arXiv subjects

Amy R. Ward

Publications and source records attributed to Amy R. Ward.

10 recordsLinked to original sources

arXiv2023

Matching Impatient and Heterogeneous Demand and Supply

Service platforms must determine rules for matching heterogeneous demand (customers) and supply (workers) that arrive randomly over time and may be lost if forced to wait too long for a match. Our objective is to maximize the cumulative value of matches, minus costs incurred when demand and supply wait. We develop a fluid model, that approximates the evolution of the stochastic model, and captures explicitly the nonlinear dependence between the amount of demand and supply waiting and the distribution of their patience times, also known as reneging or abandonment times in the literature. The fluid model invariant states approximate the steady-state mean queue-lengths in the stochastic system, and, therefore, can be used to develop an optimization problem whose optimal solution provides matching rates between demand and supply types that are asymptotically optimal (on fluid scale, as demand and supply rates grow large). We propose a discrete review matching policy that asymptotically achieves the optimal matching rates. We further show that when the aforementioned matching optimization problem has an optimal extreme point solution, which occurs when the patience time distributions have increasing hazard rate functions, a state-independent priority policy, that ranks the edges on the bipartite graph connecting demand and supply, is asymptotically optimal. A key insight from this analysis is that the ranking critically depends on the patience time distributions, and may be different for different distributions even if they have the same mean, demonstrating that models assuming, e.g., exponential patience times for tractability, may lack robustness. Finally, we observe that when holding costs are zero, a discrete review policy, that does not require knowledge of inter-arrival and patience time distributions, is asymptotically optimal.

math.OC↗

arXiv2023

Asymptotically Optimal Idling in the GI/GI/N+GI Queue

We formulate a control problem for a GI/GI/N+GI queue, whose objective is to trade off the long-run average operational costs (i.e., abandonment costs and holding costs) with server utilization costs. To solve the control problem, we consider an asymptotic regime in which the arrival rate and the number of servers grow large. The solution to an associated fluid control problem motivates that non-idling service disciplines are not in general optimal, unless some arrivals are turned away. We propose an admission control policy designed to ensure servers have sufficient idle time that we show is asymptotically optimal.

math.OC↗

arXiv2021

Stationary Distribution Convergence of the Offered Waiting Processes in Heavy Traffic under General Patience Time Scaling

We study a sequence of single server queues with customer abandonment (GI/GI/1+GI) under heavy traffic. The patience time distributions vary with the sequence, which allows for a wider scope of applications. It is known ([20, 18]) that the sequence of scaled offered waiting time processes converges weakly to a reflecting diffusion process with non-linear drift, as the traffic intensity approaches one. In this paper, we further show that the sequence of stationary distributions and moments of the offered waiting times, with diffusion scaling, converge to those of the limit diffusion process. This justifies the stationary performance of the diffusion limit as a valid approximation for the stationary performance of the GI/GI/1+GI queue. Consequently, we also derive the approximation for the abandonment probability for the GI/GI/1+GI queue in the stationary state.

math.PR↗

arXiv2019

Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic

A result of Ward and Glynn (2005) asserts that the sequence of scaled offered waiting time processes of the $GI/GI/1+GI$ queue converges weakly to a reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $GI/GI/1+GI$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in Ward and Glynn (2005). In comparison to Kingman's classical result in Kingman (1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a $GI/GI/1$ queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue stationary behavior.

math.PR↗

arXiv2017

Conditioned Functional Limits and Applications to Queues

We consider a renewal process that is conditioned on the number of events in a fixed time horizon. We prove that a centered and scaled version of this process converges to a Brownian bridge, as the number of events grows large, which relies on first establishing a functional strong law of large numbers result to determine the centering. These results are consistent with the asymptotic behavior of a conditioned Poisson process. We prove the limit theorems over triangular arrays of exchangeable random variables, obtained by conditionning a sequence of independent and identically distributed renewal processes. We construct martingale difference sequences with respect to these triangular arrays, and use martingale convergence results in our proofs. To illustrate how these results apply to performance analysis in queueing, we prove that the workload process of a single server queue with conditioned renewal arrival process can be approximated by a reflected diffusion having the sum of a Brownian Bridge and Brownian motion as input to its regulator mapping.

math.PR↗

arXiv2016

On the Control of Fork-Join Networks

Networks in which the processing of jobs occurs both sequentially and in parallel are prevalent in many application domains, such as computer systems, healthcare, manufacturing, and project management. The parallel processing of jobs gives rise to synchronization constraints that can be a main reason for job delay. In comparison with feedforward queueing networks that have only sequential processing of jobs, the approximation and control of networks that have synchronization constraints is less understood. One well-known modeling framework in which synchronization constraints are prominent is the fork-join processing network. Our objective is to find scheduling rules for fork-join processing networks with multiple job types in which there is first a fork operation, then activities that can be performed in parallel, and then a join operation. The difficulty is that some of the activities that can be performed in parallel require a shared resource. We solve the scheduling problem for that shared server (that is, which type of job to prioritize each time the server becomes available) when that server is in heavy traffic and prove an asymptotic optimality result.

math.PR↗

arXiv2014

On Transitory Queueing

We introduce a framework and develop a theory of transitory queueing models. These are models that are not only non-stationary and time-varying but also have other features such as the queueing system operates over finite time, or only a finite population arrives. Such models are relevant in many real-world settings, from queues at post-offces, DMV, concert halls and stadia to out-patient departments at hospitals. We develop fluid and diffusion limits for a large class of transitory queueing models. We then introduce three specific models that fit within this framework, namely, the Delta(i)/GI/1 model, the conditioned G/GI/1 model, and an arrival model of scheduled traffic with epoch uncertainty. We show that asymptotically these models are distributionally equivalent, i.e., they have the same fluid and diffusion limits. We note that our framework provides the first ever way of analyzing the standard G/GI/1 model when we condition on the number of arrivals. In obtaining these results, we provide generalizations and extensions of the Glivenko-Cantelli and Donskers Theorem for empirical processes with triangular arrays. Our analysis uses the population acceleration technique that we introduce and develop. This may be useful in analysis of other non-stationary and non-ergodic queuing models.

math.PR↗

arXiv2014

Staffing Call Centers with Uncertain Arrival Rates and Co-sourcing

In a call center, staffing decisions must be made before the call arrival rate is known with certainty. Then, once the arrival rate becomes known, the call center may be over-staffed, in which case staff are being paid to be idle, or under-staffed, in which case many callers hang-up in the face of long wait times. Firms that have chosen to keep their call center operations in-house can mitigate this problem by co-sourcing; that is, by sometimes outsourcing calls. Then, the required staffing $N$ depends on how the firm chooses which calls to outsource in real-time, after the arrival rate realizes and the call center operates as a $M/M/N+M$ queue with an outsourcing option. Our objective is to find a joint policy for staffing and call outsourcing that minimizes the long run average cost of this two-stage stochastic program when there is a linear staffing cost per unit time and linear costs associated with abandonments and outsourcing. We propose a policy that uses a square-root safety staffing rule, and outsources calls in accordance with a threshold rule that characterizes when the system is "too crowded". Analytically, we establish that our proposed policy is asymptotically optimal, as the mean arrival rate becomes large, when the level of uncertainty in the arrival rate is of the same order as the inherent system fluctuations in the number of waiting customers for a known arrival rate. Through an extensive numerical study, we establish that our policy is extremely robust. In particular, our policy performs remarkably well over a wide range of parameters, and far beyond where it is proved to be asymptotically optimal.

math.PR↗

arXiv2014

Routing and Staffing when Servers are Strategic

Traditionally, research focusing on the design of routing and staffing policies for service systems has modeled servers as having fixed (possibly heterogeneous) service rates. However, service systems are generally staffed by people. Furthermore, people respond to workload incentives; that is, how hard a person works can depend both on how much work there is, and how the work is divided between the people responsible for it. In a service system, the routing and staffing policies control such workload incentives; and so the rate servers work will be impacted by the system's routing and staffing policies. This observation has consequences when modeling service system performance, and our objective is to investigate those consequences. We do this in the context of the M/M/N queue, which is the canonical model for large service systems. First, we present a model for "strategic" servers that choose their service rate in order to maximize a trade-off between an "effort cost", which captures the idea that servers exert more effort when working at a faster rate, and a "value of idleness", which assumes that servers value having idle time. Next, we characterize the symmetric Nash equilibrium service rate under any routing policy that routes based on the server idle time. We find that the system must operate in a quality-driven regime, in which servers have idle time, in order for an equilibrium to exist, which implies that the staffing must have a first-order term that strictly exceeds that of the common square-root staffing policy. Then, within the class of policies that admit an equilibrium, we (asymptotically) solve the problem of minimizing the total cost, when there are linear staffing costs and linear waiting costs. Finally, we end by exploring the question of whether routing policies that are based on the service rate, instead of the server idle time, can improve system performance.

cs.GT↗

arXiv2012

A queueing model with independent arrivals, and its fluid and diffusion limits

We introduce the {\Delta}(i)/GI/1 queue, a new queueing model. In this model, customers from a given population independently sample a time to arrive from some given distribution F. Thus, the arrival times are an ordered statistics, and the inter-arrival times are differences of consecutive ordered statistics. They are served by a single server which provides service according to a general distribution G, with independent service times. The exact model is analytically intractable. Thus, we develop fluid and diffusion limits for the various stochastic processes, and performance metrics. The fluid limit of the queue length is observed to be a reflected process, while the diffusion limit is observed to be a function of a Brownian motion and a Brownian bridge process, and is given by a 'netput' process and a directional derivative of the Skorokhod reflected fluid netput in the direction of a diffusion refinement of the netput process. We also observe what may be interpreted as a transient Little's law. Sample path analysis reveals various operating regimes where the diffusion limit switches between a free diffusion, a reflected diffusion process and the zero process, with possible discontinuities during regime switches. The weak convergence is established in the M1 topology, and it is also shown that this is not possible in the J1 topology.

math.PR↗