Large Queue Research Articles

Large deviations, moderate deviations, and queues with long-range dependent input

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Large deviations, moderate deviations, and queues with long-range dependent input

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Achieving 100% throughput in an input-queued switch

It is well known that head-of-line blocking limits the throughput of an input-queued switch with first-in-first-out (FIFO) queues. Under certain conditions, the throughput can be shown to be limited to approximately 58.6%. It is also known that if non-FIFO queueing policies are used, the throughput can be increased. However, it has not been previously shown that if a suitable queueing policy and scheduling algorithm are used, then it is possible to achieve 100% throughput for all independent arrival processes. In this paper we prove this to be the case using a simple linear programming argument and quadratic Lyapunov function. In particular, we assume that each input maintains a separate FIFO queue for each output and that the switch is scheduled using a maximum weight bipartite matching algorithm. We introduce two maximum weight matching algorithms: longest queue first (LQF) and oldest cell first (OCF). Both algorithms achieve 100% throughput for all independent arrival processes. LQF favors queues with larger occupancy, ensuring that larger queues will eventually be served. However, we find that LQF can lead to the permanent starvation of short queues. OCF overcomes this limitation by favoring cells with large waiting times.

The concept of relevant time scales and its application to queuing analysis of self-similar traffic (or is Hurst naughty or nice?)

Recent traffic analyses from various packet networks have shown the existence of long-range dependence in bursty traffic. In evaluating its impact on queuing performance, earlier investigations have noted how the presence of long-range dependence, or a high value of the Hurst parameter H , is often associated with surprisingly large queue sizes. As a result, a common impression has been created of expecting queuing performance to be worse as H increases, but this impression can be misleading. In fact, there are examples in which larger values of H are associated with smaller queues. So the question is how can one tell whether queuing performance would improve or degrade as H rises? In this paper, we show that the relative queuing performance can be assessed by identifying a couple of time scales. First, in comparing a high- H process with a low- H process, there is a unique time scale t m at which the variances of the two processes match (assuming exact, second-order self similarity for both processes). Second, there are time scales t qi that are most relevant for queuing the arrivals of process i . If both of the queuing scales t qi exceed the variance-matching scale t m , then the high- H queue is worse; if the queuing scales are smaller, then the low- H queue is worse. However, no firm prediction can be made in the remaining case of t m falling between the two queuing scales. Numerical examples are given to demonstrate our results.

Efficient data structures for Time Warp simulation queues

Abstract Optimistic simulation is besides the conservative methods the most promising approach to distributed discrete-event simulation (DDEVS). In the optimistic Time Warp strategy, every sub-simulator has to handle in addition to its main event queue (input queue) two other queues, holding information about sent messages and past states, respectively. We present efficient data structures and algorithms for the three queue types in Time Warp simulation. It shows that for two queue types the simple doubly linked list (DLL) approach suffices, for the input queue however, significant speed-up can be achieved by enhancing a data structure originally developed for sequential simulation. We show an improvement from an O(n) to an O(1) average case behavior for some operations by moving from the DLL to the new data structure without increasing the average case and worst case order of the other operations. Empirical results show that the new data structure yields substantial faster operation for enqueuing items into the list, which — regardless of the overhead introduced by the new data structure to the other queue operations — results in queue access acceleration for typical Time Warp simulations even for medium input queue sizes of about 100. For large queue sizes of about 10000 the new data structure yields more than 15 times faster overall queue operation performance.

Rough asymptotics for tandem non-homogeneous M/G/\infty queues via Poissonized empirical processes

We obtain a large deviations principle and moderate deviations principle for the joint distribution of the queue length processes and departure process for tandem M / G / \infty queues. The results are obtained by applying a result providing necessary and sufficient conditions on a class of functions for a large deviations principle and moderate deviations principle to hold for a Poissonized empirical process over the class of functions. As an application, we examine how large queue lengths and numbers of departures are built up.

Call establishment overload in large ATM networks

The call establishment process in asynchronous transfer mode (ATM) networks can constitute a source of overload in the switches of the network. The purpose of this paper is to use analytic modeling and simulation to investigate this effect. The call establishment overload can be particularly significant when the network will contain a relatively large number of switches, due to the inaccuracy of the information available to the switch accepting or rejecting the call, and because of the multiple paths which need to be assessed and then utilized to establish the call. We simulate the call set-up phase of an ATM connection, including path selection, bandwidth reservation and call rejection, and simulate the flow of call establishment messages to estimate by simulation and analytically the queue lengths of call establishment processing at the input, output and intermediate switches. Average usage of bandwidth at the links, and the probability that a call request is rejected due to insufficient capacity at some link, are measured via simulation and also predicted using an analytical model. Simulation results show that if bandwidth is not reserved at input nodes, very large queues of call establishment jobs can build up in the switches of the network. A simplified analytical model is derived to estimate the queue lengths of call establishment jobs at each node of the network when advanced reservation with perfect information is implemented. Simulation results are compared to approximate model predictions for each individual switching node in a 100-node (switch) ATM network model. It is shown that the analytical model dealing with a large ATM network can qualitatively predict the locations of queue build-up of jobs for call establishment processing, with accurate queue length predictions for being made in the case of light traffic load. The analytical model provides a lower bound to queue lengths, and remains within the same order of magnitude when the call request traffic is high.

Fuzzy allocation of manufacturing resources

Fuzzy logic is investigated as a resource allocation tool in a manufacturing environment. Comparison is drawn between random, largest queue, and fuzzy allocation techniques using a discrete event simulation model as a test platform. Results indicate a fuzzy controller to be a feasible alternative.

A comparative study of parallel and sequential priority queue algorithms

Priority queues are used in many applications including real-time systems, operating systems, and simulations. Their implementation may have a profound effect on the performance of such applications. In this article, we study the performance of well-known sequential priority queue implementations and the recently proposed parallel access priority queues. To accurately assess the performance of a priority queue, the performance measurement methodology must be appropriate. We use the Classic Hold, the Markov Model, and an Up/Down access pattern to measure performance and look at both the average access time and the worst-case time that are of vital interest to real-tiem applicatons. Our results suggest that the best choice for priority queue algorithms depends heavily on the application. For queue sizes smaller than 1,000 elements, the Splay Tree, the Skew Heap, and Henriksen's algorithm show good average access times. For large queue sized of 5,000 elements or more, the Calendar Queue and the Lazy Queue offer good average access times but have very long worst-case access times. The Skew Heap and the splay Tree exhibit the best worst-case access times. Among the parallel access priority queues tested, the Parallel Access Skew Heap provides the best performance on small shares memory multiprocessors.

The effect of multiple time scales and subexponentiality in MPEG video streams on queueing behavior

Guided by the empirical observation that real-time MPEG video streams exhibit both multiple time scale and subexponential characteristics, we construct a video model that captures both of these characteristics and is amenable to queueing analysis. We investigate two fundamental approaches for extracting the model parameters: using sample path and second-order statistics-based methods. The model exhibits the following two canonical queueing behaviors. When strict stability conditions are satisfied, i.e., the conditional mean of each scene is smaller than the capacity of the server, precise modeling of the interscene dynamics (long-term dependency) is not essential for the accurate prediction of small to moderately large queue sizes. In this case, the queue length distribution is determined using quasistationary (perturbation theory) analysis. When weak stability conditions are satisfied, i.e., the conditional mean of at least one scene type is greater than the capacity of the server, the dominant effect for building a large queue size is the subexponential (long-tailed) scene length distribution. In this case, precise modeling of intrascene statistics is of secondary importance for predicting the large queueing behavior. A fluid model, whose arrival process is obtained from the video data by replacing scene statistics with their means, is shown to asymptotically converge to the exact queue distribution. Using the transition scenario of moving from one stability region to the other by a change in the value of the server capacity, we synthesize recent queueing theoretic advances and ad hoc results in video modeling, and unify a broad range of seemingly contradictory experimental observations found in the literature. As a word of caution for the widespread usage of second-order statistics modeling methods, we construct two processes with the same second-order statistics that produce distinctly different queueing behaviors.

Extreme and high-level sojourns of the single server queue in heavy traffic

This study characterizes the behavior of large queue lengths in heavy traffic. We show that the distribution of the maximum queue length in a random time interval for a queueing systems in heavy traffic converges to a novel extreme value distribution. We also study the processes that record the times that the queue length exceeds a high level and the cumulative time the queue is above the level. We show that these processes converge in distribution to compound Poisson processes.

A numerically efficient method for the MAP/D/1/K queue via rational approximations

The Markovian Arrival Process (MAP), which contains the Markov Modulated Poisson Process (MMPP) and the Phase-Type (PH) renewal processes as special cases, is a convenient traffic model for use in the performance analysis of Asynchronous Transfer Mode (ATM) networks. In ATM networks, packets are of fixed length and the buffering memory in switching nodes is limited to a finite numberK of cells. These motivate us to study the MAP/D/1/K queue. We present an algorithm to compute the stationary virtual waiting time distribution for the MAP/D/1/K queue via rational approximations for the deterministic service time distribution in transform domain. These approximations include the well-known Erlang distributions and the Padé approximations that we propose. Using these approximations, the solution for the queueing system is shown to reduce to the solution of a linear differential equation with suitable boundary conditions. The proposed algorithm has a computational complexity independent of the queue storage capacityK. We show through numerical examples that, the idea of using Padé approximations for the MAP/D/1/K queue can yield very high accuracy with tractable computational load even in the case of large queue capacities.

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.

End-station performance under leaky bucket traffic shaping

The shaping of traffic at its source is a prominent congestion control solution in ATM networks. The with a is a very popular traffic-shaping approach. By sizing the leaky bucket and spacer parameters, an end station shapes its traffic to conform to a good behaviour contract with the network. The leaky bucket delays the transmission of a selected number of cells, while the spacer forces a minimum time distance between transmitted cells. When a contracted transmission rate is close to the traffic generation rate of the end station, this traffic stream will utilize most of the bandwidth available on its virtual connection. The high utilization of such a connection leads to large queues of frames and cells at the end station, with high adverse consequences for end-to-end delay and jitter. Based on the negotiated traffic parameters and source traffic characteristics, we study, through simulations, the contribution of the leaky bucket plus cell spacer subsystem to the delay and jitter in an end station. We also study the distribution of the size of the bursts of cells leaving the end station and entering the network. Finally, we derive the theoretical upper bound for the size of bursts of the cells.

Mean time for the development of large workloads and large queue lengths in the GI/G/1 queue

We consider the GI/G/1 queue described by either the workload U(t) (unfinished work) or the number of customers N(t) in the system. We compute the mean time until U(t) reaches excess of the level K, and also the mean time until N(t) reaches N0. For the M/G/1 and GI/M/1 models, we obtain exact contour integral representations for these mean first passage times. We then compute the mean times asymptotically, as K and N0 → ∞, by evaluating these contour integrals. For the general GI/G/1 model, we obtain asymptotic results by a singular perturbation analysis of the appropriate backward Kolmogorov equation(s). Numerical comparisons show that the asymptotic formulas are very accurate even for moderate values of K and N0.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue II: Positive recurrence and logarithmic limits

We continue our investigation of the batch arrival-heterogeneous multiserver queue begun in Part I. In a general setting we prove the positive Harris recurrence of the system, and with no additional conditions we prove logarithmic tail limits for the stationary queue length and waiting time distributions.

The probability of large queue lengths and waiting times in a heterogeneous multiserver queue I: Tight limits

We consider a multiserver queuing process specified by i.i.d. interarrival time, batch size and service time sequences. In the case that different servers have different service time distributions we say the system is heterogeneous. In this paper we establish conditions for the queuing process to be characterized as a geometrically Harris recurrent Markov chain, and we characterize the stationary probabilities of large queue lengths and waiting times. The queue length is asymptotically geometric and the waiting time is asymptotically exponential. Our analysis is a generalization of the well-known characterization of the GI/G/1 queue obtained using classical probabilistic techniques of exponential change of measure and renewal theory.

Asymptotic properties of first passage times for tandem jackson networks I: buildup of large queue lengths

We consider d M/M/1 queues in series. Asymptotic expansions are constructed for the mean time until the total customer population in the network reaches a large number K. Our approach employs formal asymptotic techniques such as boundary layer theory and matched asymptotic expansions. We give formulas which show that the mean time behaves as αK β exp(γK) for K → ∞, where α, β, γ are explicitly given in terms of the arrival and service rates