Advanced Wireless Networks - 4G Technologies

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236 EFFECTIVE CAPACITY

cell is placed in an infinite capacity first-in, first-out (FIFO) buffer. Whenever a customer leaves the fictitious queue, a cell (if any) is removed and transmitted from the FIFO buffer.

The dual leaky bucket consists of two leaky buckets: a peak rate leaky bucket with parameters (λp, Bp) and a sustainable rate leaky bucket with parameters (λs, Bs), where λp > λs. A cell is conforming with respect to the dual leaky bucket if and only if it is conforming with respect to both the peak and sustainable rate buckets. The sustainable rate λs represents the maximum long-run average rate for conforming cell streams. The peak rate λp specifies a minimum intercell spacing of 1/λp with burst tolerance Bp. In this section, we shall assume that Bp = 1, so that the intercell spacing of conforming streams is constrained to be at most 1/λp. Values of Bp > 1 allow for cell delay variation (CDV) between the customer premises and the network access point.

The set (λp, λs, Bs) constitutes a UPC descriptor that the user negotiates with the network. If the connection is accepted, the user applies a dual leaky bucket shaper to shape its cell stream according to the negotiated UPC. The dual leaky bucket limits the peak rate to λp, the long-term average rate to λs, and the maximum burst length (transmitted at peak rate)

to Bc = Bsλp/(λp − λs).

Deterministic traffic characterization is based on the queueing model of the leaky bucket [5]. Suppose that the cell stream is of finite length and that the peak rate λp is known. The cell stream is offered as input to a queue served at constant service rate μ. Let Bmax(μ) be the maximum number of cells observed in the system over the duration of the cell stream. The stream then can be characterized by the peak rate and the values Bmax(μ) for 0 < μ < λp

Equation (8.1) can be interpreted as follows. For each μ, 0 < μ < λp, the cell stream passes through a leaky bucket parameterized by [μ, Bmax(μ)] without being shaped, i.e. the cell stream conforms to the leaky bucket parameters [μ, Bmax(μ)]. The notation B(CD, μ) = Bmax(μ) is introduced to show that the choice of bucket size, for a given leak rate μ, depends on CD. The characterization is tight in the following sense. If the cell stream conforms to a given set of leaky bucket parameters (μ, B ), then necessarily B ≥ B(CD, μ). For a given leak rate μ*, the minimum bucket size for conformance is B* ≥ B(CD, μ*).

Ideal statistical traffic characterization is based on assumption that the peak rate of the cell stream λp and the mean rate λm are known. Suppose that the cell stream is offered to a leaky bucket traffic shaper with leak rate μ, where λm < μ < λp, and bucket size B.

The shaping probability is defined as the probability that an arbitrary arriving cell has to wait in the data buffer of the leaky bucket. In the x/D/1 queueing model of the leaky bucket, this event corresponds to a customer arrival to the fictitious queue when the number in the system exceeds B. If B(μ, ε) is defined as the minimum bucket size necessary to ensure

represents statistical traffic characterization.

Approximate statistical traffic characterization is based on the assumption that the cell stream is offered to a queue with constant service rate μ, where λm < μ < λp. If W is the steady-state waiting time, in the queue, for an arbitrary cell arriving in the queue, then the complementary waiting time distribution, can be approximated as (see Appendix C; please

go to www.wiley.com/go/glisic):

Two asymptotic approximations are of interest:

Also the following approximation is of interest, as suggested in Chang [6]: a ≈ bE(W ). In many cases, the approximation in Equation (8.3) with a = 1 tends to be conservative,

i.e. the bound, P(W > t) ≤ e−bt , holds under certain conditions [6, 7]. The one-parameter approximation

is the basis for the theory of effective bandwidths that will be discussed below.

More extensive discussions of this class of approximations can be found in References [7, 8]. In this section Equation (8.3) is used as a reasonable approximation for an ATM cell stream modeled as a general Markovian source. Over the relatively short observation windows of interest (on the order of seconds), these approximations are fairly robust for a large class of real traffic sources, in particular, MPEG video sequences. Therefore, parameters [a(μ), b(μ)] provide a statistical characterization of the cell stream when offered to a queue with a constant service rate μ. A more complete characterization of the cell stream records the values [a(μ), b(μ)] for all μ in the range (λm, λp) as follows:

Values of [a(μ), b(μ)] are chosen such that Equation (8.3) holds approximately for values of t in range of interest. Assuming that Equation (8.3) holds at equality, we have (see Chapter 6)

In Equation (8.7) Sa denotes the number of customers in service, Qa the number of customers in queue seen by an arbitrary customer arrival, and τr(μ) the average remaining service time for the customer in service as seen by an arriving customer conditional on there being a customer in service. Bearing in mind that

The estimation of the above parameters is discussed later in this section. Once these parameters are obtained they are used in Equation (8.6) for statistical characterization of a cell stream. The next step is to map the characterization to a suitable UPC descriptor.