# MP-DSM: A Distributed Cross Layer Network Control Protocol

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3 where λ pf (D(r) G) is the Perron Frobenius eigenvalue associated with the matrix D(r) G. The eigenvalue λ pf (D(r) G) can be interpreted as a capacity utilization measure for the system. The Perron Frobenius eigenvalue is monotonic increasing in r. When r is increased to λ pf (D(r) G) =, system capacity is fully utilized and no additional traffic can be carried by the system. Further, transmitter powers p satisfying equation are given by the right Perron Frobenius eigenvector of D(r) G. There may exist other power vectors ˆp that satisfy equation, but p ˆp. The left Perron Frobenius eigenvector is denoted by q and can be interpreted [] as moderating the effect of power increases in the network. The backwards or adjoint network has properties similar to the forward network. Theorem : The forward and backwards rate regions are identical. Proof: Let r be a feasible point in R, then there exists q such that q T q T D(r) G q G T D(r)q u D(r) G T u (7) where u = D(r)q identifies transmitter powers in the backwards network. Thus r is feasible for the adjoint network. A similar argument shows that if ŕ is feasible for the backwards network it is also feasible for the forward network. Thus the forward and backwards networks can always be operated at identical source rates r. Further, the left Perron Frobenius eigenvector q can always be found from the backwards network when both networks are operated at the same rate, explicitly, q = D(r) u. () Because D(r) is diagonal and positive, the inverse always exists. Similar results hold for the backwards network. IV. NETWORK PERFORMANCE METRICS System performance is modelled by a performance metric or utility function U that represents the value of rate vector r to the system. This function can represent the utility a user derives from using the system at a particular rate or can be implied by data protocols, service rate agreements, or other system level metrics [5], []. Implicitly many protocols can be described in this way. TCP in particular has been modelled as a concave function of the source rates [7]. In many situations U can be treated as a separable function, N U(r) = U i (r i ) (9) i= where source i has performance function U i (r i ).Inthis view each source is concerned with only its own rate r i. By assumption, a higher data rate is valued at least as much as a lower data rate and there is a diminishing return to additional data rate, U i is a concave function of r i. Data traffic controlled by TCP-Reno, for example, can be modelled [5] using U i (r i )= D s ln( + r i D s ) () where D s is the round trip delay. Voice traffic can also be modelled U i (r i )={, ri <R mini c, r i R mini. () Rates below R mini will never be selected by the system. QoS constraints can be added to this formulation by embedding them in the U i. A rate threshold for TCP data traffic might be formulated as U i (r i )= D s ln( + r i D s )+ ln(r i R mini ) () where is a constant. The second term acts a barrier preventing the network from allocating less than R mini to this source. Delay constraints can also be added to this formulation. Since the backwards network is used to acknowledge traffic sent from the forward network, its performance is indirectly linked to U. In what follows U is assumed to have continuous second derivatives. V. PROBLEM FORMULATION A cross layer approach is used to control the network. The goal is to adaptively control the network and operate it at the best set of source rates r and transmitter powers p such that the system performance metric or utility function U is maximized. As shown in [] optimal system performance lies along the rate surface. At the optimal source rates r, λ pf (D(r ) G) =. Thus only the surface of the rate region need to be searched to find the optimal operating point for the network. Formally, this problem can be expressed as: maximize r U(r) subject to λ pf (D(r) G) = () This is a convex problem in the variable r since the constraint set is convex and the objective concave. Each r Ris associated with one or more power vectors p, and optimum p are calculated from r. The backwards network is not explicit in this formulation but is assumed to be operated at a rate ŕ R sufficient to support the forward rate r //\$. (c) IEEE

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