Many-server queues with customer abandonment: numerical analysis of their diffusion models

We use multidimensional diffusion processes to approximate the dynamics of a queue served by many parallel servers. The queue is served in the first-in-first-out (FIFO) order and the customers waiting in queue may abandon the system without service. Two diffusion models are proposed in this paper. They differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. To analyze these diffusion models, we develop a numerical algorithm for computing the stationary distribution of such a diffusion process. A crucial part of the algorithm is to choose an appropriate reference density. Using a conjecture on the tail behavior of a limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations for many-server queues, sometimes for queues with as few as twenty servers

Load Balancing in the Non-Degenerate Slowdown Regime

We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a classical load balancing policy for queueing systems with multiple parallel servers. Parallel server queueing systems are regularly analysed and dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling regime. However, when jobs must be dispatched to a server upon arrival, we advocate the Non-Degenerate Slowdown regime (NDS) to compare different load-balancing rules. In this paper we identify novel diffusion approximation and timescale separation that provides insights into the performance of JSQ. We calculate the price of irrevocably dispatching jobs to servers and prove this to within 15% (in the NDS regime) of the rules that may manoeuvre jobs between servers. We also compare ours results for the JSQ policy with the NDS approximations of many modern load balancing policies such as Idle-Queue-First and Power-of-$d$-choices policies which act as low information proxies for the JSQ policy. Our analysis leads us to construct new rules that have identical performance to JSQ but require less communication overhead than power-of-2-choices.Comment: Revised journal submission versio

Scheduling a multi class queue with many exponential servers: asymptotic optimality in heavy traffic

We consider the problem of scheduling a queueing system in which many statistically identical servers cater to several classes of impatient customers. Service times and impatience clocks are exponential while arrival processes are renewal. Our cost is an expected cumulative discounted function, linear or nonlinear, of appropriately normalized performance measures. As a special case, the cost per unit time can be a function of the number of customers waiting to be served in each class, the number actually being served, the abandonment rate, the delay experienced by customers, the number of idling servers, as well as certain combinations thereof. We study the system in an asymptotic heavy-traffic regime where the number of servers n and the offered load r are simultaneously scaled up and carefully balanced: n\approx r+\beta \sqrtr for some scalar \beta. This yields an operation that enjoys the benefits of both heavy traffic (high server utilization) and light traffic (high service levels.

A diffusion model of scheduling control in queueing systems with many servers

This paper studies a diffusion model that arises as the limit of a queueing system scheduling problem in the asymptotic heavy traffic regime of Halfin and Whitt. The queueing system consists of several customer classes and many servers working in parallel, grouped in several stations. Servers in different stations offer service to customers of each class at possibly different rates. The control corresponds to selecting what customer class each server serves at each time. The diffusion control problem does not seem to have explicit solutions and therefore a characterization of optimal solutions via the Hamilton-Jacobi-Bellman equation is addressed. Our main result is the existence and uniqueness of solutions of the equation. Since the model is set on an unbounded domain and the cost per unit time is unbounded, the analysis requires estimates on the state process that are subexponential in the time variable. In establishing these estimates, a key role is played by an integral formula that relates queue length and idle time processes, which may be of independent interest.Comment: Published at http://dx.doi.org/10.1214/105051604000000963 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Breaking the dimensionality curse in multi-server queues

International audiencePh/Ph/c and and Ph/Ph/c/N queues can be viewed as a common model of multi-server facilities. We propose a simple approximate solution for the equilibrium probabilities in such queues based on a reduced state description in order to circumvent the well-known and dreaded combinatorial growth of the number of states inherent in the classical state description. The number of equations to solve in our approach increases linearly with the number of servers and phases in the service time distribution. A simple fixed-point iteration is used to solve these equations. Our approach applies both to open models with unrestricted buffer size and to queues with finite-size buffers. The results of a large number of empirical studies indicate that the overall accuracy of the proposed approximation appears very good. For instance, the median relative error for the mean number in the queue over thousands of examples is below 0.1% and the relative error exceeds 5% in less than 1.5% of cases explored. The accuracy of the proposed approximation becomes particularly good for systems with more than 8 servers, and tends to become excellent as the number of servers increases

Improving files availability for BitTorrent using a diffusion model

The BitTorrent mechanism effectively spreads file fragments by copying the rarest fragments first. We propose to apply a mathematical model for the diffusion of fragments on a P2P in order to take into account both the effects of peer distances and the changing availability of peers while time goes on. Moreover, we manage to provide a forecast on the availability of a torrent thanks to a neural network that models the behaviour of peers on the P2P system. The combination of the mathematical model and the neural network provides a solution for choosing file fragments that need to be copied first, in order to ensure their continuous availability, counteracting possible disconnections by some peers

Approximation Methods for the Standard Deviation of Flow Times in the G/G/s Queue

We provide approximation methods for the standard deviation of flow time in system for a general multi-server queue with infinite waiting capacity (G / G / s ). The approximations require only the mean and standard deviation or the coefficient of variation of the inter-arrival and service time distributions, and the number of servers. These approximations are simple enough to be implemented in manual or spreadsheet calculations, but in comparisons to Monte Carlo simulations have proven to give good approximations (within ±10%) for cases in which the coefficients of variation for the interarrival and service times are between 0 and 1. The approximations also have the desirable properties of being exact for the specific case of Markov queue model M / M / s, as well as some imbedded Markov queuing models ( Ek / M / 1 and M / Eα / 1). The practical significance of this research is that (1) many real world queuing problems involve the G / G / s queuing systems, and (2) predicting the range of variation of the time in the system (rather than just the average) is needed for decision making. For example, one job shop facility with which the authors have worked, guarantees its customers a nine day turnaround time and must determine the minimum number of machines of each type required to achieve nine days as a “worst case” time in the system. In many systems, the “worst case” value of flow time is very relevant because it represents the lead time that can safely be promised to customers. To estimate this we need both the average and standard deviation of the time in system. The usefulness of our results stems from the fact that they are computationally simple and thus provide quick approximations without resorting to complex numerical techniques or Monte Carlo simulations. While many accurate approximations for the G / G / s queue have been proposed previously, they often result in algebraically intractable expressions. This hinders attempts to derive closed-form solutions to the decision variables incorporated in optimization models, and inevitably leads to the use of complex numeric methods. Furthermore, actual application of many of these approximations often requires specification of the actual distributions of the inter-arrival time and the service time. Also, these results have tended to focus on delay probabilities and average waiting time, and do not provide a means of estimating the standard deviation of the time in the system. We also extend the approximations to computing the standard deviation of flow times of each priority class in the G / G / s priority queues and compare the results to those obtained via Monte Carlo simulations. These simulation experiments reveal good approximations for all priority classes with the exception of the lowest priority class in queuing systems with high utilization. In addition, we use the approximations to estimate the average and the standard deviation of the total flow time through queuing networks and have validated these results via Monte Carlo Simulations. The primary theoretical contributions of this work are the derivations of an original expression for the coefficient of variation of waiting time in the G / G / s queue, which holds exactly for G / M / s and M / G /1 queues. We also do some error sensitivity analysis of the formula and develop interpolation models to calculate the probability of waiting, since we need to estimate the probability of waiting for the G / G / s queue to calculate the coefficient of variation of waiting time. Technically we develop a general queuing system performance predictor, which can be used to estimate all kinds of performances for any steady state, infinite queues. We intend to make available a user friendly predictor for implementing our approximation methods. The advantages of these models are that they make no assumptions about distribution of inter-arrival time and service time. Our techniques generalize the previously developed approximations and can also be used in queuing networks and priority queues. Hopefully our approximation methods will be beneficial to those practitioners who like simple and quick practical answers to their multi-server queuing systems. Key words and Phrases: Queuing System, Standard Deviation, Waiting Time, Stochastic Process, Heuristics, G / G/ s, Approximation Methods, Priority Queue, and Queuing Networks