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Register a new userEarliest-deadline-first service in heavy-traffic acyclic networks
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This paper presents a heavy traffic analysis of the behavior of multi-class acyclic queueing networks in which the customers have deadlines. We assume the queueing system consists of J stations, and there are K different customer classes. Customers from each class arrive to the network according to independent renewal processes. The customers from each class are assigned a random deadline drawn from a deadline distribution associated with that class and they move from station to station according to a fixed acyclic route. The customers at a given node are processed according to the earliest-deadline-first (EDF) queue discipline. At any time, the customers of each type at each node have a lead time, the time until their deadline lapses. We model these lead times as a random counting measure on the real line. Under heavy traffic conditions and suitable scaling, it is proved that the measure-valued lead-time process converges to a deterministic function of the workload process.
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This paper analyzes fluid scale asymptotics of two models of generalized Jackson networks employing the earliest deadline first (EDF) policy. One applies the soft EDF policy, where deadlines are used to determine priority but jobs do not renege, and the other implements hard EDF, where jobs renege when deadlines expire, and deadlines are postponed with each migration to a new station. The arrival rates, deadline distribution and service capacity are allowed to fluctuate over time at the fluid scale. Earlier work on EDF network fluid limits, used as a tool to obtain stability of these networks, addressed only the soft version of the policy, and moreover did not contain a full fluid limit result. In this paper, tools that extend the notion of the measure-valued Skorokhod map are developed and used to establish for the first time fluid limits for both the soft and hard EDF network models.
This paper presents a second-order heavy traffic analysis of a single server queue that processes customers having deadlines using the earliest-deadline-first scheduling policy. For such systems, referred to as real-time queueing systems, performance is measured by the fraction of customers who meet their deadline, rather than more traditional performance measures, such as customer delay, queue length or server utilization. To model such systems, one must keep track of customer lead times (the time remaining until a customer deadline elapses) or equivalent information. This paper reviews the earlier heavy traffic analysis of such systems that provided approximations to the systems behavior. The main result of this paper is the development of a second-order analysis that gives the accuracy of the approximations and the rate of convergence of the sequence of real-time queueing systems to its heavy traffic limit.
A many-server queue operating under the earliest deadline first discipline, where the distributions of service time and deadline are generic, is studied at the law of large numbers scale. Fluid model equations, formulated in terms of the many-server transport equation and the recently introduced measure-valued Skorohod map, are proposed as a means of characterizing the limit. The main results are the uniqueness of solutions to these equations, and the law of large numbers scale convergence to the solutions.
We study a many-server queueing model with server vacations, where the population size dynamics of servers and customers are coupled: a server may leave for vacation only when no customers await, and the capacity available to customers is directly affected by the number of servers on vacation. We focus on scaling regimes in which server dynamics and queue dynamics fluctuate at matching time scales, so that their limiting dynamics are coupled. Specifically, we argue that interesting coupled dynamics occur in (a) the Halfin-Whitt regime, (b) the nondegenerate slowdown regime, and (c) the intermediate, near Halfin-Whitt regime; whereas the dynamics asymptotically decouple in the other heavy traffic regimes. We characterize the limiting dynamics, which are different for each scaling regime. We consider relevant respective performance measures for regimes (a) and (b) --- namely, the probability of wait and the slowdown. While closed form formulas for these performance measures have been derived for models that do not accommodate server vacations, it is difficult to obtain closed form formulas for these performance measures in the setting with server vacations. Instead, we propose formulas that approximate these performance measures, and depend on the steady-state mean number of available servers and previously derived formulas for models without server vacations. We test the accuracy of these formulas numerically.
The focus of this paper is on the asymptotics of large-time numbers of customers in time-periodic Markovian many-server queues with customer abandonment in heavy traffic. Limit theorems are obtained for the periodic number-of-customers processes under the fluid and diffusion scalings. Other results concern limits for general time-dependent queues and for time-homogeneous queues in steady state.
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