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Register a new userThe Limit of Stationary Distributions of Many-Server Queues in the Halfin-Whitt Regime
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We consider the so-called GI/GI/N queue, in which a stream of jobs with independent and identically distributed service times arrive as a renewal process to a common queue that is served by $N$ identical parallel servers in a first-come-first-serve manner. We introduce a new representation for the state of the system and, under suitable conditions on the service and interarrival distributions, establish convergence of the corresponding sequence of centered and scaled stationary distributions in the so-called Halfin-Whitt asymptotic regime. In particular, this resolves an open question posed by Halfin and Whitt in 1981. We also characterize the limit as the stationary distribution of an infinite-dimensional two-component Markov process that is the unique solution to a certain stochastic partial differential equation. Previous results were essentially restricted to exponential service distributions or service distributions with finite support, for which the corresponding limit process admits a reduced finite-dimensional Markovian representation. We develop a different approach to deal with the general case when the Markovian representation of the limit is truly infinite-dimensional. This approach is more broadly applicable to a larger class of networks.
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A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.
We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure- valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide first-order approximation for this model. The fluid model solution, which is proved to uniquely exists, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed.
This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the time-homogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.
The large-time behavior of a nonlinearly coupled pair of measure-valued transport equations with discontinuous boundary conditions, parameterized by a positive real-valued parameter $lambda$, is considered. These equations describe the hydrodynamic or fluid limit of many-server queues with reneging (with traffic intensity $lambda$), which model phenomena in diverse disciplines, including biology and operations research. For a broad class of reneging distributions with finite mean and service distributions with finite mean and hazard rate function that is either decreasing or bounded away from zero and infinity, it is shown that if the fluid equations have a unique invariant state, then the Dirac measure at this state is the unique random fixed point of the fluid equations, which implies that the stationary distributions of scaled $N$-server systems converge to the unique invariant state of the corresponding fluid equations. Moreover, when $lambda e 1$, it is shown that the solution to the fluid equation starting from any initial condition converges to this unique invariant state in the large time limit. The proof techniques are different under the two sets of assumptions on the service distribution. When the hazard rate function is decreasing, a reformulation of the dynamics in terms of a certain renewal equation is used, in conjunction with recursive asymptotic estimates. When the hazard rate function is bounded away from zero and infinity, the proof uses an extended relative entropy functional as a Lyapunov function. Analogous large-time convergence results are also established for a system of coupled measure-valued equations modeling a multiclass queue.
Fluid models have become an important tool for the study of many-server queues with general service and patience time distributions. The equilibrium state of a fluid model has been revealed by Whitt (2006) and shown to yield reasonable approximations to the steady state of the original stochastic systems. However, it remains an open question whether the solution to a fluid model converges to the equilibrium state and under what condition. We show in this paper that the convergence holds under a mild condition. Our method builds on the framework of measure-valued processes developed in Zhang (2013), which keeps track of the remaining patience and service times.
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