2022/06/13 by Prakirt Raj Jhunjhunwala, Siva Theja Maguluri, Jhunjhunwala, Prakirt Raj +1
Business, Management and Accounting · Computer Science · Engineering · #Advanced Queuing Theory Analysis #Age of Information Optimization #Reliability and Maintenance Optimization
paper · pdf · doi:10.48550/arxiv.2206.06504
This paper studies the heavy-traffic joint distribution of queue lengths in two stochastic processing networks (SPN), viz., an input-queued switch operating under the MaxWeight scheduling policy and a two-server parallel server system called the N-system. These two systems serve as representatives of SPNs that do not satisfy the so-called Complete Resource Pooling (CRP) condition, and consequently exhibit a multidimensional State Space Collapse (SSC). Except in special cases, only mean queue lengths of such non-CRP systems is known in the literature. In this paper, we develop the Transform method to study the joint distribution of queue lengths in non-CRP systems. The key challenge is in solving an implicit functional equation involving the Laplace transform of the heavy-traffic limiting distribution. For the N-system and a special case of an input-queued switch involving only three queues, we obtain the exact limiting heavy-traffic joint distribution in terms of a linear combination of two iid exponentials. For the general n× n input-queued switch that has n2 queues, under a conjecture on uniqueness of the solution of the functional equation, we obtain an exact joint distribution of the heavy-traffic limiting queue-lengths in terms of a non-linear combination of 2n iid exponentials.