857 research outputs found

    Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic

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    A result of Ward and Glynn (2005) asserts that the sequence of scaled offered waiting time processes of the GI/GI/1+GIGI/GI/1+GI 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 GI/GI/1+GIGI/GI/1+GI 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 GI/GI/1GI/GI/1 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

    Simple and explicit bounds for multi-server queues with 1/(1ρ)1/(1 - \rho) (and sometimes better) scaling

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    We consider the FCFS GI/GI/nGI/GI/n queue, and prove the first simple and explicit bounds that scale as 11ρ\frac{1}{1-\rho} (and sometimes better). Here ρ\rho 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 AA and SS respectively) have finite rrth moment for some r>2.r > 2. Let μA\mu_A (respectively μS\mu_S) denote 1E[A]\frac{1}{\mathbb{E}[A]} (respectively 1E[S]\frac{1}{\mathbb{E}[S]}). Then our bounds (also for higher moments) are simple and explicit functions of E[(AμA)r],E[(SμS)r],r\mathbb{E}\big[(A \mu_A)^r\big], \mathbb{E}\big[(S \mu_S)^r\big], r, and 11ρ\frac{1}{1-\rho} 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 11ρ\frac{1}{1-\rho} in certain asymptotic regimes. More precisely, they scale as 11ρ\frac{1}{1-\rho} multiplied by an inverse polynomial in n(1ρ)2.n(1 - \rho)^2. These results formalize the intuition that bounds should be tighter in light traffic as well as certain heavy-traffic regimes (e.g. with ρ\rho fixed and nn 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

    Simulating Tail Probabilities in GI/GI.1 Queues and Insurance Risk Processes with Subexponentail Distributions

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    This paper deals with estimating small tail probabilities of thesteady-state waiting time in a GI/GI/1 queue with heavy-tailed (subexponential) service times. The problem of estimating infinite horizon ruin probabilities in insurance risk processes with heavy-tailed claims can be transformed into the same framework. It is well-known that naive simulation is ineffective for estimating small probabilities and special fast simulation techniques like importance sampling, multilevel splitting, etc., have to be used. Though there exists a vast amount of literature on the rare event simulation of queuing systems and networks with light-tailed distributions, previous fast simulation techniques for queues with subexponential service times have been confined to the M/GI/1 queue. The general approach is to use the Pollaczek-Khintchine transformation to convert the problem into that of estimating the tail distribution of a geometric sum of independent subexponential random variables. However, no such useful transformation exists when one goes from Poisson arrivals to general interarrival-time distributions. We describe and evaluate an approach that is based on directly simulating the random walk associated with the waiting-time process of the GI/GI/1 queue, using a change of measure called delayed subexponential twisting -an importance sampling idea recently developed and found useful in the context of M/GI/1 heavy-tailed simulations

    Large deviations analysis for the M/H2/n+MM/H_2/n + M queue in the Halfin-Whitt regime

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    We consider the FCFS M/H2/n+MM/H_2/n + M 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 limxx2log(P(W>x))\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x)) 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

    Combined analysis of transient delay characteristics and delay autocorrelation function in the Geo(X)/G/1 queue

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    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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