EXPERIMENTOS ALTERNOS
OBTENCIÓN DEL ANARANJADO DE METILO
This Chapter presents a short overview of key mathematical concepts and re- sults that are used throughout the thesis. To begin with, define a discrete time system of K queues (users), with qk(t) the length of each queue at the begin-
ning of timeslot t, ak(t) the corresponding traffic process (i.e. the amount of
data coming into the queue at timeslot t) and µk(t) the service process, that
is the amount of data that can get transmitted at timeslot t. Note that in the context of wireless communications, the latter depends on the state of the wireless channels at t and the scheduling and resource allocation algorithms em- ployed. In addition, the real amount of data transmitted to user k at timeslot t is min{qk(t), µk(t)}, since the service offered can be greater than the amount of
data already in the queue (e.g. in the case of a user having a very good chan- nel). For the rest of the thesis, queues will be measured in bits and arrival and service processes in bits per timeslot unless specified otherwise. In general, the arrival processes can be correlated across time but they are independent across users and ergodic, with a finite mean arrival rate λkand variance σk2. Since this
thesis is devoted to study the downlink of wireless systems, traffic to each queue is single hop, i.e. the data is transmitted from queue k directly to the intended receiver and no routing/relaying are examined.
2.1
Functional Limit Theorems
In this section we present some basic functional limit theorems, namely the functional law of large numbers and the functional central limit theorem. They can be seen as analogues of the law of large numbers and central limit theorem in stochastic processes. For the first we have:
2.1. Functional Limit Theorems
Theorem 2.1.1 (Functional Law of Large Numbers). Assume x(τ ), τ = 0, 1, 2, ... is a sequence of i.i.d. random variables with mean µ. Then
1 n bntc X τ =0 x(τ ) → µt with probability 1.
An important process in the following is the Brownian Motion, or Wiener process. It is defined as follows:
Definition 2.1.1 (Standard Wiener process). A continuous time stochastic process w(t) is called Standard Wiener process (or standard Brownian motion) if it satisfies the following conditions:
Its sample paths are continuous with probability 1 w(0) = 0
Its increments are mutually independent w(t) − w(s) ∼ N (0, t − s), ∀0 ≤ s < t < ∞
For proof of existence and constructions of such a process refer to e.g.[83]. The Wiener process is used for the analogue of Central Limit Theorem in the case of stochastic processes:
Theorem 2.1.2 (Functional Central Limit Theorem). Assume x(τ ), τ = 0, 1, 2, ... is a sequence of i.i.d. random variables with mean µ and variance σ2. Then1
1 √ n bntc X τ =0 x(τ ) − µ σ w −→ w(t),
where w(t) is a standard Wiener process .
The functional laws presented above and their extensions have been used to derive asymptotic models of queuing networks for performance evaluation and optimization, see e.g. [84]. More specifically, application of the Functional Law of Large Numbers usually leads to an asymptotic model that depends on the first order statistics of the system (i.e. mean arrival and service rates). In this case an Ordinary Differential Equation describes the evolution of the queue lengths. This method has been heavily used for stability analysis of networks, beginning with the work in [85] and applied to wireless networks with scheduling in many works, e.g. [55], [53]. In addition, control policies for the network can
2.2. Stability of Queuing Systems
be derived from such models [86]. On the other hand, asymptotics based on the Functional Central Limit Theorem lead to the evolutions of the queue lengths being described by SDEs [83, 87], constrained in the nonnegative orthant. The advantage of these models is that, in addition to the mean behaviour of the system, they capture its stochastic behaviour as well, keeping the second order statistics of the arrival and service (and routing if applicable) processes. The usual interpretation of these models is that they describe a network where the arrival rates at the queues2 are pushed very close to the corresponding service
rates, with the gap closing down as O(1/√n) 3, where n → ∞ is the scaling parameter in the Functional Central Limit Theorem. Due to this interpreta- tion, they are often referred to as ”Heavy Traffic Approximations”. This kind of asymptotic models has been extensively used mainly for performance evalu- ation of queuing networks due to its ability to capture the stochastic behaviour while actually simplifying the system model; in practice it is shown that the asymptotic models describe the original systems rather accurately, even in the case where the system is not very heavily loaded.
2.2
Stability of Queuing Systems
Stability, which is the focus of Chapters 4 and 5, is a very important aspect of queuing systems. Formally, its definition is as follows:
Definition 2.2.1 (Strong Stability). A system is said to be strongly stable if
lim sup T →∞ 1 T T −1 X t=0 E{qk(t)} < ∞, ∀k ∈ {1, .., K}
Intuitively stability implies that the mean queue length of every queue in the system is finite, further implying finite delays in single hop systems. The figure of interest in this thesis is strong stability, therefore in the remainder of the manuscript ”stable” will imply ”strongly stable” unless stated otherwise. If the arrivals and service rate processes are such that the Markov chain is irreducible and aperiodic with a single communicating class, strong stability is equivalent to positive recurrence of the chain [15].
The above definition holds for a fixed mean arrival rates and resource allo- cation policy, and leads to the concept of a stability region.
2Or at the queues with the heaviest load in terms of multi-hop networks
3Another similar interpretation is that the load of the queue grows as ρ = 1 − b/√n for a
2.2. Stability of Queuing Systems
Definition 2.2.2 (Stability Region). The stability region Λ of a resource allo- cation policy is defined as the set of vectors of mean arrival rates for which the system is stable under this policy. Furthermore, an algorithm that achieves the maximum possible stability region is called throughput optimal.
For the rest of the thesis, when describing stability regions we will mean that the system is stable in the interior of the calculated region. The behaviour on the boundary is not examined - usually for the boundary points the system is stable in at least a weaker sense, i.e. mean rate stable [15]. In short, and rather informally, a system is stable when the mean service rate of each user is bigger than the mean arrival rate of the corresponding traffic process. In the case of full channel knowledge, it is known that a throughput optimal algorithm is the so- called MaxWeight [13], which, in the case of a single-hop system, maximizes the quantityPK
k=1qk(t)µk(t) (see also [88]). Other throughput optimal algorithms
can be obtained using the same concept but with different weights (appropriate functions of the queue lengths and/or delays experiences by the data unit at the head of the queues) in each service rate [12, 55, 27, 28, 53, 89].
One basic tool used to prove stability of a system comes from the general theory of stability of stochastic systems [90] and it is based on methods including Lyapunov functions. The following definitions are in order:
Definition 2.2.3 (Lyapunov function). A function V : RK → R is said to be
a Lyapunov function if it has the following properties V (x) ≥ 0, ∀x ∈ RK
It is non-decreasing in any of its arguments V (x) → +∞, as ||x|| → +∞
Definition 2.2.4 (Lyapunov drift). The (one-step) drift of a Lyapunov function V for the system q(t) is defined as
∆V (x) = E {V (q(t + 1)) − V (q(t))|q(t) = x} .
In our context, the expectation is with respect to the arrival processes, chan- nel processes and possible randomizations of the resource allocation algorithms (which affect the service processes). A main result of stochastic stability theory is the so-called Foster-Lyapunov criterion which connects the stability of the system with the drift of a Lyapunov function:
2.2. Stability of Queuing Systems
function V (x) and a bounded set B ⊂ RK
+ such that
∆V (x) < ∞, ∀x ∈ B, ∆V (x) < 0, ∀x /∈ B. Then the system is stable.
A Lyapunov function often used in practice is the quadratic function V (x) =
1 2
PK
k=1x 2
k4. In this case another sufficient condition for stability, which follows
from the Foster-Lyapunov criterion, is the following: Theorem 2.2.2. Set V (x) = 12PK
k=1x 2
k. If there exist constants B < +∞ and
> 0 such that ∆V (q(t)) ≤ B − K X k=1 qk(t),
then the system q(t) is stable.
This sufficient condition was in fact used to prove stability of the MaxWeight algorithm in [13]. It is also widely used in more general optimization and control problems in wireless networks, starting with the works of [91] and [14]. The reader is referred to [15] for a comprehensive treatment of the techniques based on Lyapunov functions for stochastic network optimization.
4Or V (x) =PK