Stochastic Models

Agapie, Alexandru; Fuenten, Thomas aus der

doi: 10.1080/15326340802427364pmid: N/A

Let ℤ d represent a set of voters who can hold either of two opinions, zero or one. At each moment of time a voter is chosen at random and changes her opinion according to the opinion of her neighborhood. If the transition probability to one for a site with no one in its current neighborhood is zero, the Markov chain has two attractors at all zeros and all ones. Otherwise—and this is the case we tackle—the chain is ergodic. Previous work on the 1-D model with neighborhood two/three led to a closed form expression for the stationary distribution, function of the number of 0–1 borders within the configuration. We now extrapolate the results to the five-neighborhood case and to the 2-D voter model.

Abbad, Mohammed; Rahhali, Khalid

doi: 10.1080/15326340802427406pmid: N/A

In this article, weighted reward Markov decision processes (semi-infinite WMDP) with finite state and countable action spaces are considered. The “weighted reward” refers to an appropriately normalized convex combination of the discounted and long-run average reward criteria. This criterion allows the controller to trade-off short-term rewards versus long-run rewards. We prove that under some conditions, the supremum in the class of general strategies is equivalent to the supremum in the class of relatively simple “ultimately deterministic” strategies. These are strategies that behave just like deterministic stationary strategies, after a certain point in time. We present an iterative algorithm for computing a δ-optimal simple ultimately deterministic strategy. The steps of the algorithm are based on the one developed in Ref. for finite WMDP.

Abramov, Vyacheslav M.

doi: 10.1080/15326340802427430pmid: N/A

This article studies a single-server queueing system with autonomous service and ℓ priority classes. Arrival and departure processes are governed by marked point processes. There are ℓ buffers corresponding to priority classes, and upon arrival a unit of the kth priority class occupies a place in the kth buffer. Let N (k), k = 1,2,…, ℓ denote the quota for the total kth buffer content. The values N (k) are assumed to be large, and queueing systems both with finite and infinite buffers are studied. In the case of a system with finite buffers, the values N (k) characterize buffer capacities. This article discusses a circle of problems related to optimization of performance measures associated with overflowing the quota of buffer contents, particularly buffers models. Our approach to this problem is new, and the presentation of our results is simple and clear for real applications.

Iyer, Srikanth K.; Manjunath, D.; Yogeshwaran, D.

doi: 10.1080/15326340802427448pmid: N/A

Let P: = {X i } i≥1 be a stationary Poisson point process in ℜ d , {C i } i≥1 be a sequence of i.i.d. random sets in ℜ d , and be i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the Markov–Poisson–Boolean model . 𝒞 t represents the coverage process at time t. We first obtain limit laws for k-coverage of an area at an arbitrary instant. We then obtain the limit laws for the k-coverage seen by a particle as it moves along a one-dimensional path.

Jonckheere, M.; Leskelä, L.

doi: 10.1080/15326340802427471pmid: N/A

This article studies multi-class loss systems with two layers of servers, where each server at the first layer is dedicated to a certain customer class, while the servers at the second layer can handle all customer classes. The routing of customers follows an overflow scheme, where arriving customers are preferentially directed to the first layer. Stochastic comparison and coupling techniques are developed for studying how the system is affected by the packing of customers, altered service rates, and altered server configurations. This analysis leads to computationally fast upper and lower bounds for the performance of the system.

Boxma, Onno; Wal, Jan van der; Yechiali, Uri

doi: 10.1080/15326340802427497pmid: N/A

This article considers a batch service polling system. We first study the case in which the server visits the queues cyclically, considering three different service regimes: gated, exhaustive, and globally gated. We subsequently analyze the case (the so-called “Israeli Queue”) in which the server first visits the queue with the “oldest” customer. In both cases, queue lengths and waiting times are the main performance measures under consideration.

Bouallouche-medjkoune, Louiza; Aissani, Djamil

doi: 10.1080/15326340802427521pmid: N/A

The main purpose of this article is to use the strong stability method to approximate the characteristics of the M 2/G 2/1 priority system with non-preemptive priority by those of the M/G/1 queue, when the arrival intensity of the priority stream is sufficiently small. This last queue is simpler and more exploitable. For this, we clarify the stability conditions and next obtain stability quantitative estimates with an exact computation of constants. From these theoretical results, we elaborate an algorithm allowing to verify the approximation conditions and to provide the made numerical error. In order to have an idea about the efficiency of this approach, we consider a concrete example whose results are compared with those obtained by our simulator.