1,066 research outputs found
Simple and explicit bounds for multi-server queues with (and sometimes better) scaling
We consider the FCFS queue, and prove the first simple and explicit
bounds that scale as (and sometimes better). Here
denotes the corresponding traffic intensity. Conceptually, our results can be
viewed as a multi-server analogue of Kingman's bound. Our main results are
bounds for the tail of the steady-state queue length and the steady-state
probability of delay. The strength of our bounds (e.g. in the form of tail
decay rate) is a function of how many moments of the inter-arrival and service
distributions are assumed finite. More formally, suppose that the inter-arrival
and service times (distributed as random variables and respectively)
have finite th moment for some Let (respectively )
denote (respectively ). Then
our bounds (also for higher moments) are simple and explicit functions of
, and
only. Our bounds scale gracefully even when the number of
servers grows large and the traffic intensity converges to unity
simultaneously, as in the Halfin-Whitt scaling regime. Some of our bounds scale
better than in certain asymptotic regimes. More precisely,
they scale as multiplied by an inverse polynomial in These results formalize the intuition that bounds should be tighter
in light traffic as well as certain heavy-traffic regimes (e.g. with
fixed and large). In these same asymptotic regimes we also prove bounds for
the tail of the steady-state number in service.
Our main proofs proceed by explicitly analyzing the bounding process which
arises in the stochastic comparison bounds of amarnik and Goldberg for
multi-server queues. Along the way we derive several novel results for suprema
of random walks and pooled renewal processes which may be of independent
interest. We also prove several additional bounds using drift arguments (which
have much smaller pre-factors), and make several conjectures which would imply
further related bounds and generalizations
Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic
A result of Ward and Glynn (2005) asserts that the sequence of scaled offered
waiting time processes of the queue converges weakly to a
reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the
traffic intensity approaches one. As a consequence, the stationary distribution
of a ROU process, which is a truncated normal, should approximate the scaled
stationary distribution of the offered waiting time in a queue;
however, no such result has been proved. We prove the aforementioned
convergence, and the convergence of the moments, in heavy traffic, thus
resolving a question left open in Ward and Glynn (2005). In comparison to
Kingman's classical result in Kingman (1961) showing that an exponential
distribution approximates the scaled stationary offered waiting time
distribution in a queue in heavy traffic, our result confirms that
the addition of customer abandonment has a non-trivial effect on the queue
stationary behavior.Comment: 29 page
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
Performance analysis of a fluid queue with random service rate in discrete-time
We consider a fluid queue in discrete time with random service rate. Such a queue has been used in several recent studies on wireless networks where the packets can be arbitrarily fragmented. We provide conditions on finiteness of moments of stationary delay, its Laplace-Stieltjes transform and various approximations under heavy traffic. Results are extended to the case where the wireless link can transmit in only a few slots during a frame
Combined analysis of transient delay characteristics and delay autocorrelation function in the Geo(X)/G/1 queue
We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state
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