Cost sharing of cooperating queues in a Jackson network

We consider networks of queues in which the independent operators of individual queues may cooperate to reduce the amount of waiting. More specifically, we focus on Jackson networks in which the total capacity of the servers can be redistributed over all queues in any desired way. If we associate a cost to waiting that is linear in the queue lengths, it is known how the operators should share the available service capacity to minimize the long run total cost. We answer the question whether or not (the operators of) the individual queues will indeed cooperate in this way, and if so, how they will share the cost in the new situation. One of the results is an explicit cost allocation that is beneficial for all operators. The approach used also works for other cost functions, such as the server utilization

Simple models of network access, with applications to the design of joint rate and admission control

At the access to networks, in contrast to the core, distances and feedback delays, as well as link capacities are small, which has network engineering implications that are investigated in this paper. We consider a single point in the access network which multiplexes several bursty users. The users adapt their sending rates based on feedback from the access multiplexer. Important parameters are the user's peak transmission rate p, which is the access line speed, the user's guaranteed minimum rate r, and the bound ε on the fraction of lost data. Two feedback schemes are proposed. In both schemes the users are allowed to send at rate p if the system is relatively lightly loaded, at rate r during periods of congestion, and at a rate between r and p, in an intermediate region. For both feedback schemes we present an exact analysis, under the assumption that the users' job sizes and think times have exponential distributions. We use our techniques to design the schemes jointly with admission control, i.e., the selection of the number of admissible users, to maximize throughput for given p, r, and ε. Next we consider the case in which the number of users is large. Under a specific scaling, we derive explicit large deviations asymptotics for both models. We discuss the extension to general distributions of user data and think times

A new, analysis-based, change of measure for tandem queues

In this paper, we introduce a simple analytical approximation for the overflow probability of a two-node tandem queue. From this, we derive a change of measure, which turns out to have good performance in almost the entire parameter space. The form of our new change of measure sheds an interesting new light on earlier changes of measure for the same problem, because here the transition zone from one measure to another - that they all have - arises naturally.\u

Alternative proof and interpretations for a recent state-dependent importance sampling scheme

Recently, a state-dependent change of measure for simulating overflows in the two-node tandem queue was proposed by Dupuis et al. (Ann. Appl. Probab. 17(4):1306–1346, 2007), together with a proof of its asymptotic optimality. In the present paper, we present an alternative, shorter and simpler proof. As a side result, we obtain interpretations for several of the quantities involved in the change of measure in terms of likelihood ratios

Transient Behaviour in Highly Dependable Markovian Systems: New Regimes, Multiple Paths

In recent years, probabilistic analysis of highly dependable Markovian systems has received considerable attention. Such systems typically consist of several component types, subject to failures, with spare components for replacement while repair is taking place. System failure occurs when all (spare) components of one or several types have failed. In this work we try to estimate the probability of system failure before some fixed time bound $\tau$ via stochastic simulation. Obviously, in a highly dependable system, system failure is a rare event, so we apply importance sampling (IS) techniques, based on knowledge of the behaviour of the system and the way the rare event occurs. In our talk we discern several interesting ways in which the rare event can occur, each of which has its own way of affecting the efficiency of an importance sampling technique

On a generic class of two-node queueing systems

This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on–off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics

Diffusion parameters of flows in stable queueing networks

We consider open multi-class queueing networks with general arrival processes, general processing time sequences and Bernoulli routing. The network is assumed to be operating under an arbitrary work-conserving scheduling policy that makes the system stable. An example is a generalized Jackson network with load less than unity and any work conserving policy. We find a simple diffusion limit for the inter-queue flows with an explicit computable expression for the covariance matrix. Specifically, we present a simple computable expression for the asymptotic variance of arrivals (or departures) of each of the individual queues and each of the flows in the network