We investigate a distributed, state-dependent, dynamic routing strategy for circuit-switched loss networks which we have called Aggregated-Least-Busy-Alternative (ALBA). The networks considered are symmetric and fully connected, the offered calls form Poisson streams and routes have at most two links. In ALBA(K), the states of each link are lumped intoK (K ≥ 2) aggregates and the route of each call is determined by local information on the aggregate-states of the links of the alternate routes. The last aggregate is always the set of states reserved for direct traffic and defined by the trunk reservation parameter. The particular case of ALBA in which there is no aggregation is Least-Busy-Alternative (LBA); ALBA(2) represents the other extreme of aggregation. We consider two separate asymptotic scalings based on Fixed Point Models for ALBA(K) which were obtained and investigated in an earlier paper. In the first, it is assumed that the number of network nodes, the offered traffic and trunk group size are all large; their ratios have been chosen to reflect practical interest. The results show that there exists a threshold which delineates fundamentally different behavior: for offered traffic below the threshold, the network loss probability decreases exponentially with increasing network size, while above the threshold the decrease is only polynomial. In the related second asymptotic scaling, the asymptotically optimum trunk reservation parameter is obtained as the solution of a simple equation. Such asymptotically optimal designs are compared to the outputs of exhaustive numerical searches for some realistically sized networks and found to perform very well.

State-dependent routing on symmetric loss networks with truck reservations. II: asymptotics, optimal design

Let x, e > 0, uo < ... u<? d' and h > O be real numbers. Let f be a real valued function and let A (h; u, w)f (x) h-d be a difference quotient associated with a generalized Riemann derivative. Set I = (x + uoh, x + Ud+eh) and let f have its ordinary (d 1)st derivative continuous on the closure of I and its dth ordinary derivative f('I) existent on 1. A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a t E I such that the difference quotient is equal to f (d)( )) is given for d = 1 and d = 2. The condition is sufficient for all d. It is used to show that many generalized Riemann derivatives that are "good" for numerical analysis do not satisfy this mean value theorem. 1. Results. Let f be a real valued function of a real variable. The dth Riemann derivative of f is Rdaf (X) := lim = (,)( 1) f (x ?+( -d/2 + i)h) h-0O hd The first two special cases R f(x) = lim -f (x h/2) + f (x + h/2) 11-0 ~~h

/pdf/mean-value-theorems-for-generalized-riemann-derivatives-501ijghpd4.pdf

Mean value theorems for generalized Riemann derivatives

In this paper we introduce generalized definitions of Peano e Riemann derivatives in order to obtain second order optimality conditions for vector optimizations problems involving C wedge(1,1) data.

On Generalized Derivatives for C Wedge(1,1) Vector Functions and Optimality Conditions

We introduce generalized definitions of Peano and Riemann directional derivatives in order to obtain second-order optimality conditions for vector optimization problems involving C1,1 data. We show that these conditions are stronger than those in literature obtained by means of second-order Clarke subdifferential.

/pdf/on-generalized-derivatives-for-c1-1-vector-optimization-3wfii24lyz.pdf

On generalized derivatives for C1,1 vector optimization problems

Marshal Ash [3] introduced the concept of (σ, τ) differentiable functions and studied the Riemann generalized derivatives In this article we study the convexity and monotonicity of (σ, τ) differentiable functions, using results by Hincin, Humke and Laczkovich, and using the Riemann generalized derivative. We give conditions such that the classic properties of differentiable functions hold also for (σ, τ) differentiable functions.

The role of Riemann generalized derivative in the study of qualitative properties of functions

Very Generalized Riemann Derivatives, Generalized Riemann Derivatives and Associated Summability Methods

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Very Generalized Riemann Derivatives, Generalized Riemann Derivatives and Associated Summability Methods