Using real-time queueing theory to control lateness in real-time systems

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Abstract
This paper presents real-time queueing theory, a new theory which embeds the ability of real-time scheduling theory to determine whether task timing requirements are met into the context of queueing models. Specifically, this paper extends the analysis developed in Lehoczky [9] to the GI/M/1 case. The paper also applies these models to study queue control strategies which can control customer lateness. Arriving customers have deadlines drawn from a general deadline distribution. The state variable for the queueing system must include the number in the queue (with supplementary variables as needed to create a Markov model) and the lead-time (deadline minus current time) of each customer; thus the state space is infinite dimensional. One can represent the state of the system as a measure on the real line and can represent that measure by its Fourier transform. Thus, a real-time queueing system can be characterized as a Markov process evolving on the space of Fourier transforms, and this paper presents a characterization of the instantaneous simultaneous lead-time profile of all the customers in the queue. This profile is complicated; however, in the heavy traffic case, a simple description of the lead-time profile emerges, namely that the lead-time profile behaves like a Brownian motion evolving on a particular manifold of Fourier transforms; the manifold depending upon the queue discipline and the customer deadline distributions. This approximation is very accurate when compared with simulations. Real-time queueing theory focuses on how well a particular queue discipline meets customer timing requirements, and focuses on the dynamic rather than the equilibrium behavior of the system. As such, it offers the potential to study control strategies to ensure that customers meet their deadlines. This paper illustrates the analysis and performance evaluation for certain queue control strategies. Generalizations to more complicated models and to queueing networks are discussed.