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

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

Revenue Maximization in an Optical Router Node Using Multiple Wavelengths

In this paper, an optical router node with multiple wavelengths is considered. We introduce revenue for successful transmission and study the ensuing revenue maximization problem. We present an efficient and accurate heuristic procedure for solving the NP-hard revenue maximization problem and investigate the advantage offered by having multiple wavelengths

Performance analysis of polling systems with retrials and glue periods

We consider gated polling systems with two special features: (i) retrials, and (ii) glue or reservation periods. When a type-$i$ customer arrives, or retries, during a glue period of station $i$, it will be served in the next visit period of the server to that station. Customers arriving at station $i$ in any other period join the orbit of that station and retry after an exponentially distributed time. Such polling systems can be used to study the performance of certain switches in optical communication systems. For the case of exponentially distributed glue periods, we present an algorithm to obtain the moments of the number of customers in each station. For generally distributed glue periods, we consider the distribution of the total workload in the system, using it to derive a pseudo conservation law which in its turn is used to obtain accurate approximations of the individual mean waiting times. We also consider the problem of choosing the lengths of the glue periods, under a constraint on the total glue period per cycle, so as to minimize a weighted sum of the mean waiting times

Revenue maximization in optical router nodes

In this paper, a few models for optical router nodes are considered. The stations (ports) of such a node try to transmit packets. Successful transmission of a packet of type j at station i gives a profit γij, but there is also a positive probability that such a packet is dropped, causing a penalty θij. Consider one fixed cycle (frame), in which each station is assigned some visit time. The goal is to choose the visit times in such a way that the revenue is maximized. In our first model there is only one wavelength, and we take the finiteness of buffers into account. The revenue maximization problem is shown to be separable concave, thus allowing application of a very efficient algorithm. In our second model we allow multiple wavelengths. We aim to maximize the revenue by optimally assigning stations to wavelengths and, for each wavelength, by optimally choosing the visit times of the allocated stations within the cycle. This gives rise to a mixed integer linear programming problem (MILP) which is NP-hard. To solve this problem fast and efficiently we provide a three-step heuristic. It consists of (i) solving a separable concave optimization problem, then (ii) allocating the stations to wavelengths using a simple bin packing algorithm, and finally (iii) solving another set of separable concave optimization problems. We present numerical results to investigate the effectiveness of the heuristic and the advantages of having multiple wavelengths. Finally, some model variants are briefly discussed