Yunan Liu

A Network of Time-Varying Many-Server Fluid Queues with Customer Abandonment

To describe the congestion in large-scale service systems, we introduce and analyze a non-Markovian open network of many-server fluid queues with customer abandonment, proportional routing, and time-varying model elements. Proportions of the fluid completing service from each queue are immediately routed to the other queues, with the fluid not routed to one of the queues being immediately routed out of the network. The fluid queue network serves as an approximation for the corresponding non-Markovian open network of many-server queues with Markovian routing, where all model elements may be time varying. We establish the existence of a unique vector of (net) arrival rate functions at each queue and the associated time-varying performance. In doing so, we provide the basis for an efficient algorithm, even for networks with many queues.

A many-server fluid limit for the Gt/GI/st+GI queueing model experiencing periods of overloading

Abstract A many-server heavy-traffic functional weak law of large numbers is established for the G t / G I / s t + G I queueing model, which has customer abandonment (the + G I ), time-varying arrival rate and staffing (the subscript t ) and non-exponential service and patience distributions (the two G I ’s). This limit provides support for a previously proposed deterministic fluid approximation, and extends a previously established limit for the special case of exponential service times.

Large-time asymptotics for the Gt/Mt/st+GIt many-server fluid queue with abandonment

We previously introduced and analyzed the Gt/Mt/st+GIt many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper, we establish an asymptotic loss of memory (ALOM) property for that fluid model, i.e., we show that there is asymptotic independence from the initial conditions as time t evolves, under regularity conditions. We show that the difference in the performance functions dissipates over time exponentially fast, again under the regularity conditions. We apply ALOM to show that the stationary G/M/s+GI fluid queue converges to steady state and the periodic Gt/Mt/st+GIt fluid queue converges to a periodic steady state as time evolves, for all finite initial conditions.

Many-server Heavy-traffic Limit for Queues with Time-varying Parameters

+GI queueing model, having time-varying arrival rate and staffing, ageneral arrival process satisfying a FCLT, exponential service timesand customer abandonment according to a general probability dis-tribution. The FCLT provides theoretical support for the approxi-mating deterministic fluid model the authors analyzed in a previouspaper and a refined Gaussian process approximation, using varianceformulas given here. The model is assumed to alternate between un-derloaded and overloaded intervals, with critical loading only at theisolated switching points. The proof is based on a recursive analy-sis of the system over these successive intervals, drawing heavily onprevious results for infinite-server models. The FCLT requires carefultreatment of the initial conditions for each interval.

STABILIZING PERFORMANCE IN NETWORKS OF QUEUES WITH TIME-VARYING ARRIVAL RATES

This paper investigates extensions to feed-forward queueing networks of an algorithm to set staffing levels (the number of servers) to stabilize performance % at Quality of Service (QoS) targets in an M t / GI / s t + GI multi-server queue with a time-varying arrival rate. The model has a non-homogeneous Poisson process (NHPP), customer abandonment, and non-exponential service and patience distributions. For a single queue, simulation experiments showed that the algorithm successfully stabilizes abandonment probabilities and expected delays over a wide range of Quality-of-Service (QoS) targets. A limit theorem showed that stable performance at fixed QoS targets is achieved asymptotically as the scale increases (by letting the arrival rate grow while holding the service and patience distributions fixed). Here we extend that limit theorem to a feed-forward queueing network. However, these fixed QoS targets provide low QoS as the scale increases. Hence, these limits primarily support the algorithm with a low QoS target. For a high QoS target, effectiveness depends on the NHPP property, but the departure process never is exactly an NHPP. Thus, we investigate when a departure process can be regarded as approximately an NHPP. We show that index of dispersion for counts is effective for determining when a departure process is approximately an NHPP in this setting. In the important common case when all queues have high QoS targets, we show that both: (i) the departure process is approximately an NHPP from this perspective and (ii) the algorithm is effective.

STAFFING A SERVICE SYSTEM WITH NON-POISSON NON-STATIONARY ARRIVALS

Motivated by non-Poisson stochastic variability found in service system arrival data, we extend established service system staffing algorithms using the square-root staffing formula to allow for non-Poisson arrival processes. We develop a general model of the non-Poisson non-stationary arrival process that includes as a special case the non-stationary Cox process (a modification of a Poisson process in which the rate itself is a non-stationary stochastic process), which has been advocated in the literature. We characterize the impact of the non-Poisson stochastic variability upon the staffing through the heavy-traffic limit of the peakedness (ratio of the variance to the mean in an associated stationary infinite-server queueing model), which depends on the arrival process through its central limit theorem behavior. We provide simple formulas to quantify the performance impact of the non-Poisson arrivals upon the staffing decisions, in order to achieve the desired service level. We conduct simulation experiments with non-stationary Markov-modulated Poisson arrival processes with sinusoidal arrival rate functions to demonstrate that the staffing algorithm is effective in stabilizing the time-varying probability of delay at designated targets.

Nearly Periodic Behavior in the Overloaded G/D/s + GI Queue

Under general conditions, the number of customers in a GI/D/s + GI many-server queue at time t converges to a unique stationary distribution as t → ∞. However, simulations show that the sample paths routinely exhibit nearly periodic behavior over long time intervals when the system is overloaded and s is large, provided that the system does not start in steady state. Moreover, the precise periodic behavior observed depends critically on the initial conditions. We provide insight into the transient behavior by studying the deterministic fluid model, which arises as the many-server heavy-traffic limit. The limiting fluid model also has a unique stationary point, but that stationary point is not approached from any other initial state as t → ∞. Instead, the fluid model performance approaches one of its uncountably many periodic steady states, depending on the initial conditions. Simulation experiments confirm that the time-dependent performance of the stochastic queueing model is well approximated by the flu...

Many-server Gaussian limits for overloaded non-Markovian queues with customer abandonment

Extending Ward Whitt’s pioneering work “Fluid Models for Multiserver Queues with Abandonments, Operations Research, 54(1) 37–54, 2006,” this paper establishes a many-server heavy-traffic functional central limit theorem for the overloaded

Many-Server Queues with Time-Varying Arrivals, Customer Abandonment, and non-Exponential Distributions

G{/}GI{/}n+GI