Advanced Wireless Networks - 4G Technologies

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queue length Q(t) could be nonzero. As already discussed in Section 8.1 [analogous to Equation (8.5)] or by using the theory of large deviations [14] , it can be shown that the probability of Q(t) exceeding a threshold B satisfies

where f (x) ≈ g(x) means that limx→∞ f (x)/g(x) = 1. For smaller values of B, the following approximation, analogous to Equation (8.3), is more accurate [15]:

where both γ (r) and θB (r) are functions of channel capacity r. According to the theory,

γ (r) = Pr {Q(t) ≥ 0} is the probability that the buffer is nonempty for randomly chosen time t, while the QoS exponent θB is the solution of α(θB ) = r. Thus, the pair of functions {γ (r), θB (r)} model the source. Note that θB (r) is simply the inverse function corresponding to the effective bandwidth function α(u).

If the quantity of interest is the delay D(t) experienced by a source packet arriving at time t, then with same reasoning the probability of D(t) exceeding a delay bound Dmax satisfies

where θ (r) = θB (r) × r [16]. Thus, the key point is that for a source modeled by the pair {γ (r), θ (r)}, which has a communication delay bound of Dmax, and can tolerate a delaybound violation probability of at most ε, the effective bandwidth concept shows that the constant channel capacity should be at least r, where r is the solution to ε = γ (r)e−θ (r)Dmax . In terms of the traffic envelope (t) (Figure 8.7), the slope λ(ss) = r and σ (s) = r Dmax.

In Section 8.1 a simple and efficient algorithm to estimate the source model functions γ (r) and θ (r) was discussed. In the following section, we use the duality between traffic modeling {γ (r), θ (r)}, and channel modeling to present an EC link model, specified by a pair of functions {γ (c)(μ), θ (c)(μ)}. The intention is to use {γ (c)(μ), θ (c)(μ)} as the channel duals of the source functions {γ (r), θ (r)}. Just as the constant channel rate r is used in source traffic modeling, we use the constant source traffic rate μ in modeling the channel. Furthermore, we adapt the source estimation algorithm from Section 8.1 to estimate the link model parameters {γ (c)(μ), θ (c)(μ)}.

the service provided by the channel. Note that the channel service ˜ ( ) is different from the

S t

actual service ( ) received by the source; ˜ ( ) only depends on the instantaneous channel

S t S t

capacity and thus is independent of the arrival A(t). Paralleling the development of Equation (8.21) and (8.22) we assume that

exists for all u ≥ 0. This assumption is valid, for example, for a stationary Markov-fading process r(t). Then, the EC function of r(t) is defined as

Consider a queue of infinite buffer size supplied by a data source of constant data rate μ (see Figure 8.6). The theory of effective bandwidth can be easily adapted to this case. The difference is that, whereas in the previous case the source rate was variable while the channel capacity was constant, now the source rate is constant while the channel capacity is variable. Similar to Equation (8.25), it can be shown that the probability of D(t) exceeding a delay bound Dmax satisfies

where {γ (c)(μ), θ (c)(μ)} are functions of source rate μ. This approximation is accurate for large Dmax, but we will see later in the simulation results, that this approximation is also accurate even for smaller values of Dmax.

For a given source rate μ, γ (c)(μ) = Pr {Q(t) ≥ 0} is again the probability that the buffer is nonempty at a randomly chosen time t, while the QoS exponent θ (c)(μ) is defined as θ (μ) = μα−1(μ), where α−1(·) is the inverse function of α(c)(u). Thus, the pair of functions {γ (c)(μ), θ (c)(μ)} model the link.

So, if a link that is modeled by the pair {γ (c)(μ), θ (c)(μ)} is used, a source that requires a communication delay bound of Dmax, and can tolerate a delay-bound violation probability of at most ε, needs to limit its data rate to a maximum of μ, where μ is the solution to

If the channel-fading process r(t) is stationary and ergodic, then a simple algorithm to estimate the functions {γ (c)(μ), θ (c)(μ)} is similar to the one described in Section 8.1. Paralleling Equation (8.7) we have

where τs(μ) is the average remaining service time of a packet being served. Note that τs(μ) is zero for a fluid model (assuming infinitesimal packet size). Now, the delay D(t) is the sum of the delay incurred due to the packet already in service, and the delay in waiting for the queue Q(t) to clear which results in Equation (8.29), using Little’s theorem. Substituting Dmax = 0 in Equation (2.28) results in Equation (3.30). As in Section 8.1, solving Equation (8.29) for θ (c)(μ) gives similarly to Equation (8.8a)

According to Equation (8.30) and (8.31) , as in Section 8.1, the functions γ and θ can be estimated by estimating Pr {D(t) > 0} , τs(μ), and E [Q(t)]. The latter can be estimated by taking a number of samples, say N , over an interval of length T , and recording the following quantities at the nth sampling epoch: Sn the indicator of whether a packets is in service (Sn {0, 1}), Qn the number of bits in the queue (excluding the packet in service),

250 EFFECTIVE CAPACITY

and Tn the remaining service time of the packet in service (if there is one in service). Based on the same measurements, as in Section 8.1,

given the delay-bound Dmax and the target delay-bound violation probability ε of a connec-

8.2.3 Physical layer vs link-layer channel model

In Jack’s model of a Rayleigh flat-fading channel, the Doppler spectrum S( f ) is given as

where fm is the maximum Doppler frequency, fc is the carrier frequency, and F = f − fc. Below we show how to calculate the EC for this channel [11].

Denote a sequence of N measurements of the channel gain, spaced at a time interval δ apart, by x = [x0, x1, · · · , xN −1], where {xn , n [0, N − 1]} are the complex-valued Gaussian distributed channel gains (|xn | are, therefore, Rayleigh distributed). For simplicity, the constant noise variance will be included in the definition of xn . The measurement xn is a realization of a random variable sequence denoted by Xn , which can be written as the vector X = [X0, X1, · · · , X N −1]. The pdf of a random vector X for the Rayleigh-fading channel is

where R is the covariance matrix of the random vector X, det(R) the determinant of matrix R, and xH the conjugate transpose (Hermitan) of x. To calculate the EC, we start with

Received

signal

Receiver

Figure 8.8 Queueing model used for simulations.

8.2.4 Performance examples

The discrete-time system depicted in Figure 8.8 is simulated. The data source generates packets at a constant rate μ which are first sent to the (infinite) buffer at the transmitter, whose queue length is Qn , where n refers to the nth sample interval. The head-of-line packet in the queue is transmitted over the fading channel at data rate rn . The fading channel has a random channel gain xn (the noise variance is absorbed into xn ). We use a fluid model that is the size of a packet is infinitesimal. A perfect knowledge of the channel gain xn (the SNR, really) at the transmitter side is assumed. Therefore, as described in Chapter 4, it can use rate-adaptive transmissions and strong channel coding to transmit packets without errors. Thus, the transmission rate rn is equal to the instantaneous (time-varying) capacity of the fading channel, defined by the Shannon law, rn = Bc log2(1 + |xn |2) where Bc is the channel bandwidth.

The average SNR is fixed in each simulation run. We define rg as the capacity of an equivalent AWGN channel, which has the same average SNR, i.e. rg = Bc log2(1 + SNRavg) where SNRavg is the average SNR, i.e. E |xn |2. Then, rn /rg relation

Simulation parameters as in Wu and Negi [11] were used. Channel samples xn are generated by the following AR(1) (autoregressive) model: xn = kxn−1 + vn where the modeling error vn is zero-mean complex Gaussian with unit variance per dimension and is statistically independent of xn−1. The coefficient k can be determined by the following procedure:

(1) compute the coherence time Tc by Tc ≈ 9/16π fm, where the coherence time is defined as the time over which the time autocorrelation function of the fading process is above 0.5; (2)

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