Fall 2009
Due: Nov. 3
1. CSMA in continuous-time: Consider an ad hoc network whose scheduling algorithm operates
in continuous-time. Assume that if a link l interferes
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 ar
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, asymptot
Problem Set 5 Solution
Problem 1
(i) Assume the state X(t) is the set of all ON links, and take values
S1 , S2 , ., SK .
In a suciently small interval, only one link changes it states. Assume link
l i
ECE 567: Communication Network Analysis
Fall 2012
Quiz I
R. Srikant
Oct 30 2012, 11:00-12:15
Each question is worth 25 points.
1. Consider a discrete time queue in which a departure occurs rst, then a
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 tw
3.1. PROBLEMS
Solution
33
1. Note that
M0 (k + 1) = E e
k
l=1
a(l)
a(0) = 0
= E e
k
l=2
a(l)
a(0) = 0, a(1) = 0 p00 + E e
= E e
k
l=2
a(l)
a(1) = 0 p00 + E e
k
l=2
a(l)
k
l=2
a(l)
a(0) = 0, a(1) = 1 p
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 co
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 generate
ECE 567: Communication Network Analysis
Fall 2012
Quiz I
R. Srikant
Oct 30 2012, 11:00-12:15
Each question is worth 25 points.
1. Consider a discrete time queue in which a departure occurs rst, then a
R. Srikants Notes on Modeling and Control of High-Speed Networks
1
1
Large deviations
Consider a sequence of i.i.d. random variables cfw_Xi . The central limit theorem provides an estimate of the prob
R. Srikants ECE567 Notes on Downlink Scheduling
Let us consider the formulation of a two-user downlink transmission (Figure 1) with the following assumptions.
Two users want to make downlink transmi
1
Discrete-time Markov Chains
Let cfw_Xk be a discrete-time stochastic process which takes on values in a countable set S , called the state space. cfw_Xk is called a Discrete-time Markov chain (or
Previously, we studied how to design a stable and simple control mechanism to asymptotically solve the relaxed utility maximization problem. We also mentioned that by using an appropriate barrier func
ECE 567: Solutions of Problem Set 1
Osama Al-Hamad, Navid Aghasadeghi and Vineet Abhishek
September 8, 2009
Problem 1
Proof: cfw_xr be a max-min fair allocation every source has at least one bottlene
ECE 567: Solutions of homework 2
Ibtissam Ezzeddine, Javad Ghaderi, and I-Hong Hou
September 29, 2009
Problem 1
a) At the equilibrium:
xr = 0 xr = wr /qr
and
pl = 0
which yields
(i) pl = 0 if yl cl
(i
Solution for Problem Set 3
October 15, 2009
1.
(a)
In the following derivation for
that
i
mod
2n
2,
i > 0,
mod
clearly 1. Then for
P (S (n) = i|Y (n) = i, Y (n 1), Y (n 2), . . . , Y (1), assume 0 i n
Problem Set 3
ECE567
Each problem is worth 10 points.
Fall 2009
Due: Oct. 7
1. Consider a mobile radio that is moving on the integer
random walk. Let S (n) denote the position of the mobile
follows: S
Problem Set 4
ECE567
Each problem is worth 10 points.
Do any four of the six problems.
Fall 2009
Due: Oct. 20
1. Consider an N N VoQ switch where the arrivals into queue (i, j ) are Bernoulli with mea
Problem Set 6
ECE 567
R. Srikant
Fall 2009
Due: Nov. 19
1. Consider the M/G/B/B loss model, where the service-time distribution is Erlang with K
stages, each with mean 1/K. Show that the blocking prob