8,235 research outputs found
Relaxation time for the discrete D/G/1 queue
When queueing models are used for performance analysis of some stochastic system, it is usually assumed that the system is in steady-state. Whether or not this is a realistic assumption depends on the speed at which the system tends to its steady-state. A characterization of this speed is known in the queueing literature as relaxation time. The discrete D/G/1 queue has a wide range of applications. We derive relaxation time asymptotics for the discrete D/G/1 queue in a purely analytical way, mostly relying on the saddle point method. We present a simple and useful approximate upper bound which is sharp in case the load on the system is not very high. A sharpening of this upper bound, which involves the complementary error function, is then developed and this covers both the cases of low and high loads. For the discrete D/G/1 queue, the stationary waiting time distribution can be expressed in terms of infinite series that follow from Spitzer’s identity. These series involve convolutions of the probability distribution of a discrete random variable, which makes them suitable for computation. For practical purposes, though, the infinite series should be truncated. The relaxation time asymptotics can be applied to determine an appropriate truncation level based on a sharp estimate of the error caused by truncating
Bits Through Bufferless Queues
This paper investigates the capacity of a channel in which information is
conveyed by the timing of consecutive packets passing through a queue with
independent and identically distributed service times. Such timing channels are
commonly studied under the assumption of a work-conserving queue. In contrast,
this paper studies the case of a bufferless queue that drops arriving packets
while a packet is in service. Under this bufferless model, the paper provides
upper bounds on the capacity of timing channels and establishes achievable
rates for the case of bufferless M/M/1 and M/G/1 queues. In particular, it is
shown that a bufferless M/M/1 queue at worst suffers less than 10% reduction in
capacity when compared to an M/M/1 work-conserving queue.Comment: 8 pages, 3 figures, accepted in 51st Annual Allerton Conference on
Communication, Control, and Computing, University of Illinois, Monticello,
Illinois, Oct 2-4, 201
Stability Analysis of GI/G/c/K Retrial Queue with Constant Retrial Rate
We consider a GI/G/c/K-type retrial queueing system with constant retrial
rate. The system consists of a primary queue and an orbit queue. The primary
queue has identical servers and can accommodate the maximal number of
jobs. If a newly arriving job finds the full primary queue, it joins the orbit.
The original primary jobs arrive to the system according to a renewal process.
The jobs have general i.i.d. service times. A job in front of the orbit queue
retries to enter the primary queue after an exponentially distributed time
independent of the orbit queue length. Telephone exchange systems, Medium
Access Protocols and short TCP transfers are just some applications of the
proposed queueing system. For this system we establish minimal sufficient
stability conditions. Our model is very general. In addition, to the known
particular cases (e.g., M/G/1/1 or M/M/c/c systems), the proposed model covers
as particular cases the deterministic service model and the Erlang model with
constant retrial rate. The latter particular cases have not been considered in
the past. The obtained stability conditions have clear probabilistic
interpretation
Scaling of avalanche queues in directed dissipative sandpiles
We simulate queues of activity in a directed sandpile automaton in 1+1
dimensions by adding grains at the top row with driving rate .
The duration of elementary avalanches is exactly described by the distribution
, limited either by the system size or by
dissipation at defects . Recognizing the probability
as a distribution of service time of jobs arriving at a server with frequency
, the model represents a new example of the server
queue in the queue theory. We study numerically and analytically the tail
behavior of the distributions of busy periods and energy dissipated in the
queue and the probability of an infinite queue as a function of driving rate.Comment: 11 pages, 9 figures; To appear in Phys. Rev.
Queuing with future information
We study an admissions control problem, where a queue with service rate
receives incoming jobs at rate , and the decision maker is
allowed to redirect away jobs up to a rate of , with the objective of
minimizing the time-average queue length. We show that the amount of
information about the future has a significant impact on system performance, in
the heavy-traffic regime. When the future is unknown, the optimal average queue
length diverges at rate , as . In sharp contrast, when all future arrival and service times are revealed
beforehand, the optimal average queue length converges to a finite constant,
, as . We further show that the finite limit of
can be achieved using only a finite lookahead window starting from the current
time frame, whose length scales as , as
. This leads to the conjecture of an interesting duality between
queuing delay and the amount of information about the future.Comment: Published in at http://dx.doi.org/10.1214/13-AAP973 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On Sojourn Times in the Finite Capacity Queue with Processor Sharing
We consider a processor shared queue that can accommodate at most a
finite number of customers. We give an exact expression for the sojourn
time distribution in the finite capacity model, in terms of a Laplace
transform. We then give the tail behavior, for the limit , by
locating the dominant singularity of the Laplace transform.Comment: 10 page
Restless bandit marginal productivity indices II: multiproject case and scheduling a multiclass make-to-order/-stock M/G/1 queue
This paper develops a framework based on convex optimization and economic ideas to formulate and solve approximately a rich class of dynamic and stochastic resource allocation problems, fitting in a generic discrete-state multi-project restless bandit problem (RBP). It draws on the single-project framework in the author's companion paper "Restless bandit marginal productivity indices I: Single-project case and optimal control of a make-to-stock M/G/1 queue", based on characterization of a project's marginal productivity index (MPI). Our framework significantly expands the scope of Whittle (1988)'s seminal approach to the RBP. Contributions include: (i) Formulation of a generic multi-project RBP, and algorithmic solution via single-project MPIs of a relaxed problem, giving a lower bound on optimal cost performance; (ii) a heuristic MPI-based hedging point and index policy; (iii) application of the MPI policy and bound to the problem of dynamic scheduling for a multiclass combined MTO/MTS M/G/1 queue with convex backorder and stock holding cost rates, under the LRA criterion; and (iv) results of a computational study on the MPI bound and policy, showing the latter's near-optimality across the cases investigated
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