219 research outputs found
On the speed of convergence to stationarity of the Erlang loss system
We consider the Erlang loss system, characterized by servers, Poisson arrivals and exponential service times, and allow the arrival rate to be a function of We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and display some bounds for the total variation distance between the time-dependent and stationary distributions. We also pay attention to time-dependent rates
Delay versus Stickiness Violation Trade-offs for Load Balancing in Large-Scale Data Centers
Most load balancing techniques implemented in current data centers tend to
rely on a mapping from packets to server IP addresses through a hash value
calculated from the flow five-tuple. The hash calculation allows extremely fast
packet forwarding and provides flow `stickiness', meaning that all packets
belonging to the same flow get dispatched to the same server. Unfortunately,
such static hashing may not yield an optimal degree of load balancing, e.g.,
due to variations in server processing speeds or traffic patterns. On the other
hand, dynamic schemes, such as the Join-the-Shortest-Queue (JSQ) scheme,
provide a natural way to mitigate load imbalances, but at the expense of
stickiness violation.
In the present paper we examine the fundamental trade-off between stickiness
violation and packet-level latency performance in large-scale data centers. We
establish that stringent flow stickiness carries a significant performance
penalty in terms of packet-level delay. Moreover, relaxing the stickiness
requirement by a minuscule amount is highly effective in clipping the tail of
the latency distribution. We further propose a bin-based load balancing scheme
that achieves a good balance among scalability, stickiness violation and
packet-level delay performance. Extensive simulation experiments corroborate
the analytical results and validate the effectiveness of the bin-based load
balancing scheme
On the rate of convergence to stationarity of the M/M/N queue in the Halfin-Whitt regime
We prove several results about the rate of convergence to stationarity, that
is, the spectral gap, for the M/M/n queue in the Halfin-Whitt regime. We
identify the limiting rate of convergence to steady-state, and discover an
asymptotic phase transition that occurs w.r.t. this rate. In particular, we
demonstrate the existence of a constant s.t. when a certain
excess parameter , the error in the steady-state approximation
converges exponentially fast to zero at rate . For , the
error in the steady-state approximation converges exponentially fast to zero at
a different rate, which is the solution to an explicit equation given in terms
of special functions. This result may be interpreted as an asymptotic version
of a phase transition proven to occur for any fixed n by van Doorn [Stochastic
Monotonicity and Queueing Applications of Birth-death Processes (1981)
Springer]. We also prove explicit bounds on the distance to stationarity for
the M/M/n queue in the Halfin-Whitt regime, when . Our bounds scale
independently of in the Halfin-Whitt regime, and do not follow from the
weak-convergence theory.Comment: Published in at http://dx.doi.org/10.1214/12-AAP889 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Transient handover blocking probabilities in road covering cellular mobile networks
This paper investigates handover and fresh call blocking probabilities for subscribers moving along a road in a traffic jam passing through consecutive cells of a wireless network. It is observed and theoretically motivated that the handover blocking probabilities show a sharp peak in the initial part of a traffic jam roughly at the moment when the traffic jam starts covering a new cell. The theoretical motivation relates handover blocking probabilities to blocking probabilities in the M/D/C/C queue with time-varying arrival rates. We provide a numerically efficient recursion for these blocking probabilities. \u
Queues and risk processes with dependencies
We study the generalization of the G/G/1 queue obtained by relaxing the
assumption of independence between inter-arrival times and service
requirements. The analysis is carried out for the class of multivariate matrix
exponential distributions introduced in [12]. In this setting, we obtain the
steady state waiting time distribution and we show that the classical relation
between the steady state waiting time and the workload distributions re- mains
valid when the independence assumption is relaxed. We also prove duality
results with the ruin functions in an ordinary and a delayed ruin process.
These extend several known dualities between queueing and risk models in the
independent case. Finally we show that there exist stochastic order relations
between the waiting times under various instances of correlation
Transient bayesian inference for short and long-tailed GI/G/1 queueing systems
In this paper, we describe how to make Bayesian inference for the transient behaviour and busy period in a single server system with general and unknown distribution for the service and interarrival time. The dense family of Coxian distributions is used for the service and arrival process to the system. This distribution model is reparametrized such that it is possible to define a non-informative prior which allows for the approximation of heavytailed distributions. Reversible jump Markov chain Monte Carlo methods are used to estimate the predictive distribution of the interarrival and service time. Our procedure for estimating the system measures is based in recent results for known parameters which are frequently implemented by using symbolical packages. Alternatively, we propose a simple numerical technique that can be performed for every MCMC iteration so that we can estimate interesting measures, such as the transient queue length distribution. We illustrate our approach with simulated and real queues
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