Wireless Network Design: Optimization Models and Solution ...

332 Nitin Salodkar and Abhay Karandikar
i in slot n. Let R i n ∈ U denote the number of packets that user i should transmit in
slot n if it is scheduled. Then U i n = I i nR i n, where I i n is an indicator variable that is set
to 1 if the user i is scheduled in slot n, otherwise it is set to 0.
The queue dynamics for user i can be expressed as (for U i n):
Q i n+1 = max(0,Qi n + A i n+1 − Ii nR i n). (14.27)
Since the scheduler can at most schedule all the bits in a buffer in any slot, U i n ≤
Q i n. Moreover, Q i n+1 ≥ Ai n+1 , ∀n. We assume that the scheduler can choose Ui n based
on the queue length Q i n, the channel state X i n and the number of arrivals (source
arrival state) A i n. More generally, the scheduler can determine U i n based on entire
history of queue lengths, channel states and source arrival states.
Different formulations make various assumptions on the arrival process {A i n} and
control action process {U i n}. We state these assumptions later while formulating
different scheduling problems.
The average queue length of a user i can be expressed as:
¯Q i = limsup
M→∞
M 1
M ∑ Q
n=1
i n. (14.28)
Average delay ¯D i can be treated to be equivalent to the average queue length ¯Q i
because of the Little’s law (Chapter 3, [9]) as follows:
¯Q i = ā i ¯D i , (14.29)
where ā i is the average packet arrival rate of user i. Due to this relationship, queue
length measure is usually considered to be synonymous with delay measure.
Similarly, the sum throughput over a long period of time can be expressed as:
¯T = liminf
M→∞
1
M
M N
∑ ∑
n=1 i=1
I i nR i n. (14.30)
The average power consumed by a user i over a long period of time can be expressed
as:
¯P i = limsup
M→∞
M 1
M ∑ P(X
n=1
i n,I i nR i n), (14.31)
where P(X i n,I i nR i n) is the power required by i when the channel state is X i n and user
transmits I i nR i n packets.
A number of scheduling algorithms have been proposed in the literature that focus
on the above measures or variations of these measures. Broadly, these algorithms
can be classified into three types:
1. Maximize sum throughput subject to fairness constraint.
2. Maximize sum throughput subject to delay and queue stability constraint.