Jorma Virtamo

M/M/1-PS queue and size-aware task assignment

We consider a distributed server system in which heterogeneous servers operate under the processor sharing (PS) discipline. Exponentially distributed jobs arrive to a dispatcher, which assigns each task to one of the servers. In the so-called size-aware system, the dispatcher is assumed to know the remaining service requirements of some or all of the existing jobs in each server. The aim is to minimize the mean sojourn time, i.e., the mean response time. To this end, we first analyze an M/M/1-PS queue in the framework of Markov decision processes, and derive the so-called size-aware relative value of state, which sums up the deviation from the average rate at which sojourn times are accumulated in the infinite time horizon. This task turns out to be non-trivial. The exact analysis yields an infinite system of first order differential equations, for which an explicit solution is derived. The relative values are then utilized to develop efficient dispatching policies by means of the first policy iteration (FPI). Numerically, we show that for the exponentially distributed job sizes the myopic approach, ignoring the future arrivals, yields an efficient and robust policy when compared to other heuristics. However, in the case of highly asymmetric service rates, an FPI based policy outperforms it. Additionally, the size-aware relative value of an M/G/1-PS queue is shown to be sensitive with respect to the form of job size distribution, and indeed, the numerical experiments with constant job sizes confirm that the optimal decision depends on the job size distribution.

Performance analysis of finite-source retrial queues operating in random environments

This paper is concerned with the performance analysis of finite-source retrial queues with heterogeneous sources operating in random environments. All random variables involved in the model construction are assumed to be exponentially distributed with a parameter depending on the source index and on the state of the corresponding random environment. The novelty of the investigation is the involvement of the random environments, which makes the system rather complicated. The MOSEL tool is used to formulate and solve the problem and the main performance measures are derived and graphically displayed.

Criticality condition for information floating with random walk of nodes

In an opportunistic content sharing system referred to as floating content, information is copied between mobile nodes upon node encounters inside an area which is called the anchor zone. We study the conditions under which information can be sustained in such a system. The anchor zone is assumed to be a circular disk, and a random walk type mobility model is adopted. First, we consider the one-speed case where all the nodes have a common velocity. Using the transport equation, adopted from nuclear reactor theory, we derive the criticality condition that defines a lower limit for the product of node density, communication distance and the radius of the anchor zone necessary for information floating. The dependence of this criticality parameter on the mean step size of the random walk is numerically established. Complemented by the asymptotic behavior, found by diffusion theory, an accurate approximation formula is derived. While the velocity of the nodes does not appear at all in the criticality condition of the one-speed system, in general, the shape of the velocity distribution has an important effect: the higher the spread of the distribution, the lower the criticality threshold is. This effect is analyzed and discussed.

Maximum weight independent sets in an infinite plane with uni- and bidirectional interference models

We study the maximum weight independent sets of links between nodes distributed randomly in an infinite plane. Different definitions of the weight of a link are considered, leading to slight variations of what is essentially a spatial reuse problem in wireless multihop networks. A simple interference model is assumed with the interference radius equaling the transmission radius. In addition to unidirectional interference from a transmitter to the receivers of other links, also an RTS/CTS-type bidirectional handshake is considered. We study both the case where the transmission radius is fixed and tunable through power control. With a fixed transmission radius, we derive asymptotic results for the low- and high-density regimes. The main contribution is in the numerical results for the maximum weight, establishing some previously unknown parameters of stochastic geometry. The results are obtained by the Moving Window Algorithm that is able to find the maximum weight independent set in a strip of limited height but unlimited length. By studying the results as a function of the height of the strip, we are able to extrapolate to the infinite plane.

Window based estimation of the intensity of a doubly stochastic poisson process

Tasks related to advanced management of telecommunications networks, such as optimal dynamic routing or dynamic allocation of bandwidth for virtual paths in an ATM network, call for a method to estimate the current call arrival intensities. Such an estimate must be based on the observed history of the call arrival instants. In this paper, we study window type estimators with the purpose of giving a clear idea of their accuracy. In particular, the simple straight case and exponential windows are considered and applied to an example where the correlation function of the intensity process is of the exponential type - a system for which the exponential window, in fact, is the optimal window. It turns out that the accuracy is controlled by a single dimensionless traffic parameter. The achievable accuracy is little affected by the shape of the window function. It also turns out that a Bayesian approach, applied in an earlier study, gives only a slight improvement over the window based estimator. More important instead is the proper choice of the size of the window. In order to find the optimal size, one has to know some basic characteristics of the intensity process. We briefly address the problem how these can be inferred from the long term statistics of the counting process.

Continuum Percolation Threshold for Permeable Aligned Cylinders and Opportunistic Networking

We consider the critical percolation threshold for aligned cylinders, which provides a lower bound for the required node degree for the permanence of information in opportunistic networking. The height of a cylinder corresponds to the time a node is active in its current location. By means of Monte Carlo simulations, we obtain an accurate numerical estimate for the critical reduced number density, ηc ≈ 0.3312(1) for constant height cylinders. This threshold is the same for all ratios of the height to the diameter of the base, and corresponds to the mean node degree of 1.3248 in opportunistic networking, which is clearly below the percolation threshold of 4.51 above which a gigantic connected component emerges in the network.

On the achievable forwarding capacity of an infinite wireless network

We consider the problem of finding the maximum directed packet flow that can be sustained in an infinite wireless multihop network. This ability of the network to relay traffic is called the forwarding capacity, and the problem appears when the spatial scales corresponding to the end-to-end paths (routing) and the neighboring nodes (forwarding) are strongly separated in a massively dense network. We assume a Boolean interference model. The infinite network is approximated with a finite but large network where the node locations form a spatial Poisson process. We study two constructive approaches to tighten the lower bound for the forwarding capacity by a significant amount. In path scheduling the packets traverse the network using predefined paths that do not interfere with each other, and coordination is thus required only between the nodes of a path. In greedy maximum weight scheduling, the transmissions are scheduled greedily according to queue-length based weights of the links. In addition to a fixed transmission radius, we consider greedy maximum weight scheduling with a transmission radius adjustable up to a given maximum. We are able to produce numerical results that characterize the achievable forwarding capacity under global coordination of the transmissions, providing, e.g., concrete points of reference for practical distributed implementations.

Meeting Soft Deadlines in Single- and Multi-server Systems

We consider single-and multi-server systems, where jobs have a maximum waiting time (deadline) defined, e.g., by a service level agreement. A fixed cost is associated with deadline violations and the task is to minimize the long-run cumulative costs. Job sizes (service durations) are observed upon arrival, and current queue backlogs are known. For a single FCFS server, the optimization task is to find the optimal admission policy that may reject a job upon arrival if admitting it would cause in future one or more deadlines to be violated (in expectation). For parallel FCFS servers, the policy must (i) either accept or reject a job upon arrival, and if accepted, (ii) assign it to one of the servers. We derive efficient deadline-aware policies in the MDP framework. For a single server, we obtain the optimal admission policy. For dispatching to parallel servers, we develop efficient heuristic admission and dispatching policies, whose performances are evaluated by means of numerical examples. Additionally, we give some exact closed-form results for heavy-traffic limits.

Optimal transmission modes by simulated annealing

We address the problem of finding efficient combinations of transmitting links between nodes distributed as a spatial Poisson process in an infinite plane using a stochastic optimization method called simulated annealing. A simple Boolean interference model with the interference radius equaling the transmission radius is used to verify the operation of the method. The same approach is then applied to SINR-determined data rates. The obtained numerical results shed light on the spatial reuse problem in wireless multihop networks. In particular, for the SINR-based interference model we obtain new results. We characterize the asymptotic behavior of the sum capacity of the optimal combination of transmitting links and the fraction of transmitting nodes in the low and high interference regimes. Additionally, the numerical results establish, in the interference-limited case of high node densities, the sum capacity (spectral efficiency) to be approximately equal to 1.2 bit/s/Hz per node.

Impact of Multidirectional Forwarding on the Capacity of Large Wireless Networks

We consider the capacity problem in large wireless multihop networks by separating the problem into macroscopic and microscopic level subproblems. At the macroscopic level, the task is to determine the routes that are geometric curves. At the microscopic level, we need to forward the traffic according to its directional distribution that results from the macroscopic level routes. Previous studies have treated the macroscopic level problem simply as that of load balancing, implying the use of time sharing to serve the traffic flowing in different directions. We study how a scheduling that truly interleaves the traffic flows in different directions affects the macroscopic level problem. We restrict the macroscopic routes to certain simple path sets. This together with earlier results on the microscopic level forwarding capacity allows us to obtain new results on the macroscopic level capacity that demonstrate the gains from multidirectional forwarding.