System-theoretical algorithmic solution to waiting times in semi-Markov queues

Cataloged from PDF version of article.Markov renewal processes with matrix-exponential semi-Markov kernels provide a generic tool for modeling auto-correlated interarrival and service times in queueing systems. In this paper, we study the steady-state actual waiting time distribution in an infinite capacity single-server semi-Markov queue with the auto-correlation in interarrival and service times modeled by Markov renewal processes with matrix-exponential kernels. Our approach is based on the equivalence between the waiting time distribution of this semi-Markov queue and the output of a linear feedback interconnection system. The unknown parameters of the latter system need to be determined through the solution of a SDC (Spectral-Divide-and-Conquer) problem for which we propose to use the ordered Schur decomposition. This approach leads us to a completely matrix-analytical algorithm to calculate the steady-state waiting time which has a matrix-exponential distribution. Besides its unifying structure, the proposed algorithm is easy to implement and is computationally efficient and stable. We validate the effectiveness and the generality of the proposed approach through numerical examples. © 2009 Elsevier B.V. All rights reserve

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

We investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival. We focus on the probability that this bivariate reserve process survives indefinitely. The infinite- horizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with reflecting boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process. Under assumptions on the arrival process and the claim amounts, and using Wiener-Hopf factor- ization with one parameter, we explicitly determine the Laplace-Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution. Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.Comment: 24 pages, 6 figure

Stability of Service under Time-of-Use Pricing

We consider "time-of-use" pricing as a technique for matching supply and demand of temporal resources with the goal of maximizing social welfare. Relevant examples include energy, computing resources on a cloud computing platform, and charging stations for electric vehicles, among many others. A client/job in this setting has a window of time during which he needs service, and a particular value for obtaining it. We assume a stochastic model for demand, where each job materializes with some probability via an independent Bernoulli trial. Given a per-time-unit pricing of resources, any realized job will first try to get served by the cheapest available resource in its window and, failing that, will try to find service at the next cheapest available resource, and so on. Thus, the natural stochastic fluctuations in demand have the potential to lead to cascading overload events. Our main result shows that setting prices so as to optimally handle the {\em expected} demand works well: with high probability, when the actual demand is instantiated, the system is stable and the expected value of the jobs served is very close to that of the optimal offline algorithm.Comment: To appear in STOC'1

Receiver-Based Flow Control for Networks in Overload

We consider utility maximization in networks where the sources do not employ flow control and may consequently overload the network. In the absence of flow control at the sources, some packets will inevitably have to be dropped when the network is in overload. To that end, we first develop a distributed, threshold-based packet dropping policy that maximizes the weighted sum throughput. Next, we consider utility maximization and develop a receiver-based flow control scheme that, when combined with threshold-based packet dropping, achieves the optimal utility. The flow control scheme creates virtual queues at the receivers as a push-back mechanism to optimize the amount of data delivered to the destinations via back-pressure routing. A novel feature of our scheme is that a utility function can be assigned to a collection of flows, generalizing the traditional approach of optimizing per-flow utilities. Our control policies use finite-buffer queues and are independent of arrival statistics. Their near-optimal performance is proved and further supported by simulation results.Comment: 14 pages, 4 figures, 5 tables, preprint submitted to IEEE INFOCOM 201