subdirección de administración y servicios

Inspired by the theory of EB, the authors in [10] proposed the concept of effective ca- pacity (EC), as a dual of EB. Let_{c(t), t= 0,1,2, . . ._}be a discrete-time service pro- cess, which is stationary and ergodic. C(t1, t2) =

t2

P

t=t1+1

c(t) denotes the partial sum. Assume that the G¨artner-Ellis limit of C(t), expressed as lim

t→∞

1 t log E

eθC(0,t) ₌

ΛC(θ), exists and is a differentiable convex function for allθ ∈R [16]. Consider that

the arrival rate is a constant, i.e., a. Therefore, by applying Theorem 4 in Section 2.1.3, one can get that

ΛC(−θ∗) =−ΛA(θ∗) =−θ∗a, (2.18)

whereθ∗ is the unique delay QoS exponent satisfying (2.16). From (2.18), it is noted that₋ΛC(−θ

∗₎

θ∗ =a, which can be considered as the effective capacity of the service

process, on the condition that the queue overflow probability can be guaranteed with a decay rate θ∗. Finally, −ΛC(−θ

∗₎

θ∗ , denoted by Ec(θ), can be calculated from the

G¨artner-Ellis limit: Ec(θ) = − ΛC(−θ∗) θ∗ =−_tlim_→∞ 1 θtlog E e−θC(0,t). (2.19) Furthermore, let us define ǫ as the required queue overflow probability limit. In other words, the maximum queue overflow probability that can be afforded is given as ǫ. In this case, by applying (2.16), the minimum decay rateθ∗ _{can be calculated}

as θ∗ ₌

−(logǫ)/x. Inserting the minimum decay rate θ∗ _{into (2.19), a maximum}

value of Ec(θ∗) satisfying the queue overflow probability limit can be found, since

EC is a monotonically decreasing function with the delay QoS exponent. Hence, one can say that, in order to guarantee a required queue overflow probability limit, the calculated effective capacityEc(θ∗) represents the maximum constant arrival rate

that the service process can support.

capacity reduces to Ec(θ) =− lim t→∞ 1 θtlog E " e−θ t P i=1 c(i)#! (2.20) =₋ lim t→∞ 1 θtlog E " _t Y i=1 e−θc(i) #! (2.21) =₋ lim t→∞ 1 θtlog t Y i=1 E_e−θc(i) ! (2.22) =− lim t→∞ 1 θtlog E e−θc(i)t (2.23) =−1 θ log E e−θc(i). (2.24)

From (2.21) to (2.22), it is due to the reason that the sequence{c(t), t= 0,1,2, . . .}is uncorrelated. From (2.22) to (2.23), it is because that the service process is stationary and ergodic. Apparently, when the service process is uncorrelated, the EC expression in (2.24) only depends on marginal statistics, which is much simpler than the general expression given in (2.19), where the higher-order statistics are required [11]. Since the block fading channel generates an i.i.d., hence uncorrelated, service process, it can greatly simplify the EC expressions [11].

Note that the above introduction of EC assumes the constant arrival rate. Actu- ally, it can be generalized to investigate the delay QoS performance of any stationary arrival process [11]. By rewriting (2.15) in Theorem 4 in Section 2.1.3, we can get that if there exists a unique θ∗ >0 such that

ΛA(θ∗) θ∗ =− ΛC(−θ∗) θ∗ , (2.25) then we have lim x→∞ log (Pr (q(∞)≥x)) x =−θ ∗ . (2.26) Since ΛA(θ ∗₎ θ∗ =Eb(θ

∗_{) denotes the EB and}

−ΛC(−θ

∗₎

θ∗ = Ec(θ

∗_{) is the EC, hence,}

(2.25) and (2.26) indicate that the EB function intersects with the EC function at the point where the delay QoS exponent isθ∗_{. Here, θ}∗ _{is the one which guarantees}

the queue overflow probability limit.

To thoroughly understand the relationship between EB and EC when the time- varying arrival process and service process are considered, Fig. 2.1 is included which shows the curves of EB and EC versus the delay QoS exponent θ [11]. Set µ =

Figure 2.1: EC and EB, as functions of the delay QoS exponent θ. lim

θ→0Eb(θ), andµc = limθ→0Ec(θ). From Fig. 2.1, it shows that when the minimum value

of EB is larger than the maximum value of EC, i.e., µa > µc, there is no solution

for θ∗ _{> 0 existing. In this case, the service process cannot support the required}

delay QoS for the given arrival process, which is consistent with the conclusion from queueing theory that, if E_[a(t)]>E_{[c(t)], both queue length and the queueing delay}

will approach to infinity. This is because that, when θ → 0, the EB is equal to the average arrival rate of the traffic process, i.e., µa = lim

θ→0Eb(θ) =

E_{[a(t)]. Meanwhile,}

when θ _→ 0, the EC is equal to the average service rate of the service process, i.e., µc = lim

θ→0Ec(θ) =

E_{[c(t)] [11].}

In the above analysis, the buffer overflow probability was considered as the delay QoS measurements. When the focus is on the delay experienced by a source packet arriving at timet, defined byD(t), an expression analogous to (2.26) can be estimated as [10, 53]

Pr (D(t)> Dmax)≈Pr (q(t)>0)e−θµDmax, (2.27)

where Dmax denote the delay bound, and Pr (q(t)>0) is the probability of a non-

empty buffer, which can be approximated by the ratio of the average arrival rate and the average service rate [11], i.e., E[a(t)]

E_[c(t)]. Furthermore, from [11], we note that

Considering the delay violation probability in (2.27) as a function of θ, one can notice that the parameter θ plays an important role for statistical QoS guarantees, by indicating the exponential decay rate of the delay QoS violation probability [11]. A smaller θ corresponds to a slower decay rate, which implies that the system can tolerate a looser QoS guarantee, while a larger θ indicates a faster decay rate, which means that a more stringent QoS requirement can be supported. In particular, when θ _→0, the system can tolerate an arbitrarily long delay. When θ _{→ ∞}, it indicates that the system cannot tolerate any delay [12].