Efficient rare-event simulation for perpetuities

We consider perpetuities of the form

D=_B1exp(_Y1)+_B2exp(_Y1+_Y2)+?,$D=B_{1}exp\left(Y_1\right)+B_{2}exp(Y_1+Y_2)+?\text{,}$

where the _Yj$Y_j$’s and _Bj$B_j$’s might be i.i.d. or jointly driven by a suitable Markov chain. We assume that the _Yj$Y_j$’s satisfy the so-called Cramér condition with associated root _θ∗∈(0,∞)${\theta }_{\ast }\in (0,\infty )$ and that the tails of the _Bj$B_j$’s are appropriately behaved so that D$D$ is regularly varying with index _θ∗${\theta }_{\ast }$. We illustrate by means of an example that the natural state-independent importance sampling estimator obtained by exponentially tilting the _Yj$Y_j$’s according to _θ∗${\theta }_{\ast }$ fails to provide an efficient estimator (in the sense of appropriately controlling the relative mean squared error as the tail probability of interest gets smaller). Then, we construct estimators based on state-dependent importance sampling that are rigorously shown to be efficient.     

The cross-entropy method with patching for rare-event simulation of large Markov chains

There are various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks. In this work, we present a general state-dependent importance sampling method which partitions the state space and applies the cross-entropy method to each partition. We investigate two versions of our algorithm and apply them to several examples of reliability and queueing models. In all these examples we compare our method with other importance sampling schemes. The performance of the importance sampling schemes is measured by the relative error of the estimator and by the efficiency of the algorithm. The results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator.     

Approximations for multi-server queues: System interpolations

This paper provides a unifying method of generating and/or evaluating approximations for the principal congestion measures in aGI/G/s queueing system. The main focus is on the mean waiting time, but approximations are also developed for the queue-length distribution, the waiting-time distribution and the delay probability for the Poisson arrival case. The approximations have closed forms that combine analytical solutions of simpler systems, and hence they are referred to as system-interpolation approximations or, simply, system interpolations. The method in this paper is consistent with and generalizes system interpolations previously presented for the mean waiting time in theGI/G/s queue.     

Simulation methods for queues: An overview

This paper gives an overview of those aspects of simulation methodology that are (to some extent) peculiar to the simulation of queueing systems. A generalized semi-Markov process framework for describing queueing systems is used through much of the paper. The main topics covered are: output analysis for simulation of transient and steady-state quantities, variance reduction methods that exploit queueing structure, and gradient estimation methods for performance parameters associated with queueing networks.The research of this author was supported by the U.S. Army Research Office under Contract DAAG29-84-K-0030.The research of this author was supported by the U.S. Army Research Office under Contract DAAG29-84-K-0030 and National Science Foundation Grant DCR-85-09668.     

Refining diffusion approximations for queues

This note illustrates the need to refine diffusion approximations for queues. Diffusion approximations are developed in several different ways for the mean waiting time in a GI/G/1 queue, yielding different results, all of which fail obvious consistency checks with bounds and exact values.     

Stochastic comparisons for bulk queues

We consider two important classes of single-server bulk queueing models: M^(X)/G^(Y)/1 with Poisson arrivals of customer groups, and G^(X)/m^(Y)1 with batch service times having exponential density. In each class we compare two systems and prove that one is more congested than the other if their basic random variables are stochastically ordered in an appropriate manner. However, it must be recognized that a system that appears congested to customers might be working efficiently from the system manager's point of view. We apply the results of this comparison to (i) the family {M/G^(s)/1,s 1} of systems with Poisson input of customers and batch service times with varying service capacity; (ii) the family {G^(s)/1,s 1} of systems with exponential customer service time density and group arrivals with varying group size; and (iii) the family {M/D/s,s 1} of systems with Poisson arrivals, constant service time and varying number of servers. Within each family, we find the system that is the best for customers, but this turns out to be the worst for the manager (or vice versa). We also establish upper (or lower) bounds for the expected queue length in steady state and the expected number of batches (or groups) served during a busy period. The approach of the paper is based on the stochastic comparison of random walks underlying the models.This research was partially supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University.     

Stochastic decomposition for retrial queues

Summary This paper deals with the stochastic decomposition property for retrial queues. This property is connected with similar results for vacation models. As applications, the moments of the number of customers in orbit and the rate of convergence under high retrial intensity can be obtained. This work was supported under Grant PR161/93-4777     

V-uniform ergodicity for fluid queues

In this paper, we show that a positive recurrent ?uid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a ?uid queue and a quasi-birth-and-death process. As a byproduct of V-uniform ergodicity, we derive computable bounds on the exponential moments of the busy period.     

Using the M/G/1 queue under processor sharing for exact simulation of queues

In Sigman (J. Appl. Probab. 48A:209–216, 2011b), a first exact simulation algorithm was presented for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ<1 (super stable case). The key idea involves dominated coupling from the past while using the M/G/1 queue under the processor sharing (PS) discipline as a sample-path upper bound, taking advantage of its time-reversibility properties so as to be able to simulate it backwards in time. Here, we expand upon this method and give several examples of other queueing models for which this method can be used to exactly simulate from their stationary distributions. Examples include sojourn times for single-server queues under various service disciplines, tandem queues, and multi-class networks with general routing.     

Heavy-traffic limits for nearly deterministic queues: stationary distributions

We establish heavy-traffic limits for stationary waiting times and other performance measures in G _n /G _n /1 queues, where G _n indicates that an original point process is modified by cyclic thinning of order n, i.e., the thinned process contains every nth point from the original point process. The classical example is the Erlang E _n /E _n /1 queue, where cyclic thinning of order n is applied to both the interarrival times and the service times, starting from a “base” M/M/1 model. The models G _n /D/1 and D/G _n /1 are special cases of G _n /G _n /1. Since waiting times before starting service in the G/D/n queue are equivalent to waiting times in an associated G _n /D/1 model, where the interarrival times are the sum of n consecutive interarrival times in the original model, the G/D/n model is a special case as well. As n→∞, the G _n /G _n /1 models approach the deterministic D/D/1 model. We obtain revealing limits by letting ρ _n ↑1 as n→∞, where ρ _n is the traffic intensity in model n.     

On bounds for bulk arrival queues

This paper gives a simple and effective approach pf deriving bounds for bulk arrival queues by making use of the bounds for single arrival queues. With this approach, upper bounds of mean actual/virtual waiting times and mean queue length at random epochs can be derived for the bulk arrival queues GI^X/G/1 and GI^X/G/c (lower bounds can be derived in a similar way). The merit of this approach is shown by comparing the bounds obtained with some existing results in the literature.     

Two-parameter heavy-traffic limits for infinite-server queues

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables Q ^e (t,y) and Q ^r (t,y) representing the number of customers in the system at time t that have elapsed service times less than or equal to time y, or residual service times strictly greater than y. We also consider W ^r (t,y) representing the total amount of work in service time remaining to be done at time t+y for customers in the system at time t. The two-parameter stochastic-process limits in the space D([0,∞),D) of D-valued functions in D draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (Adv. Appl. Probab. 23, 188–209, 1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (Queueing Syst. 25, 235–280, 1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.     

Bounds and heuristics for assembly-like queues

We consider an assembly system with exponential service times, and derive bounds for its average throughput and inventories. We also present an easily computed approximation for the throughput, and compare it to an existing approximation.     

Rate conservation principle for discrete-time queues

We present a point-process approach to stationary discrete-time queues where arrivals and services are synchronized. The introduction of a Palm distribution enables us to discuss the ASTA (Arrivals See Time Averages) property, and to derive the rate conservation principle. By applying the principle to the discrete-time queues we present qualitative relationships between customer- and time-stationary distributions, including Little's and Brumelle's formulas.     

Some limit theorems for regenerative queues

We consider a single server queue with the interarrival times and the service times forming a regenerative sequence. This traffic class includes the standard models: iid, periodic, Markov modulated (e.g., BMAP model of Lucantoni [18]) and their superpositions. This class also includes the recently proposed traffic models in high speed networks, exhibiting long range dependence. Under minimal conditions we obtain the rates of convergence to stationary distributions, finiteness of stationary moments, various functional limit theorems and the continuity of stationary distributions and moments. We use the continuity results to obtain approximations for stationary distributions and moments of an MMPP/GI/1 queue where the modulating chain has a countable state space. We extend all our results to feed-forward networks where the external arrivals to each queue can be regenerative. In the end we show that the output process of a leaky bucket is regenerative if the input process is and hence our results extend to a queue with arrivals controlled by a leaky bucket. This revised version was published online in June 2006 with corrections to the Cover Date.     

Light traffic for workload in queues

General exact light traffic limit theorems are given for the distribution of steadystate workloadV, in open queueing networks having as input a general stationary ergodic marked point process {(t _n ,K _n )n0 (where t_n denotes the arrival time and K_n the routing and service times of the nth customer). No independence assumptions of any kind are required of the input. As the light traffic regime, it is only required that the Palm distribution for the exogenous interarrival time converges weakly to infinity (while the service mechanism is not allowed to change much). As is already known in the context of a single-server queue, work is much easier to deal with mathematically in light traffic than is customer delayD, and consequently, our results are far more general than existing results forD. We obtain analogous results for multi-channel and infinite-channel queues. In the context of open queueing networks, we handle both the total workload in the network as well as the workload at isolated nodes.Research supported in part by the Japan Society for the Promotion of Science during the author's fellowship in Tokyo, and by NSF Grant DDM 895 7825.     

Fast simulation of buffer overflows in tandem networks ofGI/GI/1 queues

Simply because of their rarity, the estimation of the statistics of buffer overflows in well-dimensioned queueing networks via direct simulation is extremely costly. One technique that can be used to reduce this cost is importance sampling, and it has been shown previously that large deviations theory can be used in conjunction with importance sampling to minimize the required simulation time. In this paper, we obtain results on the fast simulation of tandem networks of queues, and derive an analytic solution to the problem of finding an optimal simulation system for a class of tandem networks ofGI/GI/1 queues.Work supported by Australian Telecommunications and Electronics Research Board (ATERB). The authors wish to acknowledge the funding of the activities of the Cooperative Research Centre for Robust and Adaptive Systems by the Australian Commonwealth Government under the Cooperative Research Centres Program.     

Processor-sharing queues: Some progress in analysis

This paper reviews some recent results based on new techniques used in the analysis of main processor-sharing queueing systems. These results include the solutions of the problems of determining the sojourn time distributions and the distributions of the number of jobs in the M/G/1/t8 queue under egalitarian and feedback (foreground-background) processor-sharing disciplines. A brief discussion of some related results is also given.     

Server allocation for zero buffer tandem queues

In this paper we consider the problem of allocating servers to maximize throughput for tandem queues with no buffers. We propose an allocation method that assigns servers to stations based on the mean service times and the current number of servers assigned to each station. A number of simulations are run on different configurations to refine and verify the algorithm. The algorithm is proposed for stations with exponentially distributed service times, but where the service rate at each station may be different. We also provide some initial thoughts on the impact on the proposed allocation method of including service time distributions with different coefficients of variation.