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Register a new userHeavy traffic analysis for EDF queues with reneging
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This paper presents a heavy-traffic analysis of the behavior of a single-server queue under an Earliest-Deadline-First (EDF) scheduling policy in which customers have deadlines and are served only until their deadlines elapse. The performance of the system is measured by the fraction of reneged work (the residual work lost due to elapsed deadlines) which is shown to be minimized by the EDF policy. The evolution of the lead time distribution of customers in queue is described by a measure-valued process. The heavy traffic limit of this (properly scaled) process is shown to be a deterministic function of the limit of the scaled workload process which, in turn, is identified to be a doubly reflected Brownian motion. This paper complements previous work by Doytchinov, Lehoczky and Shreve on the EDF discipline in which customers are served to completion even after their deadlines elapse. The fraction of reneged work in a heavily loaded system and the fraction of late work in the corresponding system without reneging are compared using explicit formulas based on the heavy traffic approximations. The formulas are validated by simulation results.
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Motivated by growing applications in two-sided markets, we study a parallel matching queue with reneging. Demand and supply units arrive to the system and are matched in an FCFS manner according to a compatibility graph specified by an N-system. If they cannot be matched upon arrival, they queue and may abandon the system as time goes by. We derive explicit product forms of the steady state distributions of this system by identifying a partial balance condition.
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.
In this note, we apply Steins method to analyze the steady-state distribution of queueing systems in the traditional heavy-traffic regime. Compared to previous methods (e.g., drift method and transform method), Steins method allows us to establish stronger results with simple and template proofs. In particular, we consider discrete-time systems in this note. We first introduce the key ideas of Steins method for heavy-traffic analysis through a single-server system. Then, we apply the developed template to analyze both load balancing problems and scheduling problems. All these three examples demonstrate the power and flexibility of Steins method in heavy-traffic analysis. In particular, we can see that one appealing property of Steins method is that it combines the advantages of both the drift method and the transform method.
Motivated by a web-server model, we present a queueing network consisting of two layers. The first layer incorporates the arrival of customers at a network of two single-server nodes. We assume that the inter-arrival and the service times have general distributions. Customers are served according to their arrival order at each node and after finishing their service they can re-enter at nodes several times (as new customers) for new services. At the second layer, active servers act as jobs which are served by a single server working at speed one in a Processor-Sharing fashion. We further assume that the degree of resource sharing is limited by choice, leading to a Limited Processor-Sharing discipline. Our main result is a diffusion approximation for the process describing the number of customers in the system. Assuming a single bottleneck node and studying the system as it approaches heavy traffic, we prove a state-space collapse property. The key to derive this property is to study the model at the second layer and to prove a diffusion limit theorem, which yields an explicit approximation for the customers in the system.
This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times.
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