Fast Jackson-Type Networks with Dynamic Routing

We propose a new class of models of queueing networks with load-balanced dynamic routing. The paper extends earlier works, including [FC], [FMcD], [VDK], where systems with no feedback were considered. The main results are: (a) a sufficient condition for positive recurrence of the arising Markov process and (b) a limiting mean-field picture where the process becomes deterministic and is described by a system of non-linear ODEs

Stability criteria for controlled queueing networks

We give criteria for the stability of a very general queueing model under different levels of control. A complete classification of stability (or positive recurrence), transience and null-recurrence is presented for the two queue model. The stability and instability results are extended for models with N > 3 queues. We look at a broad class of models which can have the following features: Customers arrive at one, several or all of the queues from the outside with exponential inter arrival times. We often have the case where a arrival stream can be routed so that under different routing schemes each queue can have external arrivals, i.e. we assume we have some control over the routing of the arrivals. We also consider models where the arrival streams are fixed. We view the service in a more abstract way, in that we allow a number к of different service configurations. Under every such service configuration service is provided to some or all of the queues, length of service time can change from one service configuration to another and we can change from one configuration to another according two some control policy. The service times are assumed to be exponentially distributed. The queueing models we consider are networks where, after completion at one queue, a customer might be fed back into another queue where it will be served another time often under with a different service time. These feedback probabilities change with the service configurations. Our interest is in different types of control policies which allow us to change the routing of arrivals and configurations of the service from time to time so that the controlled queue length process (which in most cases is Markov) is stable. The semi-martingale or Lyapunov function methods we use give necessary and sufficient conditions for the stability classification. We will look at some two queue models with different inter arrival and service times where the queueing process is still Markov

Min Cost Flow in balancierten Netzwerken mit konvexer Kostenfunktion

Standard matching problems can be stated in terms of skew symmetric networks. On skew symmetric networks matching problems can be solved using network flow techniques. We consider the problem of minimizing a separable convex objective function over a skew-symmetric network with a balanced flow. We call this problem the Convex Balanced Min Cost Flow (Convex BMCF) problem. We start with 2 examples of Convex BMCF problems. The first problem is a problem from condensed matter physics: We want to simulate a so called super-rough phase using methods from graph-theory. This problem has previously been studied by Blasum, Hochstaettler, Rieger a and Moll. The second problem is a typical example for the minconvex-problems previously studied by Apollonio and Sebo and Berger and Hochstaettler [9]. We review the results for skew-symmetric networks by Jungnickel and Fremuth-Paeger and Kocay and Stone. Using these results we present several algorithms to solve the Convex BMCF problem. We present the first complete version of the Primal-Dual algorithm previously studied by Fremuth-Paeger and Jungnickel. However, we only consider the case of positive costs. We also show how to apply this algorithm to the Convex BMCF problem. Then we extend the Shortest Admissible Path Approach of Jungnickel and Fremuth-Paeger [23, p. 12] to a complete algorithm for linear as well as convex cost problems on skew symmetric networks. In the same manner we show how to adapt the Capacity Scaling algorithm by Ahuja and Orlin to skew symmetric networks and balanced flows. The capacity scaling algorithm is weakly polynomial. Another possibility for a weakly polynomial algorithm is the Balanced Out-of-Kilter algorithm. This algorithm is based on Fulkerson’s Out-of-Kilter algorithm and Minoux’s adaptation of the algorithm for convex costs. We show that augmentation on valid paths is not always necessary and introduce the idea of slightly different networks. Using the same ideas for the Balanced Capacity Scaling we obtain an Enhanced Capacity Scaling algorithm. The Enhanced Capacity Scaling algorithm as well as the Balanced Out-of-Kilter algorithm are the fastest algorithms presented here with a complexity of roughly O(m2log2U). Finally we show how to solve the problem from condensed matter physics using the new idea of anti-balanced flows on skew-symmetric networks. Using the Balanced Successive Shortest Path algorithm we also obtain a new complexity limit for the minconvex problem. This improves the complexity bound of Berger [8] by a factor of m in the case of separable convex costs with positive slope. In the appendix of this thesis we consider dual approaches for the Convex BMCF problem. The Balanced Relaxation algorithm, based on the Relaxation algorithm by Bertsekas [13], does not determine a balanced flow as the resulting flow will not necessarily be integral. This way we only determine fractional matchings. As the algorithm is also slow this algorithm is probably of limited use. A better ansatz seems to be the Cancel and Tighten method by Karzanov and McCormick. We review their results and end with some ideas on how to implement a balanced version of this algorithm

A load-balanced network with two servers

A load-balanced network with two queues Q 1 and Q 2 is considered. Each queue receives a Poisson stream of customers at rate i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p i *, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system

A load-balanced network with two servers

A load-balanced network with two queues Q 1 and Q 2 is considered. Each queue receives a Poisson stream of customers at rate i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p i *, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system