Multiclass queueing systems in heavy traffic: an asymptotic approach based on distributional and conservation laws

We propose a new approach to analyze multiclass queueing systems in heavy traffic based on what we consider as fundamental laws in queueing systems, namely distributional and conservation laws. Methodologically, we extend the distributional laws from single class queueing systems to multiple classes and combine them with conservation laws to find the heavy traffic behavior of the following systems: a)EGI/G/1 queue under FIFO, b) EGI/G/1 queue with priorities, c) Polling systems with general arrival distributions. Compared with traditional heavy traffic analysis via Brownian processes, our approach gives more insight to the asymptotics used, solves systems that traditional heavy traffic theory has not fully addressed, and more importantly leads to closed form answers, which compared to simulation are very accurate even for moderate traffic

Optimization of multiclass queueing networks with changeover times via the achievable region approach: Part I, the single-station case

We address the performance optimization problem in a single-station multiclass queueing network with changeover times by means of the achievable region approach. This approach seeks to obtain performance bounds and scheduling policies from the solution of a mathematical program over a relaxation of the system's performance region. Relaxed formulations (including linear, convex, nonconvex and positive semidefinite constraints) of this region are developed by formulating equilibrium relations satisfied by the system, with the help of Palm calculus. Our contributions include: (1) new constraints formulating equilibrium relations on server dynamics; (2) a flow conservation interpretation of the constraints previously derived by the potential function method; (3) new positive semidefinite constraints; (4) new work decomposition laws for single-station multiclass queueing networks, which yield new convex constraints; (5) a unified buffer occupancy method of performance analysis obtained from the constraints; (6) heuristic scheduling policies from the solution of the relaxations.Multiclass queueing networks, optimal scheduling, achievable region, changeover times, polling systems, stochastic scheduling

Dynamic load balancing algorithm complexity

This paper presents a theoretical analysis of the asymptotic complexity inherent in a load balancing algorithm in a loosely-coupled network, where processor communication is achieved by message passing. The load balancing complexity depends on the network topology and the overhead of processor communication for each polling strategy. The best, worst, and average case analysis of the load balancing algorithms for the various polling topologies are presented. The polling strategies considered are local, global, and random polling. The complexity is presented as a function of the number of processors in the network

Heavy traffic analysis of a polling model with retrials and glue periods

We present a heavy traffic analysis of a single-server polling model, with the special features of retrials and glue periods. The combination of these features in a polling model typically occurs in certain optical networking models, and in models where customers have a reservation period just before their service period. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. As this model defies a closed-form expression for the queue length distributions, our main focus is on their heavy-traffic asymptotics, both at embedded time points (beginnings of glue periods, visit periods and switch periods) and at arbitrary time points. We obtain closed-form expressions for the limiting scaled joint queue length distribution in heavy traffic and use these to accurately approximate the mean number of customers in the system under different loads.Comment: 23 pages, 2 figure

Analysis and optimization of vacation and polling models with retrials

We study a vacation-type queueing model, and a single-server multi-queue polling model, with the special feature of retrials. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. Our main focus is on queue length analysis, both at embedded time points (beginnings of glue periods, visit periods and switch- or vacation periods) and at arbitrary time points.Comment: Keywords: vacation queue, polling model, retrials Submitted for review to Performance evaluation journal, as an extended version of 'Vacation and polling models with retrials', by Onno Boxma and Jacques Resin