1,084 research outputs found
Stability of the Greedy Algorithm on the Circle
We consider a single-server system with service stations in each point of the
circle. Customers arrive after exponential times at uniformly-distributed
locations. The server moves at finite speed and adopts a greedy routing
mechanism. It was conjectured by Coffman and Gilbert in~1987 that the service
rate exceeding the arrival rate is a sufficient condition for the system to be
positive recurrent, for any value of the speed. In this paper we show that the
conjecture holds true
Greedy walk on the real line
We consider a self-interacting process described in terms of a single-server
system with service stations at each point of the real line. The customer
arrivals are given by a Poisson point processes on the space-time half plane.
The server adopts a greedy routing mechanism, traveling toward the nearest
customer, and ignoring new arrivals while in transit. We study the trajectories
of the server and show that its asymptotic position diverges logarithmically in
time.Comment: Published at http://dx.doi.org/10.1214/13-AOP898 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Queuing for an infinite bus line and aging branching process
We study a queueing system with Poisson arrivals on a bus line indexed by
integers. The buses move at constant speed to the right and the time of service
per customer getting on the bus is fixed. The customers arriving at station i
wait for a bus if this latter is less than d\_i stations before, where d\_i is
non-decreasing. We determine the asymptotic behavior of a single bus and when
two buses eventually coalesce almost surely by coupling arguments. Three
regimes appear, two of which leading to a.s. coalescing of the buses.The
approach relies on a connection with aged structured branching processes with
immigration and varying environment. We need to prove a Kesten Stigum type
theorem, i.e. the a.s. convergence of the successive size of the branching
process normalized by its mean. The technics developed combines a spine
approach for multitype branching process in varying environment and geometric
ergodicity along the spine to control the increments of the normalized process
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Analysis of a class of distributed queues with application
Recently we have developed a class of media access control algorithms for different types of Local Area Networks. A common feature of these LAN algorithms is that they represent various strategies by which the processors in the LAN can simulate the availability of a centralized packet transport facility, but whose service incorporates a particular type of change over time known as 'moving sever' overhead. First we describe the operation of moving server systems in general, for both First-Come - First-Served and Head-of-the-Line orders of service, together with an approach for their delay analysis in which we transform the moving server queueing system into a conventional queueing system having proportional waiting times. Then we describe how the various LAN algorithms may be obtained from the ideal moving server system, and how a significant component of their performance characteristics is determined by the performance characteristics of that ideal system. Finally, we evaluate the compatibility of such LAN algorithms with separable queueing network models of distributed systems by computing the interdeparture time distribution for M/M/1 in the presence of moving server overhead. Although it is not exponential, except in the limits of low server utilization or low overhead, the interdeparture time distribution is a weighted sum of exponential terms with a coefficient of variation not much smaller than unity. Thus, we conjecture that a service centre with moving server overhead could be used to represent one of these LAN algorithms in a product form queueing network model of a distributed system without introducing significant approximation errors
Stability criteria for controlled queueing networks
We give criteria for the stability of a very general queueing model under different levels of control. A complete classification of stability (or positive recurrence), transience and null-recurrence is presented for the two queue model. The stability and instability results are extended for models with N > 3 queues. We look at a broad class of models which can have the following features: Customers arrive at one, several or all of the queues from the outside with exponential inter arrival times. We often have the case where a arrival stream can be routed so that under different routing schemes each queue can have external arrivals, i.e. we assume we have some control over the routing of the arrivals. We also consider models where the arrival streams are fixed. We view the service in a more abstract way, in that we allow a number к of different service configurations. Under every such service configuration service is provided to some or all of the queues, length of service time can change from one service configuration to another and we can change from one configuration to another according two some control policy. The service times are assumed to be exponentially distributed. The queueing models we consider are networks where, after completion at one queue, a customer might be fed back into another queue where it will be served another time often under with a different service time. These feedback probabilities change with the service configurations. Our interest is in different types of control policies which allow us to change the routing of arrivals and configurations of the service from time to time so that the controlled queue length process (which in most cases is Markov) is stable. The semi-martingale or Lyapunov function methods we use give necessary and sufficient conditions for the stability classification. We will look at some two queue models with different inter arrival and service times where the queueing process is still Markov
A Real-time Calculus Approach for Integrating Sporadic Events in Time-triggered Systems
In time-triggered systems, where the schedule table is predefined and
statically configured at design time, sporadic event-triggered (ET) tasks can
only be handled within specially dedicated slots or when time-triggered (TT)
tasks finish their execution early. We introduce a new paradigm for
synthesizing TT schedules that guarantee the correct temporal behavior of TT
tasks and the schedulability of sporadic ET tasks with arbitrary deadlines. The
approach first expresses a constraint for the TT task schedule in the form of a
maximal affine envelope that guarantees that as long as the schedule generation
respects this envelope, all sporadic ET tasks meet their deadline. The second
step consists of modeling this envelope as a burst limiting constraint and
building the TT schedule via simulating a modified Least-Laxity-First (LLF)
scheduler. Using this novel technique, we show that we achieve equal or better
schedulability and a faster schedule generation for most use-cases compared to
other approaches inspired by, e.g., hierarchical scheduling. Moreover, we
present an extension to our method that finds the most favourable schedule for
TT tasks with respect to ET schedulability, thus increasing the probability of
the computed TT schedule remaining feasible when ET tasks are later added or
changed
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