The deterministic network calculus offers an elegant framework for determining delays and backlog in a network with deterministic service guarantees to individual traffic flows. A drawback of the deterministic network calculus is that it only provides worst-case bounds. Here we present a network calculus for statistical service guarantees, which can exploit the statistical multiplexing gain of sources. We introduce the notion of an effective service curve as a probabilistic bound on the service received by an individual flw, and construct an effective service curve for a network where capacities are provisioned exclusively to aggregates of flows. Numerical examples demonstrate that the calculus is able to extract a significant amount of multiplexing gain in networks with a large number of flows.

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Abstract. We consider convex sets and functions over idempotent semifields, like the max-plus semifield. We show that if K is a conditionally complete idempotent semifield, with completion ¯ K, a convex function K n → ¯ K which is lower semi-continuous in the order topology is the upper hull of supporting functions defined as residuated differences of affine functions. This result is proved using a separation theorem for closed convex subsets of K n, which extends earlier results of Zimmermann, Samborski, and Shpiz.

Stochastic timed marked graphs are graphical models of concurrent systems such as asynchronous circuits, embedded systems, queuing networks, manufacturing systems, and many automatic control systems. Unlike earlier works in which delays must be fixed or exponential, we allow the models to include arbitrary delay distributions as long as they have finite means. For such models, one important problem is to determine the average Time Separations of Events (TSE's). For example, an efficient means of finding TSE's in such models of asynchronous circuits facilitates both performance analysis as well as performance-driven synthesis. Towards this end, we present a novel technique to obtain upper and lower bounds on the average TSE for arbitrary pairs of system events. The bounds are formulated using a finite segment of the infinite unfolding of the marked graph and can be efficiently evaluated either using statistical sampling or, in some special cases, analytical methods. The resulting bounds...

We consider the transmission of variable bit rate (VBR) video over a network offering a guaranteed service such as ATM VBR or the guaranteed service of the IETF. The guaranteed service requires that the flow accepted by the network has to be conforming with a traffic envelope #; in return, it receives a service guarantee expressed by a network service curve #. Functions # and # are derived from the parameters used for setting up the reservation, for example, from the T-SPEC and R-SPEC fields used with the Resource Reservation Protocol (RSVP). In order to satisfy the traffic envelope constraint, the output of the encoder is fed to a smoother, possibly with some look-ahead. The resulting stream is transported by the network; at the destination, the decoder waits for an initial playback delay and reads the stream from the receive buffer. We consider the problem of whether there exists one optimal strategy at the smoother which minimizes the playback delay and the receive buffer size, given the traffic envelope # and the service curve #. We show that there does exist such an optimal smoothing, and give an explicit representation for it. We also obtain a simple expression for the smallest playback delay and playback buffer size which can be achieved over all possible smoothing and playback strategies. We show that the computation of optimal smoothing and minimum playback delay do not depend on the past. We show that separate delay equalization is optimal in the CBR case, but not otherwise. We also apply the theory to the analysis of which T-SPEC should be requested by a source-destination pair, given some playback delay and buffer constraint, and given the path characteristics advertised in RSVP PATH messages.

Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the sup-norm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is non-empty, is isomorphic to a convex inf-subsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group

In this paper, we present an algebraic approach to idempotent functional analysis, which is an abstract version of idempotent analysis in the sense of [1–3]. Elements of such an approach were used, for example, in [1, 4]. The basic concepts and results are expressed in purely algebraic terms. We consider idempotent versions of certain basic results of linear functional analysis, including the theorem on the general form of a linear functional and the Hahn–Banach and Riesz–Fischer theorems. 1. Recall that an additive semigroup S with commutative addition ⊕ is called an idempotent semigroup (IS) if the relation x ⊕ x = x is fulfilled for all elements x ∈ S. If S contains a neutral element, this element is denoted by the symbol 0. Any IS is a partially ordered set with respect to the following standard order: x ≼ y if and only if x ⊕ y = y. It is obvious that this order is well defined and x ⊕ y = sup{x, y}. Thus, any IS is an upper semilattice; moreover, the concepts of IS and upper semilattice coincide [5]. An idempotent semigroup S is called a-complete (or algebraically complete) if it is complete as an ordered set, i.e., if any subset X in S has the least upper bound sup(X) denoted by ⊕X and the greatest lower bound inf(X) denoted by ∧X. This semigroup is called b-complete (or boundedly complete), if any bounded above subset X of this semigroup (including the empty subset) has the least upper bound ⊕X (in this case, any nonempty subset Y in S has the greatest lower bound ∧Y and S in a lattice). Note that any a-complete or b-complete IS has the zero element 0 that coincides with ⊕Ø, where Ø is the empty set. Certainly, a-completeness implies the b-completeness. Completion by means of cuts [5] yields an embedding S → ̂ S of an arbitrary IS S into an a-complete IS ̂ S (which is called a normal completion 1 International Sophus Lie Centre, Moscow, Russia,

We consider equations of the form Bf = g, where B is a Galois connection between lattices of functions. This includes the case where B is the Fenchel transform, or more generally a Moreau conjugacy. We characterize the existence and uniqueness of a solution f in terms of generalized subdifferentials, which extends K. Zimmermann’s covering theorem for max-plus linear equations, and give various illustrations.

This is survey of some recent results connecting random matrices, noncolliding processes and queues.

We consider a scenario where multimedia data are sent over a network offering a guaranteed service. A smoothing device writes the stream into a transmitting device with limited input buffer; at destination, the decoder waits for an initial playback delay and reads the stream from the receiver buffer. We assume that some limited look-ahead is possible at the source, and that the playback buffer size is limited. First, we consider the case where the stream is delivered from the source directly to the destination buffer. We obtain closed-form expressions of the minimal required values in the case of a CBR smoothing Then we consider the case where the stream is first transmitted through a backbone network to an intermediate server, which relays the multimedia data to the final destination via an access network. We compute the requirements on playback delay, buffer sizes and amount of look-ahead.