What is it about?
This paper studies the Asymmetric Inclusion Process (ASIP). This is a queuing model where customers move in a system with stations in tandem. This model has been used to study diverse processes such as road traffic between traffic lights, marine traffic in canals with locks (e.g. Panama Canal), and the transport of proteins within cells. The paper describes the mathematical model and derives expressions for the performance of the system, such as moments and correlations of queue lengths at the various stations and (joint) queue-length distributions. The analysis can provide answers such as how many cars will be in front of a traffic light given a setting and how far can nutrients sustain a cell.
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Why is it important?
ASIP is a tandem stochastic system that closes the gap between the tandem Jackson networks and the Asymmetric Exclusion process. It thus models phenomena that cannot be captured by either system. The ASIP’s irreversible tendency for particles to stick together makes it suitable for modeling physical systems whose behavior goes against ASEP’s exclusion principle. ASIP serves, e.g., as a lattice-gas model for unidirectional transport with irreversible aggregation. In this model’s dynamics, particles move in a unidirectional manner and each site can accommodate an arbitrary number of particles simultaneously. The inclusion principle allows these particles to form clusters that move together to the next site. ASIP is characterized by an unbounded buffer capacity and unlimited batch service. This means that at completion of service at a site all particles present at that site move as a cluster to the next site. This model can be viewed as a tandem array of growth-collapse processes.
Perspectives
The work uses an array of techniques in stochastics to evaluate the performance of a queuing model. Starting with the classical balance equations, it uses three different approximation schemes for the performance, proving that they are equivalent. In the first approximation, the dependence of batch sizes between successive stations is ignored. The second approximation ignores the dependence between queue lengths between successive stations, while the third uses replica mean-field limits. Interestingly, ordinary mean-field limits do not lead to useful approximations. We conjecture that this independence assumption becomes more and more accurate when the number of stations per layer grows. This is very similar to the famous Independence Assumption of Kleinrock for message-switching communication networks. In such networks, messages maintain their size while travelling through the network. Kleinrock ignored the (in his case obviously very strong) dependence, assuming that the corresponding service times in successive queues are all independent. His Independence Assumption results in a bad approximation for tandem queues, but the approximation becomes better and better, the larger or more complex the network becomes – just like in our case, the effect of dependence becomes negligible if a message can come from many different stations. The work easily extends to ASIP feed-forward networks with consumption.
Maria Vlasiou
University of Twente
Read the Original
This page is a summary of: ASIP tandem queues with consumption, ACM SIGMETRICS Performance Evaluation Review, February 2024, ACM (Association for Computing Machinery),
DOI: 10.1145/3649477.3649486.
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