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 hl are positive constants. This algorithm is called
th
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
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, and
yl = cl and xs xr
s such that l s.
Show that cfw_xr
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 with the transmission of link j, then j also
interfere
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)
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 is any link without active neighbors. From Si to Sj ,the
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)
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 arrivals occur in a time
slot. The service and arrival p
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
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 p01
a(1) = 1 p01
= M0 (k )p00 + M1 (k )p01 .
Similarly,
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 stat
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
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 arrivals occur in a time
slot. The service and arrival p
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 probability n Xi n P ( i=1 > x), n where = E (X1 ) and 2 =
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 transmissions at a base station. Discrete-time model. Each use
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 simply a Markov chain, when the discrete nature of the
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 function, one could obtain the exact value of the optimal s
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 bottleneck.
: proof by contradiction
Assume we have max-min fai
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
(ii) pl > 0 if yl = cl .
If we choose Ur (xr ) = wr log x
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, and
Additionally, if i = 0, the probability is
otherw
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 (0) = 0 and
S (n) + 1, w.p.
S (n + 1) =
S (n) 1, w.p.
Question 1
Consider the Lyapunov Function:
V (q ) =
1
2
2
qi,j
ij
The Lyapunov drift is:
Vk = E [V (q (k + 1) V (q (k )|q [k ] = q ]
1
V k = E [
2
(qij (k ) + aij (k ) Iij (k )+ )2 qij (k )2 |q [k ] = q ]
ij
For simplicity, lets drop index k. Therefore, w
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 mean ij .
Assume that
ik < 1 and
lj < 1, k, l.
i
j
Compute
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 probability depends only on the mean of the
service-time di