Abstract-In this paper we study a revenue maximization problem for optical routing nodes. We model the routing node as a single server polling model with the aim to assign visit periods (service windows) to the different stations (ports) such that the mean profit per cycle is maximized. Under reasonable assumptions regarding retrial and dropping probabilities of packets the optimization problem becomes a separable concave resource allocation problem, which can be solved using existing algorithms.

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.taking limits in known expressions for the Laplace-Stieltjes transform (LST) of the waiting-time distribution. Alternatively, Olsen and van der Mei [17] provide similar results, by studying the behaviour of the descendant set approach (a numerical computation method, cf. Konheim, Levy and Srinivasan [10]) in the heavy traffic limit. For the derivation of heavy traffic asymptotics for our model, however, we will use results from branching theory, mainly those presented in Quine [18]. Earlier, these results have resulted in heavy traffic asymptotics for conventional polling models, see van der Mei [15]. We will use the same method as presented in that paper, but for a different class of polling system that models the dynamics of optical networks. In addition, for some steps of the analysis, we will present new and straight forward proofs, while other steps require a different approach. Furthermore, we will derive asymptotics for the joint queue length process at arbitrary time points, as opposed to just the marginal processes as derived in [15]. Due to the additional intricacies of the model at hand, we will need to overcome many arising complex difficulties, as will become apparent later.The rest of the paper is organized as follows. In Section 2, we introduce some notation and present a theorem from [18] on multitype branching processes with immigration. In Section 3, we describe in detail the polling model with retrials and glue periods and recall from [2] how the joint queue length process at some embedded time points in this model is related to multitype branching processes with immigration. Next, we will derive heavy traffic results for our model. In Section 4, we consider the joint queue length process at the start of glue periods. In Section 5, we look at the joint queue length process at the start of visit and switch-over periods, while in Section 6, we consider the joint queue length process at ar...

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We study a queueing system with a Poisson arrival process, in which a dispatcher sends the jobs to K homogeneous queues. The dispatcher knows the size of each job, and can implement a size-aware policy. Instead of trying to optimize system performance, we propose a Size Interval Task Assignment (SITA) policy that aims to equalize the performance (mean waiting times, or mean queue lengths) of all queues by allocating the jobs to the queues according to size. Such SITA routing requires no communication between the servers and the dispatcher, and is hence easily implemented.We study existence and uniqueness of the allocation thresholds. For FCFS and PS queues in heavy traffic, those thresholds coincide with those of a dispatching rule, SITA-E, in which loads are balanced. Preliminary numerical studies suggest that a SITA dispatching policy that equalizes performance is close to optimal when the difference between the size of the largest and the smallest job is small.

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