31 research outputs found
Many-server diffusion limits for queues
This paper studies many-server limits for multi-server queues that have a
phase-type service time distribution and allow for customer abandonment. The
first set of limit theorems is for critically loaded queues, where
the patience times are independent and identically distributed following a
general distribution. The next limit theorem is for overloaded
queues, where the patience time distribution is restricted to be exponential.
We prove that a pair of diffusion-scaled total-customer-count and
server-allocation processes, properly centered, converges in distribution to a
continuous Markov process as the number of servers goes to infinity. In the
overloaded case, the limit is a multi-dimensional diffusion process, and in the
critically loaded case, the limit is a simple transformation of a diffusion
process. When the queues are critically loaded, our diffusion limit generalizes
the result by Puhalskii and Reiman (2000) for queues without customer
abandonment. When the queues are overloaded, the diffusion limit provides a
refinement to a fluid limit and it generalizes a result by Whitt (2004) for
queues with an exponential service time distribution. The proof
techniques employed in this paper are innovative. First, a perturbed system is
shown to be equivalent to the original system. Next, two maps are employed in
both fluid and diffusion scalings. These maps allow one to prove the limit
theorems by applying the standard continuous-mapping theorem and the standard
random-time-change theorem.Comment: Published in at http://dx.doi.org/10.1214/09-AAP674 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
A skill based parallel service system under FCFS-ALIS : steady state, overloads and abandonments
We consider a queueing system with servers S={m1,...,mJ}, and with customer types C={a,b,...}. A bipartite graph G describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic. Keywords: Service systems, multi type customers, multi type skill based servers, matching of infinite sequences, product form solution, first come first served policy, assign longest idle server policy, complete resource pooling, local steady state, overloaded queues, abandonment
A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment
Many-server queuing networks with general service and abandonment times have proven to be a realistic model for scenarios such as call centers and health-care systems. The presence of abandonment makes analytical treatment difficult for general topologies. Hence, such networks are usually studied by means of fluid limits. The current state of the art, however, suffers from two drawbacks. First, convergence to a fluid limit has been established only for the transient, but not for the steady state regime. Second, in the case of general distributed service and abandonment times, convergence to a fluid limit has been either established only for a single queue, or has been given by means of a system of coupled integral equations which does not allow for a numerical solution. By making the mild assumption of Coxian-distributed service and abandonment times, in this paper we address both drawbacks by establishing convergence in probability to a system of coupled ordinary differential equations (ODEs) using the theory of Kurtz. The presence of abandonments leads in many cases to ODE systems with a global attractor, which is known to be a sufficient condition for the fluid and the stochastic steady state to coincide in the limiting regime. The fact that our ODE systems are piecewise affine enables a computational method for establishing the presence of a global attractor, based on a solution of a system of linear matrix inequalities
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Asymptotic Analysis of Service Systems with Congestion-Sensitive Customers
Many systems in services, manufacturing, and technology, feature users or customers sharing a limited number of resources, and which suffer some form of congestion when the number of users exceeds the number of resources. In such settings, queueing models are a common tool for describing the dynamics of the system and quantifying the congestion that results from the aggregated effects of individuals joining and leaving the system. Additionally, the customers themselves may be sensitive to congestion and react to the performance of the system, creating feedback and interaction between individual customer behavior and aggregate system dynamics.This dissertation focuses on the modeling and performance of service systems with congestion-sensitive customers using large-scale asymptotic analyses of queueing models. This work extends the theoretical literature on congestion-sensitive customers in queues in the settings of service differentiation and observational learning and abandonment. Chapter 2 considers the problem of a service provider facing a heterogeneous market of customers who differ based on their value for service and delay sensitivity. The service provider seeks to find the revenue maximizing level of service differentiation (offering different price-delay combinations). We show that the optimal policy places the system in heavy traffic, but at substantially different levels of congestion depending on the degree of service differentiation. Moreover, in a differentiated offering, the level of congestion will vary substantially between service classes. Chapter 3 presents a new model of customer abandonment in which congestion-sensitive customers observe the queue length, but do not know the service rate. Instead, they join the queue and observe their progress in order to estimate their wait times and make abandonment decisions. We show that an overloaded queue with observational learning and abandonment stabilizes at a queue length whose scale depends on the tail of the service time distribution. Methodologically, our asymptotic approach leverages stochastic limit theory to provide simple and intuitive results for optimizing or characterizing system performance. In particular, we use the analysis of deterministic fluid-type queues to provide a first-order characterization of the stochastic system dynamics, which is demonstrated by the convergence of the stochastic system to the fluid model. This also allows us to crisply illustrate and quantify the relative contributions of system or customer characteristics to overall system performance