Tail Probabilities In Queues Research Articles

Low-Latency Multichannel ALOHA With Fast Retrial for Machine-Type Communications

In this paper, for low-latency transmissions in machine-type communications, we study multichannel ALOHA with fast retrial that can shorten the access delay by immediate retransmissions of collided packets. To analyze the access delay of multichannel ALOHA with fast retrial, we employ an erasure channel model, which is equivalent to the widely used collision model, and study the effective capacity and the quality of service (QoS) exponent. Through the asymptotic analysis, we show that the QoS exponent grows logarithmically with the system dimension (i.e., the number of subchannels), which implies that multichannel ALOHA with fast retrial can effectively shorten the average access delay by increasing the system dimension. However, the growth rate of the effective capacity is shown to be slower than linear when the system dimension increases, which becomes the cost of improved access delay. In particular, the effective capacity is proportional to ${(N/\ln N)}$ , where ${N}$ is the number of subchannels. From the QoS exponent, the tail probability of queue is obtained and compared with simulation results, which confirms that the theoretical results from the derived QoS agree with the simulation results for large systems.

Performance analysis of service systems with priority upgrades

In this paper, we study the performance of service systems with priority upgrades. We model the service system as a single-server two-class priority queue, with queue 1 as the normal queue and queue 2 as the priority queue. The queueing model of interest has various applications in healthcare services, perishable inventory and project management. We comprehensively examine the system’s stationary distribution, computational algorithm design and sensitivity analysis. We observe that when queue 2 is large, the conditional distribution of queue 1 approximates a Poisson distribution. The tail probability of queue 2 decays geometrically, while the tail probability of queue 1 decays much faster than queue 2’s. This helps us design an algorithm that computed the stationary distribution. Finally, by using the algorithm, we perform a sensitivity analysis on various system parameters, i.e., the arrival rates, service rates and the upgrade rate. The numerical study provides helpful insights into designing such service systems.

Exponential Approximations for Tail Probabilities in Queues II: Sojourn Time and Workload

We continue to focus on simple exponential approximations for steady-state tail probabilities in queues based on asymptotics. For the G/GI/1 model with i.i.d. service times that are independent of an arbitrary stationary arrival process, we relate the asymptotics for the steady-state waiting time, sojourn time, and workload. We show that the three asymptotic decay rates coincide and that the three asymptotic constants are simply related. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact numerical results for BMAP/G/1 queues, which have batch Markovian arrival processes. Numerical examples show that the exponential approximations for the tail probabilities are remarkably accurate at the 90th percentile and beyond. Thus, these exponential approximations appear very promising for applications.

Exponential Approximations for Tail Probabilities in Queues, I: Waiting Times

This paper focuses on simple exponential approximations for tail probabilities of the steady-state waiting time in infinite-capacity multiserver queues based on small-tail asymptotics. For the GI/GI/s model, we develop a heavy-traffic asymptotic expansion in powers of one minus the traffic intensity for the waiting-time asymptotic decay rate. We propose a two-term approximation for the asymptotic decay rate based on the first three moments of the interarrival-time and service-time distributions. We also suggest approximating the asymptotic constant by the product of the mean and the asymptotic decay rate. We evaluate the exponential approximations based on the exact asymptotic parameters and their approximations by making comparisons with exact results obtained numerically for the BMAP/GI/1 queue, which has a batch Markovian arrival process, and the GI/GI/s queue. Numerical examples show that the exponential approximations are remarkably accurate, especially for higher percentiles, such as the 90th percentile and beyond.

Waiting-time tail probabilities in queues with long-tail service-time distributions

We consider the standardGI/G/1 queue with unlimited waiting room and the first-in first-out service discipline. We investigate the steady-state waiting-time tail probabilitiesP(W>x) when the service-time distribution has a long-tail distribution, i.e., when the service-time distribution fails to have a finite moment generating function. We have developed algorithms for computing the waiting-time distribution by Laplace transform inversion when the Laplace transforms of the interarrival-time and service-time distributions are known. One algorithm, exploiting Pollaczek's classical contourintegral representation of the Laplace transform, does not require that either of these transforms be rational. To facilitate such calculations, we introduce a convenient two-parameter family of long-tail distributions on the positive half line with explicit Laplace transforms. This family is a Pareto mixture of exponential (PME) distributions. These PME distributions have monotone densities and Pareto-like tails, i.e., are of orderx −r forr>1. We use this family of long-tail distributions to investigate the quality of approximations based on asymptotics forP(W>x) asx→∞. We show that the asymptotic approximations with these long-tail service-time distributions can be remarkably inaccurate for typicalx values of interest. We also derive multi-term asymptotic expansions for the waiting-time tail probabilities in theM/G/1 queue. Even three terms of this expansion can be remarkably inaccurate for typicalx values of interest. Thus, we evidently must rely on numerical algorithms for determining the waiting-time tail probabilities in this case. When working with service-time data, we suggest using empirical Laplace transforms.

Experiments in concurrent stochastic simulation: The EcliPSe paradigm

This paper presents results on the performance of a novel and flexible concurrent simulation environment known as EcliPSe. The paradigm we advocate is based on the premise that replication based simulations, either nondistributed or minimally distributed, yield excellent speedups. The approach used makes concurrent simulation easily accessible to researchers because its use does not require knowledge of parallel programming. The experiments we report include Monte Carlo type simulations (e.g., estimating integrals, order-statistics), Markov-chain simulations (hitting-times, distributed algorithms), and discrete-event simulation (e.g., tail probabilities in queues, FDDI token ring performance, and simulations of high performance software testing techniques on SIMD machines).